6070:
5353:
7472:
6065:{\displaystyle {\begin{aligned}\operatorname {Var} ({\hat {Y}}_{\text{pwr}})&={\frac {1}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {y_{i}}{p_{i}}}-{\hat {Y}}_{pwr}\right)^{2}\\&={\frac {1}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {n}{n}}{\frac {y_{i}}{p_{i}}}-{\frac {n}{n}}\sum _{i=1}^{n}w_{i}y_{i}\right)^{2}={\frac {1}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left(n{\frac {y_{i}}{\pi _{i}}}-n{\frac {\sum _{i=1}^{n}w_{i}y_{i}}{n}}\right)^{2}\\&={\frac {n^{2}}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}\\&={\frac {n}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}\end{aligned}}}
14307:
6944:
13664:
16423:
8628:
7467:{\displaystyle {\begin{aligned}\operatorname {Var} ({\hat {Y}}_{{\text{pwr (known }}N{\text{)}}})&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\check {y}}_{i}{\check {y}}_{j}\right)\\&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left({\check {\Delta }}_{ii}{\check {y}}_{i}{\check {y}}_{i}\right)\\&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left((1-\pi _{i}){\frac {y_{i}}{\pi _{i}}}{\frac {y_{i}}{\pi _{i}}}\right)\\&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left(w_{i}y_{i}\right)^{2}\end{aligned}}}
17569:
14302:{\displaystyle {\begin{aligned}\operatorname {E} &={\frac {\sum \limits _{i=1}^{N}\operatorname {E} }{N}}\\&=\operatorname {E} )^{2}]-{\frac {1}{N}}\operatorname {E} )^{2}]\\&=\left({\frac {N-1}{N}}\right)\sigma _{\text{actual}}^{2}\\\operatorname {E} &={\frac {\sum \limits _{i=1}^{N}w_{i}\operatorname {E} }{V_{1}}}\\&=\operatorname {E} )^{2}]-{\frac {V_{2}}{V_{1}^{2}}}\operatorname {E} )^{2}]\\&=\left(1-{\frac {V_{2}}{V_{1}^{2}}}\right)\sigma _{\text{actual}}^{2}\end{aligned}}}
15984:
17287:
8151:
12796:
4735:
16418:{\displaystyle {\begin{aligned}\mathbf {C} &={\frac {\sum _{i=1}^{N}w_{i}}{\left(\sum _{i=1}^{N}w_{i}\right)^{2}-\sum _{i=1}^{N}w_{i}^{2}}}\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{*}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{*}\right)\\&={\frac {\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{*}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{*}\right)}{V_{1}-(V_{2}/V_{1})}}.\end{aligned}}}
10051:
8146:
11117:
10651:
9612:
14942:
17564:{\displaystyle {\begin{aligned}{\bar {\mathbf {x} }}&=\left(\mathbf {C} _{1}^{-1}+\mathbf {C} _{2}^{-1}\right)^{-1}\left(\mathbf {C} _{1}^{-1}\mathbf {x} _{1}+\mathbf {C} _{2}^{-1}\mathbf {x} _{2}\right)\\&={\begin{bmatrix}0.9901&0\\0&0.9901\end{bmatrix}}{\begin{bmatrix}1\\1\end{bmatrix}}={\begin{bmatrix}0.9901\\0.9901\end{bmatrix}}\end{aligned}}}
8623:{\displaystyle {\hat {R}}={\hat {\bar {Y}}}={\frac {\sum _{i=1}^{N}I_{i}{\frac {y_{i}}{\pi _{i}}}}{\sum _{i=1}^{N}I_{i}{\frac {1}{\pi _{i}}}}}={\frac {\sum _{i=1}^{N}{\check {y}}'_{i}}{\sum _{i=1}^{N}{\check {1}}'_{i}}}={\frac {\sum _{i=1}^{N}w_{i}y'_{i}}{\sum _{i=1}^{N}w_{i}1'_{i}}}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}1'_{i}}}={\bar {y}}_{w}}
1735:
5099:
6279:
12537:
2215:
9621:
10789:
10363:
4453:
4468:
7848:
9352:
19113:
Weighted means are typically used to find the weighted mean of historical data, rather than theoretically generated data. In this case, there will be some error in the variance of each data point. Typically experimental errors may be underestimated due to the experimenter not taking into account all
11761:
Gatz et al. mention that the above formulation was published by
Endlich et al. (1988) when treating the weighted mean as a combination of a weighted total estimator divided by an estimator of the population size, based on the formulation published by Cochran (1977), as an approximation to the ratio
11125:
We have (at least) two versions of variance for the weighted mean: one with known and one with unknown population size estimation. There is no uniformly better approach, but the literature presents several arguments to prefer using the population estimation version (even when the population size is
14689:
8911:
first-order linearization, asymptotics, and bootstrap/jackknife. The Taylor linearization method could lead to under-estimation of the variance for small sample sizes in general, but that depends on the complexity of the statistic. For the weighted mean, the approximate variance is supposed to be
17276:
17157:
11133:
For the trivial case in which all the weights are equal to 1, the above formula is just like the regular formula for the variance of the mean (but notice that it uses the maximum likelihood estimator for the variance instead of the unbiased variance. I.e.: dividing it by n instead of (n-1)).
6887:
16625:
15616:
6087:
1496:
113:
The mean for the morning class is 80 and the mean of the afternoon class is 90. The unweighted mean of the two means is 85. However, this does not account for the difference in number of students in each class (20 versus 30); hence the value of 85 does not reflect the average student grade
15410:
11130:), the version with unknown population mean is considered more stable. Lastly, if the proportion of sampling is negatively correlated with the values (i.e.: smaller chance to sample an observation that is large), then the un-known population size version slightly compensates for that.
3400:
is very small). For the following derivation we'll assume that the probability of selecting each element is fully represented by these probabilities. I.e.: selecting some element will not influence the probability of drawing another element (this doesn't apply for things such as
9110:
4259:
16929:
7816:
5257:
1880:
13479:
12791:{\displaystyle {\begin{aligned}{\hat {\sigma }}^{2}\ &={\frac {\sum \limits _{i=1}^{N}\left(x_{i}-\mu \right)^{2}}{N}}\\{\hat {\sigma }}_{\mathrm {w} }^{2}&={\frac {\sum \limits _{i=1}^{N}w_{i}\left(x_{i}-\mu ^{*}\right)^{2}}{\sum _{i=1}^{N}w_{i}}}\end{aligned}}}
4847:
11756:
11762:
mean. However, Endlich et al. didn't seem to publish this derivation in their paper (even though they mention they used it), and
Cochran's book includes a slightly different formulation. Still, it's almost identical to the formulations described in previous sections.
18378:
In the scenario described in the previous section, most frequently the decrease in interaction strength obeys a negative exponential law. If the observations are sampled at equidistant times, then exponential decrease is equivalent to decrease by a constant fraction
16694:
The above generalizes easily to the case of taking the mean of vector-valued estimates. For example, estimates of position on a plane may have less certainty in one direction than another. As in the scalar case, the weighted mean of multiple estimates can provide a
2767:
879:
Therefore, data elements with a high weight contribute more to the weighted mean than do elements with a low weight. The weights may not be negative in order for the equation to work. Some may be zero, but not all of them (since division by zero is not allowed).
1996:
13122:
19414:
8906:
depends on the variability of the random variables both in the numerator and the denominator - as well as their correlation. Since there is no closed analytical form to compute this variance, various methods are used for approximate estimation. Primarily
10046:{\displaystyle {\widehat {V({\hat {R}})}}={\frac {1}{{\hat {Z}}^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\frac {y_{i}-{\hat {R}}z_{i}}{\pi _{i}}}{\frac {y_{j}-{\hat {R}}z_{j}}{\pi _{j}}}\right)={\frac {1}{{\hat {Z}}^{2}}}\left}
4730:{\displaystyle {\hat {\bar {Y}}}_{{\text{known }}N}={\frac {{\hat {Y}}_{pwr}}{N}}={\frac {{\frac {1}{n}}\sum _{i=1}^{n}{\frac {y'_{i}}{p_{i}}}}{N}}\approx {\frac {\sum _{i=1}^{n}{\frac {y'_{i}}{\pi _{i}}}}{N}}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{N}}.}
8141:{\displaystyle R={\bar {Y}}={\frac {\sum _{i=1}^{N}{\frac {y_{i}}{\pi _{i}}}}{\sum _{i=1}^{N}{\frac {1}{\pi _{i}}}}}={\frac {\sum _{i=1}^{N}{\check {y}}_{i}}{\sum _{i=1}^{N}{\check {1}}_{i}}}={\frac {\sum _{i=1}^{N}w_{i}y_{i}}{\sum _{i=1}^{N}w_{i}}}}
11112:{\displaystyle {\widehat {V({\bar {y}}_{w})}}={\frac {1}{{\hat {N}}^{2}}}\sum _{i=1}^{n}\left((1-\pi _{i}){\frac {y_{i}-{\bar {y}}_{w}}{\pi _{i}}}\right)^{2}={\frac {1}{(\sum _{i=1}^{n}w_{i})^{2}}}\sum _{i=1}^{n}w_{i}^{2}(y_{i}-{\bar {y}}_{w})^{2}.}
4282:
15252:
10646:{\displaystyle {\widehat {V({\hat {R}})}}={\widehat {V({\bar {y}}_{w})}}={\frac {1}{{\hat {N}}^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\frac {y_{i}-{\bar {y}}_{w}}{\pi _{i}}}{\frac {y_{j}-{\bar {y}}_{w}}{\pi _{j}}}\right).}
8886:
874:
11126:
known). For example: if all y values are constant, the estimator with unknown population size will give the correct result, while the one with known population size will have some variability. Also, when the sample size itself is random (e.g.: in
17034:
711:
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18796:
must correspond to the actual decrease of interaction strength. If this cannot be determined from theoretical considerations, then the following properties of exponentially decreasing weights are useful in making a suitable choice: at step
2328:
2890:
2383:, is itself a random variable. Its expected value and standard deviation are related to the expected values and standard deviations of the observations, as follows. For simplicity, we assume normalized weights (weights summing to one).
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8747:
17923:
12119:
1341:
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11919:
3371:
14937:{\displaystyle {\begin{aligned}s_{\mathrm {w} }^{2}\ &={\frac {{\hat {\sigma }}_{\mathrm {w} }^{2}}{1-(V_{2}/V_{1}^{2})}}\\&={\frac {\sum \limits _{i=1}^{N}w_{i}(x_{i}-\mu ^{*})^{2}}{V_{1}-(V_{2}/V_{1})}},\end{aligned}}}
5347:
The above formula was taken from
Sarndal et al. (1992) (also presented in Cochran 1977), but was written differently. The left side is how the variance was written and the right side is how we've developed the weighted version:
9607:{\displaystyle {\hat {R}}={\frac {\hat {Y}}{\hat {Z}}}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}z'_{i}}}\approx R+{\frac {1}{Z}}\sum _{i=1}^{n}\left({\frac {y'_{i}}{\pi _{i}}}-R{\frac {z'_{i}}{\pi _{i}}}\right)}
19726:
17163:
17044:
6565:
1730:{\displaystyle {\bar {x}}={\frac {\sum _{i=1}^{n}\left({\dfrac {x_{i}}{\sigma _{i}^{2}}}\right)}{\sum _{i=1}^{n}{\dfrac {1}{\sigma _{i}^{2}}}}}={\frac {\sum _{i=1}^{n}\left(x_{i}\cdot w_{i}\right)}{\sum _{i=1}^{n}w_{i}}},}
18234:
but also on its past values. Commonly, the strength of this dependence decreases as the separation of observations in time increases. To model this situation, one may replace the independent variable by its sliding mean
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3286:
1021:
12382:
2528:
15830:
8922:
6274:{\displaystyle \operatorname {Var} ({\hat {\bar {Y}}}_{{\text{pwr (known }}N{\text{)}}})={\frac {1}{N^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\check {y}}_{i}{\check {y}}_{j}\right)}
15009:
10739:
10206:
2962:
440:
17292:
14694:
12202:
6344:
5094:{\displaystyle {\hat {Y}}_{pwr}={\frac {1}{n}}\sum _{i=1}^{n}{\frac {y'_{i}}{p_{i}}}=\sum _{i=1}^{n}{\frac {y'_{i}}{np_{i}}}\approx \sum _{i=1}^{n}{\frac {y'_{i}}{\pi _{i}}}=\sum _{i=1}^{n}w_{i}y'_{i}}
3823:
3733:
14499:
1750:
1177:
13304:
4125:
3913:
1948:
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12293:
9345:
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19808:
7654:
593:
521:
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18362:
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1485:
940:
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258:
Thus, the weighted mean makes it possible to find the mean average student grade without knowing each student's score. Only the class means and the number of students in each class are needed.
10116:
160:
6741:
2210:{\displaystyle \sigma _{\bar {x}}^{2}=\sum _{i=1}^{n}{w_{i}'^{2}\sigma _{i}^{2}}={\frac {\sum _{i=1}^{n}{\sigma _{i}^{-4}\sigma _{i}^{2}}}{\left(\sum _{i=1}^{n}\sigma _{i}^{-2}\right)^{2}}}.}
18527:
18063:
12013:
2659:
14678:
14437:
3179:
is considered constant, and the variability comes from the selection procedure. This in contrast to "model based" approaches in which the randomness is often described in the y values. The
114:(independent of class). The average student grade can be obtained by averaging all the grades, without regard to classes (add all the grades up and divide by the total number of students):
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19277:
13656:
9204:
7649:
14371:
14611:
20203:
9251:
46:), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in
19507:
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15453:
1988:
363:
319:
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10267:
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3607:
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19099:
The concept of weighted average can be extended to functions. Weighted averages of functions play an important role in the systems of weighted differential and integral calculus.
13530:
12529:
4448:{\displaystyle \operatorname {Var} \left({\hat {\bar {Y}}}_{{\text{known }}N}\right)={\frac {1}{N^{2}}}{\frac {n}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}}
2438:
15866:
15132:
2608:
19049:
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1249:
18409:
17657:
19265:
15140:
15059:
722:
18979:
16684:
13184:
12234:
11816:
1427:
18751:
4031:
perspective, the weights, used in the numerator of the weighted mean, are obtained from taking the inverse of the selection probability (i.e.: the inflation factor). I.e.:
2561:
17761:
17679:
16749:
15976:
601:
18441:
16724:
15096:
12927:
1211:
18837:
17739:
7582:
7513:
3173:
2381:
19139:
13296:
13260:
13222:
11156:
6478:
12436:
2226:
16462:
12302:
Because one can always transform non-normalized weights to normalized weights, all formulas in this section can be adapted to non-normalized weights by replacing all
10356:
18935:
8756:
18584:
18557:
18232:
18205:
18158:
17710:
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9278:
7843:
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3212:
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1388:
17813:
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4817:
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2775:
2642:
1183:
17934:
19114:
sources of error in calculating the variance of each data point. In this event, the variance in the weighted mean must be corrected to account for the fact that
19089:
19069:
18999:
18794:
18771:
18461:
18273:
18253:
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17833:
17574:
which makes sense: the estimate is "compliant" in the second component and the estimate is compliant in the first component, so the weighted mean is nearly .
13550:
9284:, the variance calculation would look the same. When all weights are equal to one another, this formula is reduced to the standard unbiased variance estimator.
7602:
7553:
7533:
4842:
3843:
3144:
11770:
Because there is no closed analytical form for the variance of the weighted mean, it was proposed in the literature to rely on replication methods such as the
6570:
57:. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in
8633:
17845:
19147:
3918:
3291:
11150:
linearization) is a reasonable estimation for the square of the standard error of the mean (when used in the context of measuring chemical constituents):
17271:{\displaystyle \mathbf {x} _{2}:={\begin{bmatrix}0&1\end{bmatrix}}^{\top },\qquad \mathbf {C} _{2}:={\begin{bmatrix}100&0\\0&1\end{bmatrix}}}
17152:{\displaystyle \mathbf {x} _{1}:={\begin{bmatrix}1&0\end{bmatrix}}^{\top },\qquad \mathbf {C} _{1}:={\begin{bmatrix}1&0\\0&100\end{bmatrix}}}
12021:
19622:
11824:
1261:
19996:"Statistical Analysis of Precipitation Chemistry Measurements over the Eastern United States. Part I: Seasonal and Regional Patterns and Correlations"
17038:
For example, consider the weighted mean of the point with high variance in the second component and with high variance in the first component. Then
13606:), we can determine a correction factor to yield an unbiased estimator. Assuming each random variable is sampled from the same distribution with mean
10122:, it would include many combinations of covariances that will depend on the indicator variables. If the selection probability are uncorrelated (i.e.:
9280:
by some factor would lead to the same estimator. It also means that if we scale the sum of weights to be equal to a known-from-before population size
6882:{\displaystyle \operatorname {Var} ({\hat {\bar {Y}}}_{{\text{pwr (known }}N{\text{)}}})={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left(w_{i}y_{i}\right)^{2}}
16620:{\displaystyle \mathbf {C} ={\frac {\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{*}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{*}\right)}{1-V_{2}}}.}
15611:{\displaystyle \mathbf {C} ={\frac {\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{*}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{*}\right)}{V_{1}-1}}.}
6483:
3462:), we often talk about the multiplication of the two, which is a random variable. To avoid confusion in the following section, let's call this term:
2443:
15405:{\displaystyle \mathbf {C} ={\frac {\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{*}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{*}\right)}{V_{1}}}.}
4034:
170:
15874:
2967:
16764:
12236:, when all weights except one are zero. Its minimum value is found when all weights are equal (i.e., unweighted mean), in which case we have
5262:
18592:
6349:
3225:
1033:
164:
Or, this can be accomplished by weighting the class means by the number of students in each class. The larger class is given more "weight":
10322:. This helps illustrate that this formula incorporates the effect of correlation between y and z on the variance of the ratio estimators.
109:
Afternoon class = {81, 82, 83, 84, 85, 86, 87, 87, 88, 88, 89, 89, 89, 90, 90, 90, 90, 91, 91, 91, 92, 92, 93, 93, 94, 95, 96, 97, 98, 99}
19833:
Technically, negatives may be used if all the values are either zero or negatives. This fills no function however as the weights work as
9105:{\displaystyle {\widehat {V({\bar {y}}_{w})}}={\frac {1}{(\sum _{i=1}^{n}w_{i})^{2}}}\sum _{i=1}^{n}w_{i}^{2}(y_{i}-{\bar {y}}_{w})^{2}}
117:
19955:
Gatz, Donald F.; Smith, Luther (June 1995). "The standard error of a weighted mean concentration—I. Bootstrapping vs other methods".
16924:{\displaystyle {\bar {\mathbf {x} }}=\mathbf {C} _{\bar {\mathbf {x} }}\left(\sum _{i=1}^{n}\mathbf {W} _{i}\mathbf {x} _{i}\right),}
4273:
948:
14950:
13186:
are drawn from the same distribution, then we can treat this set as an unweighted sample, or we can treat it as the weighted sample
12305:
10658:
10125:
270:
weights are relevant, any weighted mean can be expressed using coefficients that sum to one. Such a linear combination is called a
15755:
5252:{\displaystyle \operatorname {Var} ({\hat {Y}}_{pwr})={\frac {n}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}}
2895:
1875:{\displaystyle \sigma _{\bar {x}}={\sqrt {\frac {1}{\sum _{i=1}^{n}\sigma _{i}^{-2}}}}={\sqrt {\frac {1}{\sum _{i=1}^{n}w_{i}}}},}
374:
13474:{\displaystyle s^{2}\ ={\frac {\sum _{i=1}^{N}w_{i}}{\sum _{i=1}^{N}w_{i}-1}}\sum _{i=1}^{N}w_{i}\left(x_{i}-\mu ^{*}\right)^{2}}
4254:{\displaystyle {\hat {\bar {Y}}}_{{\text{known }}N}={\frac {{\hat {Y}}_{pwr}}{N}}\approx {\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{N}}}
6286:
3765:
3612:
14442:
11751:{\displaystyle {\widehat {\sigma _{\bar {x}}^{2}}}={\frac {n}{(n-1)(n{\bar {w}})^{2}}}\sum w_{i}^{2}(x_{i}-{\bar {x}}_{w})^{2}}
2645:
random variables), then the variance of the weighted mean can be estimated as the multiplication of the unweighted variance by
20092:
7811:{\displaystyle {\hat {N}}=\sum _{i=1}^{n}w_{i}I_{i}=\sum _{i=1}^{n}{\frac {I_{i}}{\pi _{i}}}=\sum _{i=1}^{n}{\check {1}}'_{i}}
3852:
1888:
1122:
20256:
20190:
19923:
19869:
18373:
9294:
2627:
12239:
8903:
3218:
is in the sample and 0 if it was not selected. This can occur with fixed sample size, or varied sample size sampling (e.g.:
13552:). In any case, the information on total number of samples is necessary in order to obtain an unbiased correction, even if
12127:
529:
457:
19555:
18281:
8630:. As we moved from using N to using n, we actually know that all the indicator variables get 1, so we could simply write:
2389:
15415:
Similarly to weighted sample variance, there are two different unbiased estimators depending on the type of the weights.
12862:
12451:
3100:
2566:
1435:
11538:
2762:{\displaystyle \operatorname {Var} ({\bar {y}}_{w})={\hat {\sigma }}_{y}^{2}{\frac {\overline {w^{2}}}{{\bar {w}}^{2}}}}
20061:"Weighted Standard Error and its Impact on Significance Testing (WinCross vs. Quantum & SPSS), Dr. Albert Madansky"
10059:
14680:, ensuring that the expected value of the estimated variance equals the actual variance of the sampling distribution.
6676:
20213:
20166:
19982:
18469:
18025:
13117:{\displaystyle s^{2}\ ={\frac {\sum \limits _{i=1}^{N}w_{i}\left(x_{i}-\mu ^{*}\right)^{2}}{\sum _{i=1}^{N}w_{i}-1}}}
11968:
11121:
A similar re-creation of the proof (up to some mistakes at the end) was provided by Thomas Lumley in crossvalidated.
19409:{\displaystyle \chi _{\nu }^{2}={\frac {1}{(n-1)}}\sum _{i=1}^{n}{\frac {(x_{i}-{\bar {x}})^{2}}{\sigma _{i}^{2}}};}
14620:
14376:
4844:-expanded with replacement estimator, or "probability with replacement" estimator). With the above notation, it is:
19768:
15655:
14504:
886:
19422:
13629:
9162:
7607:
20020:
19995:
14315:
20309:
14547:
4795:
14614:
9209:
19472:
18842:
15434:
2646:
1953:
327:
283:
19512:
12804:
9119:
7486:
The previous section dealt with estimating the population mean as a ratio of an estimated population total (
4743:
3518:
10744:
8753:
for specific values of y and w, but the statistical properties comes when including the indicator variable
7535:), and the variance was estimated in that context. Another common case is that the population size itself (
6899:
2334:
20145:
20035:
13487:
12498:
3738:
When each element of the sample is inflated by the inverse of its selection probability, it is termed the
20314:
15842:
15247:{\displaystyle \mathbf {\mu ^{*}} ={\frac {\sum _{i=1}^{N}w_{i}\mathbf {x} _{i}}{\sum _{i=1}^{N}w_{i}}}.}
15108:
10119:
869:{\displaystyle {\bar {x}}={\frac {w_{1}x_{1}+w_{2}x_{2}+\cdots +w_{n}x_{n}}{w_{1}+w_{2}+\cdots +w_{n}}}.}
20140:
Mark
Galassi, Jim Davies, James Theiler, Brian Gough, Gerard Jungman, Michael Booth, and Fabrice Rossi.
19004:
11927:
8912:
relatively accurate even for medium sample sizes. For when the sampling has a random sample size (as in
1429:, all having the same mean, one possible choice for the weights is given by the reciprocal of variance:
1220:
19549:
18382:
17595:
15626:
13587:
12296:
11775:
11143:
10272:
10219:
3465:
1352:
19238:
15035:
12952:). In the weighted setting, there are actually two different unbiased estimators, one for the case of
12438:
is used, the variance of the weighted sample is different from the variance of the unweighted sample.
19613:
18940:
17583:
17029:{\displaystyle \mathbf {C} _{\bar {\mathbf {x} }}=\left(\sum _{i=1}^{n}\mathbf {W} _{i}\right)^{-1},}
16932:
16633:
13133:
12207:
11789:
10118:
is the estimated covariance between the estimated sum of Y and estimated sum of Z. Since this is the
1400:
1027:
One can always normalize the weights by making the following transformation on the original weights:
706:{\displaystyle {\bar {x}}={\frac {\sum \limits _{i=1}^{n}w_{i}x_{i}}{\sum \limits _{i=1}^{n}w_{i}}},}
19071:
observations matter and the effect of the remaining observations can be ignored safely, then choose
2650:
28:
18703:
17836:
16686:, then the weighted mean and covariance reduce to the unweighted sample mean and covariance above.
2533:
1391:
17744:
17662:
16732:
15959:
11525:{\displaystyle {\widehat {\sigma _{{\bar {x}}_{w}}^{2}}}={\frac {n}{(n-1)(n{\bar {w}})^{2}}}\left}
3064:
perspective, we are interested in estimating the variance of the weighted mean when the different
19793:
19773:
18414:
16702:
15068:
12905:
2323:{\displaystyle {\bar {x}}=\sigma _{\bar {x}}^{2}\sum _{i=1}^{n}{\frac {x_{i}}{\sigma _{i}^{2}}}.}
1252:
1189:
24:
19942:), How to estimate the (approximate) variance of the weighted mean?, URL (version: 2021-06-08):
18800:
17715:
7558:
7489:
3149:
2357:
2337:
of the mean of the probability distributions under the assumption that they are independent and
102:
Morning class = {62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93, 98}
19783:
19778:
19117:
17587:
13268:
13227:
13189:
12949:
8881:{\displaystyle {\bar {y}}_{w}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}1'_{i}}}}
6453:
3184:
1358:
47:
15024:
As a side note, other approaches have been described to compute the weighted sample variance.
12414:
883:
The formulas are simplified when the weights are normalized such that they sum up to 1, i.e.,
19939:
19798:
16434:
10328:
2885:{\displaystyle {\hat {\sigma }}_{y}^{2}={\frac {\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}}{n-1}}}
20289:
20249:
Data
Fitting and Uncertainty (A practical introduction to weighted least squares and beyond)
20060:
19864:
Cochran, W. G. (1977). Sampling
Techniques (3rd ed.). Nashville, TN: John Wiley & Sons.
18910:
18009:{\displaystyle {\bar {x}}=\sigma _{\bar {x}}^{2}(\mathbf {J} ^{T}\mathbf {W} \mathbf {X} ),}
20007:
19964:
18562:
18535:
18210:
18183:
18136:
17688:
15725:
15646:
13555:
11771:
9256:
7821:
3438:
3411:
3376:
3190:
3067:
2338:
1366:
58:
17773:
13609:
4802:
3741:
8:
19268:
6663:{\displaystyle {\check {\Delta }}_{ii}=1-{\frac {\pi _{i}\pi _{i}}{\pi _{i}}}=1-\pi _{i}}
20011:
19968:
8742:{\displaystyle {\bar {y}}_{w}={\frac {\sum _{i=1}^{n}w_{i}y_{i}}{\sum _{i=1}^{n}w_{i}}}}
3095:
random variables. An alternative perspective for this problem is that of some arbitrary
3052:
observations. This has led to the development of alternative, more general, estimators.
20123:
20111:
19758:
19753:
19074:
19054:
18984:
18779:
18756:
18446:
18258:
18238:
18163:
18116:
18096:
18076:
17918:{\displaystyle \sigma _{\bar {x}}^{2}=(\mathbf {J} ^{T}\mathbf {W} \mathbf {J} )^{-1},}
17818:
16752:
16696:
15014:
The degrees of freedom of the weighted, unbiased sample variance vary accordingly from
13535:
12937:
12408:
7587:
7538:
7518:
4827:
3828:
3129:
3107:
271:
19225:{\displaystyle {\hat {\sigma }}_{\bar {x}}^{2}=\sigma _{\bar {x}}^{2}\chi _{\nu }^{2}}
4018:{\displaystyle {\check {y}}'_{i}=I_{i}{\check {y}}_{i}={\frac {I_{i}y_{i}}{\pi _{i}}}}
3366:{\displaystyle P(I_{i}=1|{\text{one sample draw}})=p_{i}\approx {\frac {\pi _{i}}{n}}}
20252:
20235:
20209:
20186:
20162:
20115:
19994:
Endlich, R. M.; Eymon, B. P.; Ferek, R. J.; Valdes, A. D.; Maxwell, C. (1988-12-01).
19976:
19919:
19865:
17682:
16727:
15630:
13591:
20127:
15722:(If they are not, divide the weights by their sum to normalize prior to calculating
20107:
20015:
19972:
19743:
13298:
are normalized to 1, then the correct expression after Bessel's correction becomes
12972:(where a weight equals the number of occurrences), then the unbiased estimator is:
11127:
8913:
7818:. With the above notation, the parameter we care about is the ratio of the sums of
6079:
3915:. As above, we can add a tick mark if multiplying by the indicator function. I.e.:
3402:
3219:
2630:
20275:
12114:{\displaystyle \sigma _{\bar {x}}^{2}=\sigma _{0}^{2}\sum _{i=1}^{n}{w_{i}'^{2}},}
3435:) is fixed, and the randomness comes from it being included in the sample or not (
3048:
However, this estimation is rather limited due to the strong assumption about the
1336:{\textstyle \sigma _{\bar {x}}=\sigma {\sqrt {\sum \limits _{i=1}^{n}w_{i}'^{2}}}}
19834:
19813:
19788:
19763:
19721:{\displaystyle \sigma ^{2}={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}{n-1}}.}
12446:
11914:{\displaystyle \sigma _{\bar {x}}^{2}=\sum _{i=1}^{n}{w_{i}'^{2}\sigma _{i}^{2}}}
8892:
4265:
3180:
3096:
524:
54:
39:
20:
16759:(both denoted in the same way, via superscripts); the weight matrix then reads:
9291:
The Taylor linearization states that for a general ratio estimator of two sums (
20304:
17768:
16756:
15622:
13583:
6560:{\displaystyle {\check {\Delta }}_{ij}=1-{\frac {\pi _{i}\pi _{j}}{\pi _{ij}}}}
3222:). The probability of some element to be chosen, given a sample, is denoted as
2638:
15629:, these processes changing the data's mean and variance and thus leading to a
13590:, these processes changing the data's mean and variance and thus leading to a
50:
and also occurs in a more general form in several other areas of mathematics.
20298:
20239:
17764:
15836:
15102:
11147:
8908:
7477:
4099:{\displaystyle w_{i}={\frac {1}{\pi _{i}}}\approx {\frac {1}{n\times p_{i}}}}
1117:
19943:
6078:
An alternative term, for when the sampling has a random sample size (as in
248:{\displaystyle {\bar {x}}={\frac {(20\times 80)+(30\times 90)}{20+30}}=86.}
20141:
20119:
19913:
17839:
states that the estimate of the mean having minimum variance is given by:
10741:), and when assuming the probability of each element is very small (i.e.:
6743:), and when assuming the probability of each element is very small, then:
4109:
1179:
is a special case of the weighted mean where all data have equal weights.
16936:
15942:{\displaystyle \mathbf {\mu ^{*}} =\sum _{i=1}^{N}w_{i}\mathbf {x} _{i}.}
3039:{\displaystyle {\overline {w^{2}}}={\frac {\sum _{i=1}^{n}w_{i}^{2}}{n}}}
15633:(the population count, which is a requirement for Bessel's correction).
15021:
The standard deviation is simply the square root of the variance above.
13594:(the population count, which is a requirement for Bessel's correction).
53:
If all the weights are equal, then the weighted mean is the same as the
19464:
standard error of the weighted mean (variance weights, scale corrected)
5337:{\displaystyle {\overline {wy}}=\sum _{i=1}^{n}{\frac {w_{i}y_{i}}{n}}}
2563:, then the expectation of the weighted sample mean will be that value,
18690:{\displaystyle V_{1}=\sum _{i=1}^{m}{w^{i-1}}={\frac {1-w^{m}}{1-w}},}
6443:{\displaystyle C(I_{i},I_{j})=\pi _{ij}-\pi _{i}\pi _{j}=\Delta _{ij}}
3281:{\displaystyle P(I_{i}=1\mid {\text{Some sample of size }}n)=\pi _{i}}
1105:{\displaystyle w_{i}'={\frac {w_{i}}{\sum \limits _{j=1}^{n}{w_{j}}}}}
20280:
19803:
14617:). This means that to unbias our estimator we need to pre-divide by
13127:
This effectively applies Bessel's correction for frequency weights.
12944:
in the denominator (corresponding to the sample size) is changed to
12404:
8750:
2634:
1990:. It is a special case of the general formula in previous section,
1742:
standard error of the weighted mean (with inverse-variance weights)
1395:
19809:
Standard error of a proportion estimation when using weighted data
2333:
The significance of this choice is that this weighted mean is the
942:. For such normalized weights, the weighted mean is equivalently:
19738:
43:
20205:
The First
Systems of Weighted Differential and Integral Calculus
12403:
Typically when a mean is calculated it is important to know the
11146:
methods, the following (variance estimation of ratio-mean using
11142:
It has been shown, by Gatz et al. (1995), that in comparison to
277:
Using the previous example, we would get the following weights:
12940:
for the population variance. In normal unweighted samples, the
2622:
When treating the weights as constants, and having a sample of
1363:
For the weighted mean of a list of data for which each element
1016:{\displaystyle {\bar {x}}=\sum \limits _{i=1}^{n}{w_{i}'x_{i}}}
15004:{\displaystyle \operatorname {E} =\sigma _{\text{actual}}^{2}}
12377:{\displaystyle w_{i}'={\frac {w_{i}}{\sum _{i=1}^{n}{w_{i}}}}}
2523:{\displaystyle E({\bar {x}})=\sum _{i=1}^{n}{w_{i}'\mu _{i}}.}
20036:"GNU Scientific Library – Reference Manual: Weighted Samples"
12398:
10734:{\displaystyle \forall i\neq j:\Delta _{ij}=C(I_{i},I_{j})=0}
10201:{\displaystyle \forall i\neq j:\Delta _{ij}=C(I_{i},I_{j})=0}
4820:-estimator. This estimator can be itself estimated using the
451:
15825:{\displaystyle w_{i}'={\frac {w_{i}}{\sum _{i=1}^{N}w_{i}}}}
2957:{\displaystyle {\bar {w}}={\frac {\sum _{i=1}^{n}w_{i}}{n}}}
435:{\displaystyle {\bar {x}}=(0.4\times 80)+(0.6\times 90)=86.}
20232:
Data
Reduction and Error Analysis for the Physical Sciences
20021:
10.1175/1520-0450(1988)027<1322:SAOPCM>2.0.CO;2
19748:
19102:
16428:
The reasoning here is the same as in the previous section.
15621:
This estimator can be unbiased only if the weights are not
11965:
Consequently, if all the observations have equal variance,
9347:), they can be expanded around the true value R, and give:
19:"Weighted average" redirects here. Not to be confused with
20161:(2nd ed.). Singapore: World Scientific. p. 324.
19940:
https://stats.stackexchange.com/users/249135/thomas-lumley
13582:
The estimator can be unbiased only if the weights are not
6339:{\displaystyle {\check {y}}_{i}={\frac {y_{i}}{\pi _{i}}}}
3818:{\displaystyle {\check {y}}_{i}={\frac {y_{i}}{\pi _{i}}}}
3728:{\displaystyle V=y_{i}^{2}V=y_{i}^{2}\pi _{i}(1-\pi _{i})}
14494:{\displaystyle \left(1-{\frac {V_{2}}{V_{1}^{2}}}\right)}
3126:) and dividing it by the population size – either known (
1172:{\textstyle {\frac {1}{n}}\sum \limits _{i=1}^{n}{x_{i}}}
3908:{\displaystyle {\frac {y_{i}}{p_{i}}}=n{\check {y}}_{i}}
3114:, is calculated by taking an estimation of the total of
1943:{\displaystyle \sigma _{\bar {x}}^{2}=\sigma _{0}^{2}/n}
1184:
independent and identically distributed random variables
20142:
GNU Scientific
Library - Reference manual, Version 1.15
19993:
19914:
Carl-Erik
Sarndal; Bengt Swensson; Jan Wretman (1992).
16809:{\displaystyle \mathbf {W} _{i}=\mathbf {C} _{i}^{-1}.}
16431:
Since we are assuming the weights are normalized, then
12859:(and thus are random variables), it can be shown that
12288:{\textstyle \sigma _{\bar {x}}=\sigma _{0}/{\sqrt {n}}}
9340:{\displaystyle {\hat {R}}={\frac {\hat {Y}}{\hat {Z}}}}
20272:
18068:
17536:
17507:
17471:
17237:
17188:
17118:
17069:
15438:
12242:
12197:{\textstyle 1/n\leq \sum _{i=1}^{n}{w_{i}'^{2}}\leq 1}
12130:
7555:) is unknown and is estimated using the sample (i.e.:
1264:
1125:
889:
19625:
19558:
19515:
19475:
19425:
19280:
19241:
19150:
19120:
19077:
19057:
19007:
18987:
18943:
18913:
18845:
18803:
18782:
18759:
18706:
18595:
18565:
18559:
is the sum of the unnormalized weights. In this case
18538:
18472:
18449:
18417:
18385:
18284:
18261:
18241:
18213:
18186:
18166:
18139:
18119:
18099:
18079:
18028:
17937:
17848:
17821:
17776:
17747:
17718:
17691:
17665:
17598:
17290:
17166:
17047:
16945:
16824:
16767:
16735:
16705:
16636:
16473:
16437:
15987:
15962:
15877:
15845:
15758:
15728:
15658:
15464:
15437:
15266:
15143:
15111:
15071:
15038:
14953:
14692:
14623:
14550:
14507:
14445:
14379:
14318:
13667:
13632:
13612:
13579:
has a different meaning other than frequency weight.
13558:
13538:
13490:
13307:
13271:
13230:
13192:
13136:
12981:
12908:
12865:
12807:
12540:
12501:
12454:
12417:
12308:
12210:
12024:
11971:
11930:
11827:
11792:
11598:
11541:
11159:
10792:
10747:
10661:
10655:
If the selection probability are uncorrelated (i.e.:
10366:
10331:
10275:
10222:
10128:
10062:
9624:
9355:
9297:
9259:
9212:
9165:
9122:
8925:
8759:
8636:
8154:
7851:
7824:
7657:
7610:
7590:
7561:
7541:
7521:
7492:
6947:
6902:
6752:
6679:
6673:
If the selection probability are uncorrelated (i.e.:
6573:
6486:
6456:
6352:
6289:
6090:
5356:
5265:
5114:
4850:
4830:
4805:
4746:
4471:
4285:
4128:
4037:
3921:
3855:
3831:
3768:
3744:
3615:
3521:
3468:
3441:
3414:
3379:
3294:
3228:
3193:
3152:
3132:
3070:
2970:
2898:
2778:
2662:
2569:
2536:
2446:
2392:
2360:
2229:
1999:
1956:
1891:
1753:
1602:
1544:
1499:
1438:
1403:
1369:
1223:
1192:
1036:
951:
725:
604:
588:{\displaystyle \left(w_{1},w_{2},\dots ,w_{n}\right)}
532:
516:{\displaystyle \left(x_{1},x_{2},\dots ,x_{n}\right)}
460:
377:
330:
286:
173:
120:
19605:{\displaystyle \sigma _{\bar {x}}^{2}=\sigma ^{2}/n}
18357:{\displaystyle z_{k}=\sum _{i=1}^{m}w_{i}x_{k+1-i}.}
18073:
Consider the time series of an independent variable
16939:), in terms of the covariance of the weighted mean:
3110:, the population mean, of some quantity of interest
18367:
14683:The final unbiased estimate of sample variance is:
14544:bias in the unweighted estimator (also notice that
12895:{\displaystyle {\hat {\sigma }}_{\mathrm {w} }^{2}}
12484:{\displaystyle {\hat {\sigma }}_{\mathrm {w} }^{2}}
4122:is known we can estimate the population mean using
1480:{\displaystyle w_{i}={\frac {1}{\sigma _{i}^{2}}}.}
19720:
19604:
19540:
19501:
19454:
19408:
19259:
19224:
19141:is too large. The correction that must be made is
19133:
19083:
19063:
19043:
18993:
18973:
18929:
18899:
18831:
18788:
18765:
18745:
18689:
18578:
18551:
18521:
18455:
18435:
18403:
18356:
18267:
18247:
18226:
18199:
18172:
18152:
18125:
18105:
18085:
18057:
18008:
17917:
17827:
17807:
17755:
17733:
17704:
17673:
17651:
17563:
17270:
17151:
17028:
16923:
16808:
16743:
16718:
16678:
16619:
16456:
16417:
15970:
15941:
15860:
15824:
15741:
15711:
15610:
15447:
15404:
15246:
15126:
15090:
15053:
15003:
14936:
14672:
14605:
14536:
14493:
14431:
14365:
14301:
13650:
13618:
13571:
13544:
13524:
13473:
13290:
13254:
13216:
13178:
13116:
12921:
12894:
12847:
12790:
12523:
12483:
12430:
12376:
12287:
12228:
12196:
12113:
12007:
11960:standard error of the weighted mean (general case)
11950:
11913:
11810:
11750:
11582:{\displaystyle {\bar {w}}={\frac {\sum w_{i}}{n}}}
11581:
11524:
11111:
10778:
10733:
10645:
10350:
10314:
10261:
10200:
10110:
10045:
9606:
9339:
9272:
9245:
9198:
9151:
9104:
8880:
8741:
8622:
8140:
7837:
7810:
7643:
7596:
7576:
7547:
7527:
7507:
7466:
6933:
6881:
6735:
6662:
6559:
6480:is the probability of selecting both i and j. And
6472:
6442:
6338:
6273:
6064:
5336:
5251:
5093:
4836:
4811:
4786:
4729:
4447:
4253:
4098:
4017:
3907:
3837:
3817:
3750:
3727:
3601:
3507:
3454:
3427:
3392:
3365:
3280:
3206:
3167:
3138:
3083:
3038:
2956:
2884:
2761:
2602:
2555:
2522:
2432:
2375:
2322:
2209:
1982:
1942:
1874:
1729:
1479:
1421:
1382:
1335:
1243:
1205:
1171:
1104:
1015:
934:
868:
705:
587:
515:
450:Formally, the weighted mean of a non-empty finite
434:
357:
313:
247:
154:
19509:, they cancel out in the weighted mean variance,
19094:
10111:{\displaystyle {\hat {C}}({\hat {Y}},{\hat {Z}})}
9253:would give the same estimator, since multiplying
155:{\displaystyle {\bar {x}}={\frac {4300}{50}}=86.}
20296:
20134:
15257:And the weighted covariance matrix is given by:
15061:(each set of single observations on each of the
10208:), this term would still include a summation of
7604:can be described as the sum of weights. So when
6736:{\displaystyle \forall i\neq j:C(I_{i},I_{j})=0}
4462:The general formula can be developed like this:
20093:"Extension of covariance selection mathematics"
19091:such that the tail area is sufficiently small.
18522:{\displaystyle w_{i}={\frac {w^{i-1}}{V_{1}}},}
18058:{\displaystyle \mathbf {W} =\mathbf {C} ^{-1}.}
12399:§ Correcting for over- or under-dispersion
12008:{\displaystyle \sigma _{i}^{2}=\sigma _{0}^{2}}
11765:
3288:, and the one-draw probability of selection is
2220:The equations above can be combined to obtain:
20202:Jane Grossman, Michael Grossman, Robert Katz.
20000:Journal of Applied Meteorology and Climatology
19909:
19907:
19905:
19903:
19901:
19899:
19897:
17577:
14673:{\displaystyle 1-\left(V_{2}/V_{1}^{2}\right)}
14432:{\displaystyle V_{2}=\sum _{i=1}^{N}w_{i}^{2}}
12015:, the weighted sample mean will have variance
11818:, the variance of the weighted sample mean is
3055:
2440:then the weighted sample mean has expectation
935:{\textstyle \sum \limits _{i=1}^{n}{w_{i}'}=1}
20181:G. H. Hardy, J. E. Littlewood, and G. Pólya.
19895:
19893:
19891:
19889:
19887:
19885:
19883:
19881:
19879:
19877:
19860:
19858:
19856:
19854:
19108:
15712:{\displaystyle V_{1}=\sum _{i=1}^{N}w_{i}=1.}
15027:
14537:{\displaystyle \left({\frac {N-1}{N}}\right)}
12936:For small samples, it is customary to use an
11786:For uncorrelated observations with variances
6082:), is presented in Sarndal et al. (1992) as:
261:
20086:
20084:
13285:
13272:
13249:
13231:
13211:
13193:
13173:
13137:
10786:), then the above reduced to the following:
8148:. We can estimate it using our sample with:
20159:Statistical Methods in Experimental Physics
19455:{\displaystyle {\hat {\sigma }}_{\bar {x}}}
17588:Variance § Sum of correlated variables
15956:weighted estimate of the covariance matrix
15431:weighted estimate of the covariance matrix
13651:{\displaystyle \sigma _{\text{actual}}^{2}}
12855:for normalized weights. If the weights are
12392:
11137:
9199:{\displaystyle w_{i}={\frac {1}{\pi _{i}}}}
7644:{\displaystyle w_{i}={\frac {1}{\pi _{i}}}}
4276:), then the variance of this estimator is:
4268:is one that results in a fixed sample size
1346:
20246:
19874:
19851:
18160:. In many common situations, the value of
16689:
14439:. Therefore, the bias in our estimator is
14366:{\displaystyle V_{1}=\sum _{i=1}^{N}w_{i}}
12204:. The variance attains its maximum value,
10120:covariance of two sums of random variables
4794:and it may be estimated by the (unbiased)
445:
20229:
20081:
20019:
19954:
16699:estimate. We simply replace the variance
15455:, with Bessel's correction, is given by:
14606:{\displaystyle \ V_{1}^{2}/V_{2}=N_{eff}}
13262:, and we get the same result either way.
9616:And the variance can be approximated by:
2386:If the observations have expected values
2344:
19944:https://stats.stackexchange.com/q/525770
19103:Correcting for over- or under-dispersion
17741:is the common mean to be estimated, and
15636:
9246:{\displaystyle w_{i}={\frac {1}{p_{i}}}}
20185:(2nd ed.), Cambridge University Press,
19502:{\displaystyle \sigma _{i}=\sigma _{0}}
18900:{\displaystyle {e^{-1}}(1-w)=0.39(1-w)}
18133:observations sampled at discrete times
15448:{\displaystyle \textstyle \mathbf {C} }
15065:random variables) is assigned a weight
13602:If the weights are instead non-random (
12902:is the maximum likelihood estimator of
2530:In particular, if the means are equal,
1983:{\displaystyle \sigma _{i}=\sigma _{0}}
358:{\displaystyle {\frac {30}{20+30}}=0.6}
314:{\displaystyle {\frac {20}{20+30}}=0.4}
20297:
19541:{\displaystyle \sigma _{\bar {x}}^{2}}
15418:
15032:In a weighted sample, each row vector
13597:
12848:{\displaystyle \sum _{i=1}^{N}w_{i}=1}
12411:about that mean. When a weighted mean
9152:{\displaystyle \pi _{i}\approx p_{i}n}
6074:And we got to the formula from above.
4787:{\displaystyle Y=\sum _{i=1}^{N}y_{i}}
3602:{\displaystyle E=y_{i}E=y_{i}\pi _{i}}
98:test grades in each class as follows:
20273:
20156:
20090:
18374:Exponentially weighted moving average
13484:where the total number of samples is
10779:{\displaystyle (1-\pi _{i})\approx 1}
8902:In this case, the variability of the
8895:and it is approximately unbiased for
6934:{\displaystyle (1-\pi _{i})\approx 1}
3118:over all elements in the population (
2617:
13525:{\displaystyle \sum _{i=1}^{N}w_{i}}
12963:
12524:{\displaystyle {\hat {\sigma }}^{2}}
2433:{\displaystyle E(x_{i})={\mu _{i}},}
19469:When all data variances are equal,
18069:Decreasing strength of interactions
16818:The weighted mean in this case is:
15861:{\displaystyle \mathbf {\mu ^{*}} }
15127:{\displaystyle \mathbf {\mu ^{*}} }
14812:
14000:
13714:
13002:
12678:
12578:
12491:is defined similarly to the normal
12387:
4740:The population total is denoted as
3101:selected with unequal probabilities
2603:{\displaystyle E({\bar {x}})=\mu .}
1490:The weighted mean in this case is:
1390:potentially comes from a different
1291:
1215:standard error of the weighted mean
1137:
1067:
968:
891:
667:
624:
368:Then, apply the weights like this:
13:
20290:Tool to calculate Weighted Average
20223:
20112:10.1111/j.1469-1809.1957.tb01874.x
19044:{\displaystyle \leq {e^{-n(1-w)}}}
18839:, the weight approximately equals
18430:
18392:
17592:In the general case, suppose that
17208:
17089:
16630:If all weights are the same, i.e.
14969:
14954:
14741:
14703:
14197:
14179:
14119:
14101:
14030:
13975:
13951:
13868:
13850:
13809:
13791:
13734:
13672:
12881:
12656:
12470:
11951:{\displaystyle \sigma _{\bar {x}}}
11589:. Further simplification leads to
10678:
10662:
10521:
10145:
10129:
9737:
7186:
7068:
6680:
6578:
6491:
6428:
6210:
3175:). In this context, each value of
1244:{\displaystyle \sigma _{\bar {x}}}
523:, with corresponding non-negative
14:
20326:
20266:
18404:{\displaystyle 0<\Delta <1}
17652:{\displaystyle \mathbf {X} =^{T}}
10315:{\displaystyle z'_{i}=I_{i}z_{i}}
10262:{\displaystyle y'_{i}=I_{i}y_{i}}
3515:. With the following expectancy:
3508:{\displaystyle y'_{i}=y_{i}I_{i}}
3214:) that get 1 if some observation
19769:Weighted average cost of capital
19260:{\displaystyle \chi _{\nu }^{2}}
18368:Exponentially decreasing weights
18039:
18030:
17996:
17991:
17980:
17895:
17890:
17879:
17749:
17667:
17600:
17441:
17421:
17406:
17386:
17347:
17324:
17299:
17219:
17169:
17100:
17050:
16999:
16956:
16948:
16903:
16891:
16852:
16844:
16829:
16785:
16770:
16737:
16565:
16524:
16475:
16331:
16290:
16210:
16169:
15993:
15964:
15926:
15884:
15852:
15556:
15515:
15466:
15440:
15358:
15317:
15268:
15195:
15150:
15118:
15054:{\displaystyle \mathbf {x} _{i}}
15041:
12295:, i.e., it degenerates into the
7515:) with a known population size (
69:
20234:. New York, N.Y.: McGraw-Hill.
20196:
20175:
20150:
20091:Price, George R. (April 1972).
19109:§ Weighted sample variance
18974:{\displaystyle {1-e^{-1}}=0.61}
17216:
17097:
16679:{\displaystyle w_{i}/V_{1}=1/N}
15018: − 1 down to 0.
13658:, taking expectations we have,
13179:{\displaystyle \{2,2,4,5,5,5\}}
12229:{\displaystyle \sigma _{0}^{2}}
11811:{\displaystyle \sigma _{i}^{2}}
7478:Variance of the weighted mean (
3099:of the data in which units are
1422:{\displaystyle \sigma _{i}^{2}}
20053:
20028:
19987:
19948:
19932:
19916:Model Assisted Survey Sampling
19827:
19692:
19685:
19663:
19569:
19526:
19445:
19433:
19376:
19369:
19347:
19317:
19305:
19195:
19170:
19158:
19095:Weighted averages of functions
19051:. Where primarily the closest
19035:
19023:
18894:
18882:
18873:
18861:
18817:
18804:
18740:
18728:
18000:
17975:
17963:
17944:
17900:
17874:
17859:
17796:
17777:
17725:
17640:
17607:
17303:
16960:
16856:
16833:
16402:
16374:
14980:
14960:
14921:
14893:
14869:
14842:
14792:
14759:
14733:
14222:
14213:
14209:
14203:
14188:
14185:
14144:
14135:
14131:
14125:
14110:
14107:
14075:
14066:
14039:
14036:
13986:
13967:
13957:
13893:
13884:
13880:
13874:
13859:
13856:
13834:
13825:
13821:
13815:
13800:
13797:
13772:
13763:
13743:
13740:
13700:
13688:
13678:
12873:
12648:
12552:
12509:
12462:
12253:
12035:
11941:
11838:
11781:
11739:
11726:
11703:
11673:
11666:
11654:
11651:
11639:
11612:
11548:
11508:
11501:
11479:
11459:
11446:
11434:
11421:
11389:
11386:
11380:
11358:
11343:
11321:
11308:
11295:
11263:
11243:
11236:
11224:
11221:
11209:
11175:
11097:
11084:
11061:
11013:
10978:
10933:
10907:
10888:
10847:
10822:
10810:
10800:
10767:
10748:
10722:
10696:
10610:
10562:
10524:
10456:
10431:
10419:
10409:
10389:
10383:
10374:
10189:
10163:
10105:
10099:
10084:
10075:
10069:
10035:
10029:
10014:
10005:
9999:
9987:
9965:
9959:
9950:
9937:
9918:
9912:
9903:
9877:
9828:
9777:
9740:
9672:
9647:
9641:
9632:
9388:
9378:
9362:
9330:
9320:
9304:
9093:
9080:
9057:
9009:
8974:
8955:
8943:
8933:
8767:
8644:
8608:
8382:
8337:
8183:
8178:
8161:
8040:
7998:
7864:
7793:
7664:
7568:
7499:
7319:
7300:
7230:
7211:
7189:
7112:
7093:
7071:
6990:
6968:
6958:
6922:
6903:
6798:
6776:
6771:
6759:
6724:
6698:
6581:
6494:
6382:
6356:
6297:
6254:
6235:
6213:
6136:
6114:
6109:
6097:
5488:
5389:
5377:
5367:
5149:
5131:
5121:
5104:The estimated variance of the
4858:
4515:
4486:
4481:
4310:
4305:
4172:
4143:
4138:
4110:Variance of the weighted sum (
3964:
3929:
3893:
3776:
3722:
3703:
3672:
3659:
3635:
3619:
3573:
3560:
3541:
3525:
3327:
3318:
3298:
3262:
3232:
3159:
2905:
2859:
2852:
2830:
2786:
2745:
2704:
2691:
2679:
2669:
2588:
2582:
2573:
2465:
2459:
2450:
2409:
2396:
2367:
2255:
2236:
2010:
1902:
1764:
1506:
1275:
1234:
958:
732:
611:
423:
411:
405:
393:
384:
222:
210:
204:
192:
180:
127:
86:with 20 students, one with 30
1:
19844:
19612:, formulated in terms of the
19548:, which again reduces to the
18746:{\displaystyle V_{1}=1/(1-w)}
10212:covariances for each element
3373:(If N is very large and each
3183:procedure yields a series of
2556:{\displaystyle \mu _{i}=\mu }
2349:
20230:Bevington, Philip R (1969).
19977:10.1016/1352-2310(94)00210-C
17756:{\displaystyle \mathbf {J} }
17674:{\displaystyle \mathbf {C} }
16744:{\displaystyle \mathbf {C} }
15971:{\displaystyle \mathbf {C} }
12956:and another for the case of
11766:Replication-based estimators
6042:
5938:
5276:
5233:
4429:
2983:
2734:
2335:maximum likelihood estimator
7:
19731:
18436:{\displaystyle w=1-\Delta }
18411:at each time step. Setting
17578:Accounting for correlations
17281:then the weighted mean is:
16719:{\displaystyle \sigma ^{2}}
15091:{\displaystyle w_{i}\geq 0}
13224:with corresponding weights
12922:{\displaystyle \sigma ^{2}}
3056:Survey sampling perspective
2612:
1206:{\displaystyle \sigma ^{2}}
64:
10:
20331:
20146:Sec. 21.7 Weighted Samples
19550:standard error of the mean
19106:
18907:, the tail area the value
18832:{\displaystyle (1-w)^{-1}}
18371:
17734:{\displaystyle {\bar {x}}}
17581:
15028:Weighted sample covariance
12396:
12297:standard error of the mean
7577:{\displaystyle {\hat {N}}}
7508:{\displaystyle {\hat {Y}}}
7482:-estimator for ratio-mean)
4796:Horvitz–Thompson estimator
3168:{\displaystyle {\hat {N}}}
2376:{\displaystyle {\bar {x}}}
2354:The weighted sample mean,
1356:
1353:Inverse-variance weighting
1350:
262:Convex combination example
38:is similar to an ordinary
18:
20157:James, Frederick (2006).
19799:Weighted sum of variables
19614:sample standard deviation
19134:{\displaystyle \chi ^{2}}
18093:and a dependent variable
17584:Generalized least squares
13291:{\displaystyle \{w_{i}\}}
13265:If the frequency weights
13255:{\displaystyle \{2,1,3\}}
13217:{\displaystyle \{2,4,5\}}
6473:{\displaystyle \pi _{ij}}
3256:Some sample of size
1182:If the data elements are
42:(the most common type of
20100:Annals of Human Genetics
19820:
18981:. The tail area at step
17685:relating the quantities
16931:(where the order of the
12431:{\displaystyle \mu ^{*}}
12393:Weighted sample variance
11138:Bootstrapping validation
5108:-estimator is given by:
3825:. A related quantity is
1392:probability distribution
1347:Variance-defined weights
36:weighted arithmetic mean
19957:Atmospheric Environment
19794:Weighted moving average
19774:Weighted geometric mean
16690:Vector-valued estimates
16457:{\displaystyle V_{1}=1}
13130:For example, if values
12933:Gaussian observations.
10351:{\displaystyle z_{i}=1}
4118:If the population size
1253:uncertainty propagation
446:Mathematical definition
25:Weighted geometric mean
19784:Weighted least squares
19779:Weighted harmonic mean
19722:
19662:
19606:
19542:
19503:
19456:
19410:
19343:
19261:
19226:
19135:
19085:
19065:
19045:
18995:
18975:
18931:
18930:{\displaystyle e^{-1}}
18901:
18833:
18790:
18767:
18747:
18691:
18629:
18580:
18553:
18523:
18463:normalized weights by
18457:
18437:
18405:
18358:
18318:
18269:
18249:
18228:
18201:
18174:
18154:
18127:
18107:
18087:
18059:
18010:
17919:
17829:
17809:
17757:
17735:
17706:
17675:
17653:
17565:
17272:
17153:
17030:
16996:
16925:
16888:
16810:
16745:
16720:
16680:
16621:
16505:
16458:
16419:
16271:
16150:
16111:
16066:
16027:
15972:
15943:
15913:
15862:
15826:
15808:
15743:
15713:
15692:
15612:
15496:
15449:
15406:
15298:
15248:
15227:
15182:
15128:
15092:
15055:
15005:
14938:
14831:
14674:
14607:
14538:
14495:
14433:
14413:
14367:
14352:
14303:
14019:
13733:
13652:
13620:
13573:
13546:
13526:
13511:
13475:
13420:
13380:
13347:
13292:
13256:
13218:
13180:
13118:
13094:
13021:
12923:
12896:
12849:
12828:
12792:
12770:
12697:
12597:
12525:
12485:
12432:
12378:
12358:
12289:
12230:
12198:
12165:
12115:
12085:
12009:
11952:
11915:
11873:
11812:
11752:
11583:
11526:
11113:
11045:
11001:
10881:
10780:
10735:
10647:
10511:
10490:
10352:
10316:
10263:
10202:
10112:
10047:
9727:
9706:
9608:
9533:
9467:
9421:
9341:
9274:
9247:
9200:
9153:
9106:
9041:
8997:
8882:
8851:
8805:
8743:
8725:
8682:
8624:
8571:
8525:
8472:
8426:
8374:
8329:
8272:
8215:
8142:
8124:
8081:
8032:
7990:
7943:
7896:
7839:
7812:
7785:
7737:
7693:
7645:
7598:
7578:
7549:
7529:
7509:
7468:
7422:
7294:
7176:
7058:
7037:
6935:
6883:
6841:
6737:
6664:
6561:
6474:
6444:
6340:
6275:
6200:
6179:
6066:
6002:
5898:
5795:
5732:
5649:
5575:
5447:
5338:
5304:
5253:
5193:
5095:
5067:
5016:
4960:
4909:
4838:
4813:
4788:
4773:
4731:
4694:
4634:
4574:
4449:
4389:
4255:
4221:
4114:-estimator for totals)
4100:
4019:
3909:
3839:
3819:
3752:
3729:
3603:
3509:
3456:
3429:
3394:
3367:
3282:
3208:
3169:
3140:
3085:
3040:
3014:
2958:
2937:
2886:
2829:
2763:
2604:
2557:
2524:
2491:
2434:
2377:
2345:Statistical properties
2324:
2287:
2211:
2172:
2109:
2045:
1984:
1944:
1876:
1854:
1801:
1731:
1710:
1654:
1600:
1538:
1481:
1423:
1384:
1359:Weighted least squares
1337:
1310:
1245:
1207:
1173:
1156:
1106:
1086:
1017:
987:
936:
910:
870:
707:
686:
643:
589:
517:
436:
359:
315:
249:
156:
48:descriptive statistics
29:Weighted harmonic mean
20310:Mathematical analysis
19723:
19642:
19607:
19543:
19504:
19457:
19411:
19323:
19262:
19227:
19136:
19086:
19066:
19046:
18996:
18976:
18932:
18902:
18834:
18791:
18776:The damping constant
18768:
18748:
18692:
18609:
18581:
18579:{\displaystyle V_{1}}
18554:
18552:{\displaystyle V_{1}}
18524:
18458:
18438:
18406:
18359:
18298:
18270:
18250:
18229:
18227:{\displaystyle x_{i}}
18202:
18200:{\displaystyle t_{i}}
18175:
18155:
18153:{\displaystyle t_{i}}
18128:
18108:
18088:
18060:
18011:
17920:
17830:
17810:
17758:
17736:
17707:
17705:{\displaystyle x_{i}}
17676:
17654:
17566:
17273:
17154:
17031:
16976:
16933:matrix–vector product
16926:
16868:
16811:
16746:
16721:
16681:
16622:
16485:
16464:and this reduces to:
16459:
16420:
16251:
16130:
16091:
16046:
16007:
15973:
15944:
15893:
15868:can be simplified to
15863:
15827:
15788:
15744:
15742:{\displaystyle V_{1}}
15714:
15672:
15631:loss of the base rate
15613:
15476:
15450:
15407:
15278:
15249:
15207:
15162:
15129:
15093:
15056:
15006:
14939:
14811:
14675:
14615:effective sample size
14608:
14539:
14496:
14434:
14393:
14368:
14332:
14304:
13999:
13713:
13653:
13621:
13592:loss of the base rate
13574:
13572:{\displaystyle w_{i}}
13547:
13527:
13491:
13476:
13400:
13360:
13327:
13293:
13257:
13219:
13181:
13119:
13074:
13001:
12924:
12897:
12850:
12808:
12793:
12750:
12677:
12577:
12526:
12486:
12433:
12379:
12338:
12290:
12231:
12199:
12145:
12116:
12065:
12010:
11953:
11916:
11853:
11813:
11753:
11584:
11527:
11114:
11025:
10981:
10861:
10781:
10736:
10648:
10491:
10470:
10353:
10317:
10264:
10203:
10113:
10048:
9707:
9686:
9609:
9513:
9447:
9401:
9342:
9275:
9273:{\displaystyle w_{i}}
9248:
9201:
9154:
9107:
9021:
8977:
8916:), it is as follows:
8883:
8831:
8785:
8744:
8705:
8662:
8625:
8551:
8505:
8452:
8406:
8354:
8309:
8252:
8195:
8143:
8104:
8061:
8012:
7970:
7923:
7876:
7840:
7838:{\displaystyle y_{i}}
7813:
7765:
7717:
7673:
7646:
7599:
7584:). The estimation of
7579:
7550:
7530:
7510:
7469:
7402:
7274:
7156:
7038:
7017:
6936:
6884:
6821:
6738:
6665:
6562:
6475:
6445:
6341:
6276:
6180:
6159:
6067:
5982:
5878:
5775:
5712:
5629:
5555:
5427:
5339:
5284:
5254:
5173:
5096:
5047:
4996:
4940:
4889:
4839:
4814:
4789:
4753:
4732:
4674:
4614:
4554:
4450:
4369:
4256:
4201:
4101:
4020:
3910:
3840:
3820:
3753:
3730:
3604:
3510:
3457:
3455:{\displaystyle I_{i}}
3430:
3428:{\displaystyle y_{i}}
3395:
3393:{\displaystyle p_{i}}
3368:
3283:
3209:
3207:{\displaystyle I_{i}}
3170:
3141:
3086:
3084:{\displaystyle y_{i}}
3041:
2994:
2959:
2917:
2887:
2809:
2764:
2605:
2558:
2525:
2471:
2435:
2378:
2325:
2267:
2212:
2152:
2089:
2025:
1985:
1945:
1885:Note this reduces to
1877:
1834:
1781:
1732:
1690:
1634:
1580:
1518:
1482:
1424:
1385:
1383:{\displaystyle x_{i}}
1338:
1290:
1246:
1208:
1174:
1136:
1107:
1066:
1018:
967:
937:
890:
871:
708:
666:
623:
590:
518:
437:
360:
316:
250:
157:
19623:
19556:
19513:
19473:
19423:
19278:
19239:
19148:
19118:
19075:
19055:
19005:
18985:
18941:
18911:
18843:
18801:
18780:
18757:
18753:for large values of
18704:
18593:
18563:
18536:
18470:
18447:
18415:
18383:
18282:
18259:
18239:
18211:
18207:depends not only on
18184:
18164:
18137:
18117:
18097:
18077:
18026:
17935:
17846:
17837:Gauss–Markov theorem
17819:
17808:{\displaystyle ^{T}}
17774:
17745:
17716:
17689:
17663:
17596:
17288:
17164:
17045:
16943:
16822:
16765:
16733:
16703:
16634:
16471:
16435:
15985:
15960:
15875:
15843:
15756:
15726:
15656:
15462:
15435:
15264:
15141:
15109:
15069:
15036:
14951:
14690:
14621:
14548:
14505:
14443:
14377:
14316:
13665:
13630:
13626:and actual variance
13619:{\displaystyle \mu }
13610:
13556:
13536:
13488:
13305:
13269:
13228:
13190:
13134:
12979:
12948: − 1 (see
12906:
12863:
12805:
12538:
12499:
12452:
12415:
12306:
12240:
12208:
12128:
12022:
11969:
11928:
11825:
11790:
11596:
11539:
11157:
10790:
10745:
10659:
10364:
10329:
10273:
10220:
10126:
10060:
9622:
9353:
9295:
9257:
9210:
9163:
9159:, then either using
9120:
8923:
8757:
8634:
8152:
7849:
7822:
7655:
7608:
7588:
7559:
7539:
7519:
7490:
6945:
6900:
6750:
6677:
6571:
6484:
6454:
6350:
6287:
6088:
5354:
5263:
5112:
4848:
4828:
4812:{\displaystyle \pi }
4803:
4744:
4469:
4283:
4126:
4035:
3919:
3853:
3829:
3766:
3751:{\displaystyle \pi }
3742:
3613:
3519:
3466:
3439:
3412:
3408:Since each element (
3377:
3292:
3226:
3191:
3150:
3130:
3103:(with replacement).
3068:
2968:
2896:
2776:
2660:
2647:Kish's design effect
2641:(as is the case for
2633:, all with the same
2567:
2534:
2444:
2390:
2358:
2341:with the same mean.
2339:normally distributed
2227:
1997:
1954:
1889:
1751:
1497:
1436:
1401:
1367:
1262:
1221:
1190:
1123:
1034:
949:
887:
723:
602:
530:
458:
375:
328:
284:
171:
118:
20247:Strutz, T. (2010).
20068:Analyticalgroup.com
20012:1988JApMe..27.1322E
19969:1995AtmEn..29.1185G
19580:
19537:
19400:
19295:
19269:reduced chi-squared
19256:
19221:
19206:
19181:
17974:
17870:
17438:
17403:
17364:
17341:
16802:
16126:
15771:
15643:reliability weights
15637:Reliability weights
15423:If the weights are
15000:
14979:
14791:
14751:
14713:
14664:
14568:
14501:, analogous to the
14483:
14428:
14294:
14272:
14176:
13985:
13946:
13647:
13604:reliability weights
13598:Reliability weights
12968:If the weights are
12958:reliability weights
12950:Bessel's correction
12891:
12666:
12480:
12321:
12225:
12186:
12106:
12064:
12046:
12004:
11986:
11909:
11894:
11849:
11807:
11702:
11623:
11475:
11193:
11060:
10358:the above becomes:
10288:
10235:
9586:
9553:
9490:
9444:
9056:
8874:
8828:
8749:. This will be the
8594:
8548:
8495:
8449:
8396:
8351:
7807:
5090:
5031:
4975:
4924:
4717:
4649:
4589:
4244:
3943:
3692:
3655:
3634:
3540:
3481:
3029:
2802:
2720:
2505:
2314:
2266:
2190:
2143:
2128:
2081:
2066:
2021:
1931:
1913:
1819:
1621:
1570:
1471:
1418:
1330:
1251:, can be shown via
1049:
1001:
924:
20315:Summary statistics
20251:. Vieweg+Teubner.
19759:Summary statistics
19754:Standard deviation
19718:
19602:
19559:
19538:
19516:
19499:
19462:can be called the
19452:
19406:
19386:
19281:
19257:
19242:
19222:
19207:
19185:
19151:
19131:
19081:
19061:
19041:
18991:
18971:
18927:
18897:
18829:
18786:
18763:
18743:
18687:
18576:
18549:
18519:
18453:
18433:
18401:
18354:
18265:
18255:for a window size
18245:
18224:
18197:
18170:
18150:
18123:
18103:
18083:
18055:
18006:
17953:
17915:
17849:
17825:
17805:
17753:
17731:
17702:
17671:
17649:
17561:
17559:
17551:
17522:
17496:
17419:
17384:
17345:
17322:
17268:
17262:
17201:
17149:
17143:
17082:
17026:
16921:
16806:
16783:
16753:arithmetic inverse
16741:
16716:
16697:maximum likelihood
16676:
16617:
16454:
16415:
16413:
16112:
15968:
15939:
15858:
15822:
15759:
15739:
15709:
15645:, the weights are
15608:
15445:
15444:
15402:
15244:
15124:
15088:
15051:
15001:
14986:
14963:
14934:
14932:
14777:
14726:
14697:
14670:
14650:
14603:
14554:
14534:
14491:
14469:
14429:
14414:
14363:
14299:
14297:
14280:
14258:
14162:
13960:
13932:
13648:
13633:
13616:
13569:
13542:
13522:
13471:
13288:
13252:
13214:
13176:
13114:
12938:unbiased estimator
12919:
12892:
12866:
12845:
12788:
12786:
12641:
12521:
12481:
12455:
12428:
12409:standard deviation
12374:
12309:
12285:
12226:
12211:
12194:
12167:
12111:
12087:
12050:
12025:
12005:
11990:
11972:
11958:can be called the
11948:
11924:whose square root
11911:
11895:
11875:
11828:
11808:
11793:
11748:
11688:
11602:
11579:
11522:
11452:
11163:
11109:
11046:
10776:
10731:
10643:
10348:
10312:
10276:
10259:
10223:
10198:
10108:
10043:
9604:
9574:
9541:
9478:
9432:
9337:
9289:
9270:
9243:
9196:
9149:
9102:
9042:
8878:
8862:
8816:
8739:
8620:
8582:
8536:
8483:
8437:
8375:
8330:
8138:
7835:
7808:
7786:
7641:
7594:
7574:
7545:
7525:
7505:
7464:
7462:
6931:
6894:
6879:
6733:
6660:
6557:
6470:
6440:
6336:
6271:
6062:
6060:
5334:
5249:
5091:
5078:
5019:
4963:
4912:
4834:
4824:-estimator (i.e.:
4809:
4798:, also called the
4784:
4727:
4705:
4637:
4577:
4460:
4445:
4251:
4232:
4096:
4015:
3922:
3905:
3835:
3815:
3748:
3725:
3678:
3641:
3622:
3599:
3528:
3505:
3469:
3452:
3425:
3390:
3363:
3278:
3204:
3187:indicator values (
3165:
3136:
3108:Survey methodology
3081:
3036:
3015:
2954:
2882:
2779:
2759:
2697:
2626:observations from
2618:Simple i.i.d. case
2600:
2553:
2520:
2493:
2430:
2373:
2320:
2300:
2245:
2207:
2173:
2129:
2111:
2067:
2047:
2000:
1980:
1940:
1917:
1892:
1872:
1802:
1727:
1623:
1607:
1572:
1556:
1477:
1457:
1419:
1404:
1380:
1333:
1311:
1241:
1203:
1169:
1102:
1037:
1013:
989:
932:
912:
866:
716:which expands to:
703:
585:
513:
432:
355:
311:
272:convex combination
245:
152:
16:Statistical amount
20258:978-3-8348-1022-9
20191:978-0-521-35880-4
20006:(12): 1322–1333.
19963:(11): 1185–1193.
19925:978-0-387-97528-3
19870:978-0-471-16240-7
19713:
19688:
19572:
19529:
19448:
19436:
19401:
19372:
19321:
19198:
19173:
19161:
19084:{\displaystyle w}
19064:{\displaystyle n}
18994:{\displaystyle n}
18789:{\displaystyle w}
18766:{\displaystyle m}
18682:
18514:
18456:{\displaystyle m}
18268:{\displaystyle m}
18248:{\displaystyle z}
18173:{\displaystyle y}
18126:{\displaystyle n}
18106:{\displaystyle y}
18086:{\displaystyle x}
17966:
17947:
17862:
17828:{\displaystyle n}
17728:
17683:covariance matrix
17306:
16963:
16859:
16836:
16728:covariance matrix
16612:
16406:
16128:
15820:
15603:
15425:frequency weights
15419:Frequency weights
15397:
15239:
14993:
14925:
14796:
14736:
14716:
14553:
14528:
14484:
14287:
14273:
14177:
14089:
13970:
13939:
13926:
13848:
13779:
13691:
13640:
13545:{\displaystyle N}
13398:
13320:
13112:
12994:
12970:frequency weights
12964:Frequency weights
12954:frequency weights
12876:
12857:frequency weights
12782:
12651:
12635:
12566:
12555:
12512:
12465:
12372:
12283:
12256:
12038:
11944:
11841:
11729:
11683:
11669:
11627:
11615:
11577:
11551:
11504:
11462:
11437:
11424:
11383:
11346:
11311:
11298:
11253:
11239:
11197:
11178:
11087:
11023:
10956:
10936:
10859:
10850:
10829:
10813:
10633:
10613:
10585:
10565:
10527:
10468:
10459:
10438:
10422:
10396:
10386:
10102:
10087:
10072:
10032:
10017:
10002:
9990:
9972:
9962:
9940:
9925:
9915:
9889:
9880:
9855:
9831:
9804:
9780:
9743:
9684:
9675:
9654:
9644:
9597:
9564:
9511:
9492:
9393:
9391:
9381:
9365:
9335:
9333:
9323:
9307:
9287:
9241:
9194:
9083:
9019:
8962:
8946:
8891:This is called a
8876:
8770:
8737:
8647:
8611:
8596:
8497:
8398:
8385:
8340:
8301:
8298:
8248:
8186:
8181:
8164:
8136:
8053:
8043:
8001:
7962:
7959:
7919:
7867:
7845:s, and 1s. I.e.:
7796:
7760:
7667:
7639:
7597:{\displaystyle N}
7571:
7548:{\displaystyle N}
7528:{\displaystyle N}
7502:
7400:
7368:
7344:
7272:
7233:
7214:
7192:
7154:
7115:
7096:
7074:
7015:
6986:
6978:
6971:
6892:
6819:
6794:
6786:
6779:
6774:
6639:
6584:
6555:
6497:
6334:
6300:
6257:
6238:
6216:
6157:
6132:
6124:
6117:
6112:
6045:
5980:
5941:
5876:
5858:
5820:
5764:
5710:
5692:
5627:
5614:
5590:
5553:
5535:
5491:
5476:
5425:
5407:
5386:
5380:
5332:
5279:
5236:
5171:
5134:
5042:
4991:
4935:
4887:
4861:
4837:{\displaystyle p}
4722:
4666:
4660:
4606:
4600:
4552:
4536:
4518:
4496:
4489:
4484:
4458:
4432:
4367:
4349:
4320:
4313:
4308:
4249:
4193:
4175:
4153:
4146:
4141:
4094:
4066:
4013:
3967:
3932:
3896:
3878:
3838:{\displaystyle p}
3813:
3779:
3361:
3325:
3257:
3162:
3139:{\displaystyle N}
3034:
2986:
2952:
2908:
2880:
2855:
2789:
2757:
2748:
2737:
2707:
2682:
2585:
2462:
2370:
2315:
2258:
2239:
2202:
2013:
1905:
1867:
1866:
1822:
1821:
1767:
1722:
1626:
1622:
1571:
1509:
1472:
1331:
1278:
1237:
1134:
1100:
961:
861:
735:
698:
614:
387:
347:
303:
237:
183:
144:
130:
74:Given two school
59:Simpson's paradox
20322:
20286:
20285:
20262:
20243:
20217:
20200:
20194:
20179:
20173:
20172:
20154:
20148:
20138:
20132:
20131:
20097:
20088:
20079:
20078:
20076:
20074:
20065:
20057:
20051:
20050:
20048:
20046:
20032:
20026:
20025:
20023:
19991:
19985:
19980:
19952:
19946:
19936:
19930:
19929:
19911:
19872:
19862:
19838:
19831:
19744:Central tendency
19727:
19725:
19724:
19719:
19714:
19712:
19701:
19700:
19699:
19690:
19689:
19681:
19675:
19674:
19661:
19656:
19640:
19635:
19634:
19611:
19609:
19608:
19603:
19598:
19593:
19592:
19579:
19574:
19573:
19565:
19547:
19545:
19544:
19539:
19536:
19531:
19530:
19522:
19508:
19506:
19505:
19500:
19498:
19497:
19485:
19484:
19461:
19459:
19458:
19453:
19451:
19450:
19449:
19441:
19438:
19437:
19429:
19419:The square root
19415:
19413:
19412:
19407:
19402:
19399:
19394:
19385:
19384:
19383:
19374:
19373:
19365:
19359:
19358:
19345:
19342:
19337:
19322:
19320:
19300:
19294:
19289:
19266:
19264:
19263:
19258:
19255:
19250:
19231:
19229:
19228:
19223:
19220:
19215:
19205:
19200:
19199:
19191:
19180:
19175:
19174:
19166:
19163:
19162:
19154:
19140:
19138:
19137:
19132:
19130:
19129:
19090:
19088:
19087:
19082:
19070:
19068:
19067:
19062:
19050:
19048:
19047:
19042:
19040:
19039:
19038:
19000:
18998:
18997:
18992:
18980:
18978:
18977:
18972:
18964:
18963:
18962:
18937:, the head area
18936:
18934:
18933:
18928:
18926:
18925:
18906:
18904:
18903:
18898:
18860:
18859:
18858:
18838:
18836:
18835:
18830:
18828:
18827:
18795:
18793:
18792:
18787:
18772:
18770:
18769:
18764:
18752:
18750:
18749:
18744:
18727:
18716:
18715:
18696:
18694:
18693:
18688:
18683:
18681:
18670:
18669:
18668:
18652:
18647:
18646:
18645:
18628:
18623:
18605:
18604:
18585:
18583:
18582:
18577:
18575:
18574:
18558:
18556:
18555:
18550:
18548:
18547:
18528:
18526:
18525:
18520:
18515:
18513:
18512:
18503:
18502:
18487:
18482:
18481:
18462:
18460:
18459:
18454:
18442:
18440:
18439:
18434:
18410:
18408:
18407:
18402:
18363:
18361:
18360:
18355:
18350:
18349:
18328:
18327:
18317:
18312:
18294:
18293:
18274:
18272:
18271:
18266:
18254:
18252:
18251:
18246:
18233:
18231:
18230:
18225:
18223:
18222:
18206:
18204:
18203:
18198:
18196:
18195:
18179:
18177:
18176:
18171:
18159:
18157:
18156:
18151:
18149:
18148:
18132:
18130:
18129:
18124:
18112:
18110:
18109:
18104:
18092:
18090:
18089:
18084:
18064:
18062:
18061:
18056:
18051:
18050:
18042:
18033:
18015:
18013:
18012:
18007:
17999:
17994:
17989:
17988:
17983:
17973:
17968:
17967:
17959:
17949:
17948:
17940:
17924:
17922:
17921:
17916:
17911:
17910:
17898:
17893:
17888:
17887:
17882:
17869:
17864:
17863:
17855:
17834:
17832:
17831:
17826:
17814:
17812:
17811:
17806:
17804:
17803:
17762:
17760:
17759:
17754:
17752:
17740:
17738:
17737:
17732:
17730:
17729:
17721:
17711:
17709:
17708:
17703:
17701:
17700:
17680:
17678:
17677:
17672:
17670:
17658:
17656:
17655:
17650:
17648:
17647:
17638:
17637:
17619:
17618:
17603:
17570:
17568:
17567:
17562:
17560:
17556:
17555:
17527:
17526:
17501:
17500:
17459:
17455:
17451:
17450:
17449:
17444:
17437:
17429:
17424:
17415:
17414:
17409:
17402:
17394:
17389:
17378:
17377:
17369:
17365:
17363:
17355:
17350:
17340:
17332:
17327:
17308:
17307:
17302:
17297:
17277:
17275:
17274:
17269:
17267:
17266:
17228:
17227:
17222:
17212:
17211:
17206:
17205:
17178:
17177:
17172:
17158:
17156:
17155:
17150:
17148:
17147:
17109:
17108:
17103:
17093:
17092:
17087:
17086:
17059:
17058:
17053:
17035:
17033:
17032:
17027:
17022:
17021:
17013:
17009:
17008:
17007:
17002:
16995:
16990:
16966:
16965:
16964:
16959:
16954:
16951:
16930:
16928:
16927:
16922:
16917:
16913:
16912:
16911:
16906:
16900:
16899:
16894:
16887:
16882:
16862:
16861:
16860:
16855:
16850:
16847:
16838:
16837:
16832:
16827:
16815:
16813:
16812:
16807:
16801:
16793:
16788:
16779:
16778:
16773:
16750:
16748:
16747:
16742:
16740:
16725:
16723:
16722:
16717:
16715:
16714:
16685:
16683:
16682:
16677:
16672:
16661:
16660:
16651:
16646:
16645:
16626:
16624:
16623:
16618:
16613:
16611:
16610:
16609:
16593:
16592:
16588:
16587:
16586:
16574:
16573:
16568:
16557:
16556:
16551:
16547:
16546:
16545:
16533:
16532:
16527:
16515:
16514:
16504:
16499:
16483:
16478:
16463:
16461:
16460:
16455:
16447:
16446:
16424:
16422:
16421:
16416:
16414:
16407:
16405:
16401:
16400:
16391:
16386:
16385:
16370:
16369:
16359:
16358:
16354:
16353:
16352:
16340:
16339:
16334:
16323:
16322:
16317:
16313:
16312:
16311:
16299:
16298:
16293:
16281:
16280:
16270:
16265:
16249:
16241:
16237:
16233:
16232:
16231:
16219:
16218:
16213:
16202:
16201:
16196:
16192:
16191:
16190:
16178:
16177:
16172:
16160:
16159:
16149:
16144:
16129:
16127:
16125:
16120:
16110:
16105:
16087:
16086:
16081:
16077:
16076:
16075:
16065:
16060:
16038:
16037:
16036:
16026:
16021:
16005:
15996:
15977:
15975:
15974:
15969:
15967:
15948:
15946:
15945:
15940:
15935:
15934:
15929:
15923:
15922:
15912:
15907:
15889:
15888:
15887:
15867:
15865:
15864:
15859:
15857:
15856:
15855:
15831:
15829:
15828:
15823:
15821:
15819:
15818:
15817:
15807:
15802:
15786:
15785:
15776:
15767:
15748:
15746:
15745:
15740:
15738:
15737:
15718:
15716:
15715:
15710:
15702:
15701:
15691:
15686:
15668:
15667:
15617:
15615:
15614:
15609:
15604:
15602:
15595:
15594:
15584:
15583:
15579:
15578:
15577:
15565:
15564:
15559:
15548:
15547:
15542:
15538:
15537:
15536:
15524:
15523:
15518:
15506:
15505:
15495:
15490:
15474:
15469:
15454:
15452:
15451:
15446:
15443:
15411:
15409:
15408:
15403:
15398:
15396:
15395:
15386:
15385:
15381:
15380:
15379:
15367:
15366:
15361:
15350:
15349:
15344:
15340:
15339:
15338:
15326:
15325:
15320:
15308:
15307:
15297:
15292:
15276:
15271:
15253:
15251:
15250:
15245:
15240:
15238:
15237:
15236:
15226:
15221:
15205:
15204:
15203:
15198:
15192:
15191:
15181:
15176:
15160:
15155:
15154:
15153:
15133:
15131:
15130:
15125:
15123:
15122:
15121:
15097:
15095:
15094:
15089:
15081:
15080:
15060:
15058:
15057:
15052:
15050:
15049:
15044:
15010:
15008:
15007:
15002:
14999:
14994:
14991:
14978:
14973:
14972:
14943:
14941:
14940:
14935:
14933:
14926:
14924:
14920:
14919:
14910:
14905:
14904:
14889:
14888:
14878:
14877:
14876:
14867:
14866:
14854:
14853:
14841:
14840:
14830:
14825:
14809:
14801:
14797:
14795:
14790:
14785:
14776:
14771:
14770:
14750:
14745:
14744:
14738:
14737:
14729:
14725:
14714:
14712:
14707:
14706:
14679:
14677:
14676:
14671:
14669:
14665:
14663:
14658:
14649:
14644:
14643:
14612:
14610:
14609:
14604:
14602:
14601:
14583:
14582:
14573:
14567:
14562:
14551:
14543:
14541:
14540:
14535:
14533:
14529:
14524:
14513:
14500:
14498:
14497:
14492:
14490:
14486:
14485:
14482:
14477:
14468:
14467:
14458:
14438:
14436:
14435:
14430:
14427:
14422:
14412:
14407:
14389:
14388:
14372:
14370:
14369:
14364:
14362:
14361:
14351:
14346:
14328:
14327:
14308:
14306:
14305:
14300:
14298:
14293:
14288:
14285:
14279:
14275:
14274:
14271:
14266:
14257:
14256:
14247:
14228:
14221:
14220:
14178:
14175:
14170:
14161:
14160:
14151:
14143:
14142:
14094:
14090:
14088:
14087:
14078:
14074:
14073:
14064:
14063:
14051:
14050:
14029:
14028:
14018:
14013:
13997:
13984:
13979:
13978:
13972:
13971:
13963:
13945:
13940:
13937:
13931:
13927:
13922:
13911:
13899:
13892:
13891:
13849:
13841:
13833:
13832:
13784:
13780:
13775:
13771:
13770:
13755:
13754:
13732:
13727:
13711:
13699:
13698:
13693:
13692:
13684:
13657:
13655:
13654:
13649:
13646:
13641:
13638:
13625:
13623:
13622:
13617:
13578:
13576:
13575:
13570:
13568:
13567:
13551:
13549:
13548:
13543:
13531:
13529:
13528:
13523:
13521:
13520:
13510:
13505:
13480:
13478:
13477:
13472:
13470:
13469:
13464:
13460:
13459:
13458:
13446:
13445:
13430:
13429:
13419:
13414:
13399:
13397:
13390:
13389:
13379:
13374:
13358:
13357:
13356:
13346:
13341:
13325:
13318:
13317:
13316:
13297:
13295:
13294:
13289:
13284:
13283:
13261:
13259:
13258:
13253:
13223:
13221:
13220:
13215:
13185:
13183:
13182:
13177:
13123:
13121:
13120:
13115:
13113:
13111:
13104:
13103:
13093:
13088:
13072:
13071:
13070:
13065:
13061:
13060:
13059:
13047:
13046:
13031:
13030:
13020:
13015:
12999:
12992:
12991:
12990:
12928:
12926:
12925:
12920:
12918:
12917:
12901:
12899:
12898:
12893:
12890:
12885:
12884:
12878:
12877:
12869:
12854:
12852:
12851:
12846:
12838:
12837:
12827:
12822:
12797:
12795:
12794:
12789:
12787:
12783:
12781:
12780:
12779:
12769:
12764:
12748:
12747:
12746:
12741:
12737:
12736:
12735:
12723:
12722:
12707:
12706:
12696:
12691:
12675:
12665:
12660:
12659:
12653:
12652:
12644:
12636:
12631:
12630:
12629:
12624:
12620:
12613:
12612:
12596:
12591:
12575:
12564:
12563:
12562:
12557:
12556:
12548:
12530:
12528:
12527:
12522:
12520:
12519:
12514:
12513:
12505:
12495:sample variance
12490:
12488:
12487:
12482:
12479:
12474:
12473:
12467:
12466:
12458:
12437:
12435:
12434:
12429:
12427:
12426:
12388:Related concepts
12383:
12381:
12380:
12375:
12373:
12371:
12370:
12369:
12368:
12357:
12352:
12336:
12335:
12326:
12317:
12294:
12292:
12291:
12286:
12284:
12279:
12277:
12272:
12271:
12259:
12258:
12257:
12249:
12235:
12233:
12232:
12227:
12224:
12219:
12203:
12201:
12200:
12195:
12187:
12185:
12184:
12175:
12164:
12159:
12138:
12120:
12118:
12117:
12112:
12107:
12105:
12104:
12095:
12084:
12079:
12063:
12058:
12045:
12040:
12039:
12031:
12014:
12012:
12011:
12006:
12003:
11998:
11985:
11980:
11957:
11955:
11954:
11949:
11947:
11946:
11945:
11937:
11920:
11918:
11917:
11912:
11910:
11908:
11903:
11893:
11892:
11883:
11872:
11867:
11848:
11843:
11842:
11834:
11817:
11815:
11814:
11809:
11806:
11801:
11757:
11755:
11754:
11749:
11747:
11746:
11737:
11736:
11731:
11730:
11722:
11715:
11714:
11701:
11696:
11684:
11682:
11681:
11680:
11671:
11670:
11662:
11634:
11629:
11628:
11622:
11617:
11616:
11608:
11601:
11588:
11586:
11585:
11580:
11578:
11573:
11572:
11571:
11558:
11553:
11552:
11544:
11531:
11529:
11528:
11523:
11521:
11517:
11516:
11515:
11506:
11505:
11497:
11491:
11490:
11474:
11469:
11464:
11463:
11455:
11445:
11444:
11439:
11438:
11430:
11426:
11425:
11417:
11411:
11410:
11401:
11400:
11385:
11384:
11376:
11370:
11369:
11354:
11353:
11348:
11347:
11339:
11329:
11328:
11319:
11318:
11313:
11312:
11304:
11300:
11299:
11291:
11285:
11284:
11275:
11274:
11254:
11252:
11251:
11250:
11241:
11240:
11232:
11204:
11199:
11198:
11192:
11187:
11186:
11185:
11180:
11179:
11171:
11162:
11128:Poisson sampling
11118:
11116:
11115:
11110:
11105:
11104:
11095:
11094:
11089:
11088:
11080:
11073:
11072:
11059:
11054:
11044:
11039:
11024:
11022:
11021:
11020:
11011:
11010:
11000:
10995:
10973:
10968:
10967:
10962:
10958:
10957:
10955:
10954:
10945:
10944:
10943:
10938:
10937:
10929:
10922:
10921:
10911:
10906:
10905:
10880:
10875:
10860:
10858:
10857:
10852:
10851:
10843:
10836:
10831:
10830:
10825:
10821:
10820:
10815:
10814:
10806:
10795:
10785:
10783:
10782:
10777:
10766:
10765:
10740:
10738:
10737:
10732:
10721:
10720:
10708:
10707:
10689:
10688:
10652:
10650:
10649:
10644:
10639:
10635:
10634:
10632:
10631:
10622:
10621:
10620:
10615:
10614:
10606:
10599:
10598:
10588:
10586:
10584:
10583:
10574:
10573:
10572:
10567:
10566:
10558:
10551:
10550:
10540:
10538:
10537:
10529:
10528:
10520:
10510:
10505:
10489:
10484:
10469:
10467:
10466:
10461:
10460:
10452:
10445:
10440:
10439:
10434:
10430:
10429:
10424:
10423:
10415:
10404:
10398:
10397:
10392:
10388:
10387:
10379:
10369:
10357:
10355:
10354:
10349:
10341:
10340:
10321:
10319:
10318:
10313:
10311:
10310:
10301:
10300:
10284:
10268:
10266:
10265:
10260:
10258:
10257:
10248:
10247:
10231:
10207:
10205:
10204:
10199:
10188:
10187:
10175:
10174:
10156:
10155:
10117:
10115:
10114:
10109:
10104:
10103:
10095:
10089:
10088:
10080:
10074:
10073:
10065:
10052:
10050:
10049:
10044:
10042:
10038:
10034:
10033:
10025:
10019:
10018:
10010:
10004:
10003:
9995:
9992:
9991:
9983:
9974:
9973:
9968:
9964:
9963:
9955:
9945:
9942:
9941:
9933:
9927:
9926:
9921:
9917:
9916:
9908:
9898:
9890:
9888:
9887:
9882:
9881:
9873:
9866:
9861:
9857:
9856:
9854:
9853:
9844:
9843:
9842:
9833:
9832:
9824:
9818:
9817:
9807:
9805:
9803:
9802:
9793:
9792:
9791:
9782:
9781:
9773:
9767:
9766:
9756:
9754:
9753:
9745:
9744:
9736:
9726:
9721:
9705:
9700:
9685:
9683:
9682:
9677:
9676:
9668:
9661:
9656:
9655:
9650:
9646:
9645:
9637:
9627:
9613:
9611:
9610:
9605:
9603:
9599:
9598:
9596:
9595:
9582:
9573:
9565:
9563:
9562:
9549:
9540:
9532:
9527:
9512:
9504:
9493:
9491:
9486:
9477:
9476:
9466:
9461:
9445:
9440:
9431:
9430:
9420:
9415:
9399:
9394:
9392:
9384:
9382:
9374:
9372:
9367:
9366:
9358:
9346:
9344:
9343:
9338:
9336:
9334:
9326:
9324:
9316:
9314:
9309:
9308:
9300:
9279:
9277:
9276:
9271:
9269:
9268:
9252:
9250:
9249:
9244:
9242:
9240:
9239:
9227:
9222:
9221:
9205:
9203:
9202:
9197:
9195:
9193:
9192:
9180:
9175:
9174:
9158:
9156:
9155:
9150:
9145:
9144:
9132:
9131:
9111:
9109:
9108:
9103:
9101:
9100:
9091:
9090:
9085:
9084:
9076:
9069:
9068:
9055:
9050:
9040:
9035:
9020:
9018:
9017:
9016:
9007:
9006:
8996:
8991:
8969:
8964:
8963:
8958:
8954:
8953:
8948:
8947:
8939:
8928:
8914:Poisson sampling
8887:
8885:
8884:
8879:
8877:
8875:
8870:
8861:
8860:
8850:
8845:
8829:
8824:
8815:
8814:
8804:
8799:
8783:
8778:
8777:
8772:
8771:
8763:
8748:
8746:
8745:
8740:
8738:
8736:
8735:
8734:
8724:
8719:
8703:
8702:
8701:
8692:
8691:
8681:
8676:
8660:
8655:
8654:
8649:
8648:
8640:
8629:
8627:
8626:
8621:
8619:
8618:
8613:
8612:
8604:
8597:
8595:
8590:
8581:
8580:
8570:
8565:
8549:
8544:
8535:
8534:
8524:
8519:
8503:
8498:
8496:
8491:
8482:
8481:
8471:
8466:
8450:
8445:
8436:
8435:
8425:
8420:
8404:
8399:
8397:
8392:
8387:
8386:
8378:
8373:
8368:
8352:
8347:
8342:
8341:
8333:
8328:
8323:
8307:
8302:
8300:
8299:
8297:
8296:
8284:
8282:
8281:
8271:
8266:
8250:
8249:
8247:
8246:
8237:
8236:
8227:
8225:
8224:
8214:
8209:
8193:
8188:
8187:
8182:
8174:
8172:
8166:
8165:
8157:
8147:
8145:
8144:
8139:
8137:
8135:
8134:
8133:
8123:
8118:
8102:
8101:
8100:
8091:
8090:
8080:
8075:
8059:
8054:
8052:
8051:
8050:
8045:
8044:
8036:
8031:
8026:
8010:
8009:
8008:
8003:
8002:
7994:
7989:
7984:
7968:
7963:
7961:
7960:
7958:
7957:
7945:
7942:
7937:
7921:
7920:
7918:
7917:
7908:
7907:
7898:
7895:
7890:
7874:
7869:
7868:
7860:
7844:
7842:
7841:
7836:
7834:
7833:
7817:
7815:
7814:
7809:
7803:
7798:
7797:
7789:
7784:
7779:
7761:
7759:
7758:
7749:
7748:
7739:
7736:
7731:
7713:
7712:
7703:
7702:
7692:
7687:
7669:
7668:
7660:
7650:
7648:
7647:
7642:
7640:
7638:
7637:
7625:
7620:
7619:
7603:
7601:
7600:
7595:
7583:
7581:
7580:
7575:
7573:
7572:
7564:
7554:
7552:
7551:
7546:
7534:
7532:
7531:
7526:
7514:
7512:
7511:
7506:
7504:
7503:
7495:
7481:
7473:
7471:
7470:
7465:
7463:
7459:
7458:
7453:
7449:
7448:
7447:
7438:
7437:
7421:
7416:
7401:
7399:
7398:
7386:
7378:
7374:
7370:
7369:
7367:
7366:
7357:
7356:
7347:
7345:
7343:
7342:
7333:
7332:
7323:
7318:
7317:
7293:
7288:
7273:
7271:
7270:
7258:
7250:
7246:
7242:
7241:
7240:
7235:
7234:
7226:
7222:
7221:
7216:
7215:
7207:
7203:
7202:
7194:
7193:
7185:
7175:
7170:
7155:
7153:
7152:
7140:
7132:
7128:
7124:
7123:
7122:
7117:
7116:
7108:
7104:
7103:
7098:
7097:
7089:
7085:
7084:
7076:
7075:
7067:
7057:
7052:
7036:
7031:
7016:
7014:
7013:
7001:
6989:
6988:
6987:
6984:
6979:
6977:pwr (known
6976:
6973:
6972:
6964:
6940:
6938:
6937:
6932:
6921:
6920:
6888:
6886:
6885:
6880:
6878:
6877:
6872:
6868:
6867:
6866:
6857:
6856:
6840:
6835:
6820:
6818:
6817:
6805:
6797:
6796:
6795:
6792:
6787:
6785:pwr (known
6784:
6781:
6780:
6775:
6767:
6765:
6742:
6740:
6739:
6734:
6723:
6722:
6710:
6709:
6669:
6667:
6666:
6661:
6659:
6658:
6640:
6638:
6637:
6628:
6627:
6626:
6617:
6616:
6606:
6595:
6594:
6586:
6585:
6577:
6566:
6564:
6563:
6558:
6556:
6554:
6553:
6541:
6540:
6539:
6530:
6529:
6519:
6508:
6507:
6499:
6498:
6490:
6479:
6477:
6476:
6471:
6469:
6468:
6449:
6447:
6446:
6441:
6439:
6438:
6423:
6422:
6413:
6412:
6400:
6399:
6381:
6380:
6368:
6367:
6345:
6343:
6342:
6337:
6335:
6333:
6332:
6323:
6322:
6313:
6308:
6307:
6302:
6301:
6293:
6280:
6278:
6277:
6272:
6270:
6266:
6265:
6264:
6259:
6258:
6250:
6246:
6245:
6240:
6239:
6231:
6227:
6226:
6218:
6217:
6209:
6199:
6194:
6178:
6173:
6158:
6156:
6155:
6143:
6135:
6134:
6133:
6130:
6125:
6123:pwr (known
6122:
6119:
6118:
6113:
6105:
6103:
6080:Poisson sampling
6071:
6069:
6068:
6063:
6061:
6057:
6056:
6051:
6047:
6046:
6041:
6033:
6028:
6027:
6018:
6017:
6001:
5996:
5981:
5979:
5965:
5957:
5953:
5952:
5947:
5943:
5942:
5937:
5929:
5924:
5923:
5914:
5913:
5897:
5892:
5877:
5875:
5861:
5859:
5854:
5853:
5844:
5836:
5832:
5831:
5826:
5822:
5821:
5816:
5815:
5814:
5805:
5804:
5794:
5789:
5773:
5765:
5763:
5762:
5753:
5752:
5743:
5731:
5726:
5711:
5709:
5695:
5693:
5685:
5680:
5679:
5674:
5670:
5669:
5668:
5659:
5658:
5648:
5643:
5628:
5620:
5615:
5613:
5612:
5603:
5602:
5593:
5591:
5583:
5574:
5569:
5554:
5552:
5538:
5536:
5528:
5520:
5516:
5515:
5510:
5506:
5505:
5504:
5493:
5492:
5484:
5477:
5475:
5474:
5465:
5464:
5455:
5446:
5441:
5426:
5424:
5410:
5408:
5400:
5388:
5387:
5384:
5382:
5381:
5373:
5343:
5341:
5340:
5335:
5333:
5328:
5327:
5326:
5317:
5316:
5306:
5303:
5298:
5280:
5275:
5267:
5258:
5256:
5255:
5250:
5248:
5247:
5242:
5238:
5237:
5232:
5224:
5219:
5218:
5209:
5208:
5192:
5187:
5172:
5170:
5156:
5148:
5147:
5136:
5135:
5127:
5100:
5098:
5097:
5092:
5086:
5077:
5076:
5066:
5061:
5043:
5041:
5040:
5027:
5018:
5015:
5010:
4992:
4990:
4989:
4988:
4971:
4962:
4959:
4954:
4936:
4934:
4933:
4920:
4911:
4908:
4903:
4888:
4880:
4875:
4874:
4863:
4862:
4854:
4843:
4841:
4840:
4835:
4818:
4816:
4815:
4810:
4793:
4791:
4790:
4785:
4783:
4782:
4772:
4767:
4736:
4734:
4733:
4728:
4723:
4718:
4713:
4704:
4703:
4693:
4688:
4672:
4667:
4662:
4661:
4659:
4658:
4645:
4636:
4633:
4628:
4612:
4607:
4602:
4601:
4599:
4598:
4585:
4576:
4573:
4568:
4553:
4545:
4542:
4537:
4532:
4531:
4520:
4519:
4511:
4507:
4502:
4501:
4497:
4494:
4491:
4490:
4485:
4477:
4475:
4454:
4452:
4451:
4446:
4444:
4443:
4438:
4434:
4433:
4428:
4420:
4415:
4414:
4405:
4404:
4388:
4383:
4368:
4366:
4352:
4350:
4348:
4347:
4335:
4330:
4326:
4325:
4321:
4318:
4315:
4314:
4309:
4301:
4299:
4260:
4258:
4257:
4252:
4250:
4245:
4240:
4231:
4230:
4220:
4215:
4199:
4194:
4189:
4188:
4177:
4176:
4168:
4164:
4159:
4158:
4154:
4151:
4148:
4147:
4142:
4134:
4132:
4105:
4103:
4102:
4097:
4095:
4093:
4092:
4091:
4072:
4067:
4065:
4064:
4052:
4047:
4046:
4024:
4022:
4021:
4016:
4014:
4012:
4011:
4002:
4001:
4000:
3991:
3990:
3980:
3975:
3974:
3969:
3968:
3960:
3956:
3955:
3939:
3934:
3933:
3925:
3914:
3912:
3911:
3906:
3904:
3903:
3898:
3897:
3889:
3879:
3877:
3876:
3867:
3866:
3857:
3844:
3842:
3841:
3836:
3824:
3822:
3821:
3816:
3814:
3812:
3811:
3802:
3801:
3792:
3787:
3786:
3781:
3780:
3772:
3757:
3755:
3754:
3749:
3734:
3732:
3731:
3726:
3721:
3720:
3702:
3701:
3691:
3686:
3671:
3670:
3654:
3649:
3630:
3609:; and variance:
3608:
3606:
3605:
3600:
3598:
3597:
3588:
3587:
3572:
3571:
3556:
3555:
3536:
3514:
3512:
3511:
3506:
3504:
3503:
3494:
3493:
3477:
3461:
3459:
3458:
3453:
3451:
3450:
3434:
3432:
3431:
3426:
3424:
3423:
3403:cluster sampling
3399:
3397:
3396:
3391:
3389:
3388:
3372:
3370:
3369:
3364:
3362:
3357:
3356:
3347:
3342:
3341:
3326:
3323:
3321:
3310:
3309:
3287:
3285:
3284:
3279:
3277:
3276:
3258:
3255:
3244:
3243:
3220:Poisson sampling
3213:
3211:
3210:
3205:
3203:
3202:
3174:
3172:
3171:
3166:
3164:
3163:
3155:
3146:) or estimated (
3145:
3143:
3142:
3137:
3090:
3088:
3087:
3082:
3080:
3079:
3045:
3043:
3042:
3037:
3035:
3030:
3028:
3023:
3013:
3008:
2992:
2987:
2982:
2981:
2972:
2963:
2961:
2960:
2955:
2953:
2948:
2947:
2946:
2936:
2931:
2915:
2910:
2909:
2901:
2891:
2889:
2888:
2883:
2881:
2879:
2868:
2867:
2866:
2857:
2856:
2848:
2842:
2841:
2828:
2823:
2807:
2801:
2796:
2791:
2790:
2782:
2768:
2766:
2765:
2760:
2758:
2756:
2755:
2750:
2749:
2741:
2733:
2732:
2723:
2722:
2719:
2714:
2709:
2708:
2700:
2690:
2689:
2684:
2683:
2675:
2631:random variables
2609:
2607:
2606:
2601:
2587:
2586:
2578:
2562:
2560:
2559:
2554:
2546:
2545:
2529:
2527:
2526:
2521:
2516:
2515:
2514:
2501:
2490:
2485:
2464:
2463:
2455:
2439:
2437:
2436:
2431:
2426:
2425:
2424:
2408:
2407:
2382:
2380:
2379:
2374:
2372:
2371:
2363:
2329:
2327:
2326:
2321:
2316:
2313:
2308:
2299:
2298:
2289:
2286:
2281:
2265:
2260:
2259:
2251:
2241:
2240:
2232:
2216:
2214:
2213:
2208:
2203:
2201:
2200:
2195:
2191:
2189:
2181:
2171:
2166:
2145:
2144:
2142:
2137:
2127:
2119:
2108:
2103:
2087:
2082:
2080:
2075:
2065:
2064:
2055:
2044:
2039:
2020:
2015:
2014:
2006:
1989:
1987:
1986:
1981:
1979:
1978:
1966:
1965:
1949:
1947:
1946:
1941:
1936:
1930:
1925:
1912:
1907:
1906:
1898:
1881:
1879:
1878:
1873:
1868:
1865:
1864:
1863:
1853:
1848:
1829:
1828:
1823:
1820:
1818:
1810:
1800:
1795:
1776:
1775:
1770:
1769:
1768:
1760:
1736:
1734:
1733:
1728:
1723:
1721:
1720:
1719:
1709:
1704:
1688:
1687:
1683:
1682:
1681:
1669:
1668:
1653:
1648:
1632:
1627:
1625:
1624:
1620:
1615:
1603:
1599:
1594:
1578:
1577:
1573:
1569:
1564:
1555:
1554:
1545:
1537:
1532:
1516:
1511:
1510:
1502:
1486:
1484:
1483:
1478:
1473:
1470:
1465:
1453:
1448:
1447:
1428:
1426:
1425:
1420:
1417:
1412:
1389:
1387:
1386:
1381:
1379:
1378:
1342:
1340:
1339:
1334:
1332:
1329:
1328:
1319:
1309:
1304:
1289:
1281:
1280:
1279:
1271:
1250:
1248:
1247:
1242:
1240:
1239:
1238:
1230:
1212:
1210:
1209:
1204:
1202:
1201:
1178:
1176:
1175:
1170:
1168:
1167:
1166:
1155:
1150:
1135:
1127:
1111:
1109:
1108:
1103:
1101:
1099:
1098:
1097:
1096:
1085:
1080:
1064:
1063:
1054:
1045:
1022:
1020:
1019:
1014:
1012:
1011:
1010:
997:
986:
981:
963:
962:
954:
941:
939:
938:
933:
925:
920:
909:
904:
875:
873:
872:
867:
862:
860:
859:
858:
840:
839:
827:
826:
816:
815:
814:
805:
804:
786:
785:
776:
775:
763:
762:
753:
752:
742:
737:
736:
728:
712:
710:
709:
704:
699:
697:
696:
695:
685:
680:
664:
663:
662:
653:
652:
642:
637:
621:
616:
615:
607:
594:
592:
591:
586:
584:
580:
579:
578:
560:
559:
547:
546:
522:
520:
519:
514:
512:
508:
507:
506:
488:
487:
475:
474:
441:
439:
438:
433:
389:
388:
380:
364:
362:
361:
356:
348:
346:
332:
320:
318:
317:
312:
304:
302:
288:
254:
252:
251:
246:
238:
236:
225:
190:
185:
184:
176:
161:
159:
158:
153:
145:
137:
132:
131:
123:
97:
95:
91:
85:
83:
79:
20330:
20329:
20325:
20324:
20323:
20321:
20320:
20319:
20295:
20294:
20276:"Weighted Mean"
20269:
20259:
20226:
20224:Further reading
20221:
20220:
20201:
20197:
20180:
20176:
20169:
20155:
20151:
20139:
20135:
20095:
20089:
20082:
20072:
20070:
20063:
20059:
20058:
20054:
20044:
20042:
20034:
20033:
20029:
19992:
19988:
19953:
19949:
19938:Thomas Lumley (
19937:
19933:
19926:
19912:
19875:
19863:
19852:
19847:
19842:
19841:
19835:absolute values
19832:
19828:
19823:
19818:
19814:Ratio estimator
19789:Weighted median
19764:Weight function
19734:
19702:
19695:
19691:
19680:
19679:
19670:
19666:
19657:
19646:
19641:
19639:
19630:
19626:
19624:
19621:
19620:
19594:
19588:
19584:
19575:
19564:
19563:
19557:
19554:
19553:
19532:
19521:
19520:
19514:
19511:
19510:
19493:
19489:
19480:
19476:
19474:
19471:
19470:
19440:
19439:
19428:
19427:
19426:
19424:
19421:
19420:
19395:
19390:
19379:
19375:
19364:
19363:
19354:
19350:
19346:
19344:
19338:
19327:
19304:
19299:
19290:
19285:
19279:
19276:
19275:
19251:
19246:
19240:
19237:
19236:
19216:
19211:
19201:
19190:
19189:
19176:
19165:
19164:
19153:
19152:
19149:
19146:
19145:
19125:
19121:
19119:
19116:
19115:
19111:
19105:
19097:
19076:
19073:
19072:
19056:
19053:
19052:
19016:
19012:
19011:
19006:
19003:
19002:
18986:
18983:
18982:
18955:
18951:
18944:
18942:
18939:
18938:
18918:
18914:
18912:
18909:
18908:
18851:
18847:
18846:
18844:
18841:
18840:
18820:
18816:
18802:
18799:
18798:
18781:
18778:
18777:
18758:
18755:
18754:
18723:
18711:
18707:
18705:
18702:
18701:
18671:
18664:
18660:
18653:
18651:
18635:
18631:
18630:
18624:
18613:
18600:
18596:
18594:
18591:
18590:
18570:
18566:
18564:
18561:
18560:
18543:
18539:
18537:
18534:
18533:
18508:
18504:
18492:
18488:
18486:
18477:
18473:
18471:
18468:
18467:
18448:
18445:
18444:
18416:
18413:
18412:
18384:
18381:
18380:
18376:
18370:
18333:
18329:
18323:
18319:
18313:
18302:
18289:
18285:
18283:
18280:
18279:
18260:
18257:
18256:
18240:
18237:
18236:
18218:
18214:
18212:
18209:
18208:
18191:
18187:
18185:
18182:
18181:
18165:
18162:
18161:
18144:
18140:
18138:
18135:
18134:
18118:
18115:
18114:
18098:
18095:
18094:
18078:
18075:
18074:
18071:
18043:
18038:
18037:
18029:
18027:
18024:
18023:
17995:
17990:
17984:
17979:
17978:
17969:
17958:
17957:
17939:
17938:
17936:
17933:
17932:
17903:
17899:
17894:
17889:
17883:
17878:
17877:
17865:
17854:
17853:
17847:
17844:
17843:
17820:
17817:
17816:
17799:
17795:
17775:
17772:
17771:
17748:
17746:
17743:
17742:
17720:
17719:
17717:
17714:
17713:
17696:
17692:
17690:
17687:
17686:
17666:
17664:
17661:
17660:
17643:
17639:
17633:
17629:
17614:
17610:
17599:
17597:
17594:
17593:
17590:
17580:
17558:
17557:
17550:
17549:
17543:
17542:
17532:
17531:
17521:
17520:
17514:
17513:
17503:
17502:
17495:
17494:
17489:
17483:
17482:
17477:
17467:
17466:
17457:
17456:
17445:
17440:
17439:
17430:
17425:
17420:
17410:
17405:
17404:
17395:
17390:
17385:
17383:
17379:
17370:
17356:
17351:
17346:
17333:
17328:
17323:
17321:
17317:
17316:
17309:
17298:
17296:
17295:
17291:
17289:
17286:
17285:
17261:
17260:
17255:
17249:
17248:
17243:
17233:
17232:
17223:
17218:
17217:
17207:
17200:
17199:
17194:
17184:
17183:
17182:
17173:
17168:
17167:
17165:
17162:
17161:
17142:
17141:
17136:
17130:
17129:
17124:
17114:
17113:
17104:
17099:
17098:
17088:
17081:
17080:
17075:
17065:
17064:
17063:
17054:
17049:
17048:
17046:
17043:
17042:
17014:
17003:
16998:
16997:
16991:
16980:
16975:
16971:
16970:
16955:
16953:
16952:
16947:
16946:
16944:
16941:
16940:
16907:
16902:
16901:
16895:
16890:
16889:
16883:
16872:
16867:
16863:
16851:
16849:
16848:
16843:
16842:
16828:
16826:
16825:
16823:
16820:
16819:
16794:
16789:
16784:
16774:
16769:
16768:
16766:
16763:
16762:
16736:
16734:
16731:
16730:
16710:
16706:
16704:
16701:
16700:
16692:
16668:
16656:
16652:
16647:
16641:
16637:
16635:
16632:
16631:
16605:
16601:
16594:
16582:
16578:
16569:
16564:
16563:
16562:
16558:
16552:
16541:
16537:
16528:
16523:
16522:
16521:
16517:
16516:
16510:
16506:
16500:
16489:
16484:
16482:
16474:
16472:
16469:
16468:
16442:
16438:
16436:
16433:
16432:
16412:
16411:
16396:
16392:
16387:
16381:
16377:
16365:
16361:
16360:
16348:
16344:
16335:
16330:
16329:
16328:
16324:
16318:
16307:
16303:
16294:
16289:
16288:
16287:
16283:
16282:
16276:
16272:
16266:
16255:
16250:
16248:
16239:
16238:
16227:
16223:
16214:
16209:
16208:
16207:
16203:
16197:
16186:
16182:
16173:
16168:
16167:
16166:
16162:
16161:
16155:
16151:
16145:
16134:
16121:
16116:
16106:
16095:
16082:
16071:
16067:
16061:
16050:
16045:
16041:
16040:
16039:
16032:
16028:
16022:
16011:
16006:
16004:
15997:
15992:
15988:
15986:
15983:
15982:
15963:
15961:
15958:
15957:
15930:
15925:
15924:
15918:
15914:
15908:
15897:
15883:
15879:
15878:
15876:
15873:
15872:
15851:
15847:
15846:
15844:
15841:
15840:
15813:
15809:
15803:
15792:
15787:
15781:
15777:
15775:
15763:
15757:
15754:
15753:
15733:
15729:
15727:
15724:
15723:
15697:
15693:
15687:
15676:
15663:
15659:
15657:
15654:
15653:
15641:In the case of
15639:
15590:
15586:
15585:
15573:
15569:
15560:
15555:
15554:
15553:
15549:
15543:
15532:
15528:
15519:
15514:
15513:
15512:
15508:
15507:
15501:
15497:
15491:
15480:
15475:
15473:
15465:
15463:
15460:
15459:
15439:
15436:
15433:
15432:
15421:
15391:
15387:
15375:
15371:
15362:
15357:
15356:
15355:
15351:
15345:
15334:
15330:
15321:
15316:
15315:
15314:
15310:
15309:
15303:
15299:
15293:
15282:
15277:
15275:
15267:
15265:
15262:
15261:
15232:
15228:
15222:
15211:
15206:
15199:
15194:
15193:
15187:
15183:
15177:
15166:
15161:
15159:
15149:
15145:
15144:
15142:
15139:
15138:
15117:
15113:
15112:
15110:
15107:
15106:
15076:
15072:
15070:
15067:
15066:
15045:
15040:
15039:
15037:
15034:
15033:
15030:
14995:
14990:
14974:
14968:
14967:
14952:
14949:
14948:
14931:
14930:
14915:
14911:
14906:
14900:
14896:
14884:
14880:
14879:
14872:
14868:
14862:
14858:
14849:
14845:
14836:
14832:
14826:
14815:
14810:
14808:
14799:
14798:
14786:
14781:
14772:
14766:
14762:
14752:
14746:
14740:
14739:
14728:
14727:
14724:
14717:
14708:
14702:
14701:
14693:
14691:
14688:
14687:
14659:
14654:
14645:
14639:
14635:
14634:
14630:
14622:
14619:
14618:
14591:
14587:
14578:
14574:
14569:
14563:
14558:
14549:
14546:
14545:
14514:
14512:
14508:
14506:
14503:
14502:
14478:
14473:
14463:
14459:
14457:
14450:
14446:
14444:
14441:
14440:
14423:
14418:
14408:
14397:
14384:
14380:
14378:
14375:
14374:
14357:
14353:
14347:
14336:
14323:
14319:
14317:
14314:
14313:
14296:
14295:
14289:
14284:
14267:
14262:
14252:
14248:
14246:
14239:
14235:
14226:
14225:
14216:
14212:
14171:
14166:
14156:
14152:
14150:
14138:
14134:
14092:
14091:
14083:
14079:
14069:
14065:
14059:
14055:
14046:
14042:
14024:
14020:
14014:
14003:
13998:
13996:
13989:
13980:
13974:
13973:
13962:
13961:
13948:
13947:
13941:
13936:
13912:
13910:
13906:
13897:
13896:
13887:
13883:
13840:
13828:
13824:
13782:
13781:
13766:
13762:
13750:
13746:
13728:
13717:
13712:
13710:
13703:
13694:
13683:
13682:
13681:
13668:
13666:
13663:
13662:
13642:
13637:
13631:
13628:
13627:
13611:
13608:
13607:
13600:
13563:
13559:
13557:
13554:
13553:
13537:
13534:
13533:
13516:
13512:
13506:
13495:
13489:
13486:
13485:
13465:
13454:
13450:
13441:
13437:
13436:
13432:
13431:
13425:
13421:
13415:
13404:
13385:
13381:
13375:
13364:
13359:
13352:
13348:
13342:
13331:
13326:
13324:
13312:
13308:
13306:
13303:
13302:
13279:
13275:
13270:
13267:
13266:
13229:
13226:
13225:
13191:
13188:
13187:
13135:
13132:
13131:
13099:
13095:
13089:
13078:
13073:
13066:
13055:
13051:
13042:
13038:
13037:
13033:
13032:
13026:
13022:
13016:
13005:
13000:
12998:
12986:
12982:
12980:
12977:
12976:
12966:
12913:
12909:
12907:
12904:
12903:
12886:
12880:
12879:
12868:
12867:
12864:
12861:
12860:
12833:
12829:
12823:
12812:
12806:
12803:
12802:
12785:
12784:
12775:
12771:
12765:
12754:
12749:
12742:
12731:
12727:
12718:
12714:
12713:
12709:
12708:
12702:
12698:
12692:
12681:
12676:
12674:
12667:
12661:
12655:
12654:
12643:
12642:
12638:
12637:
12625:
12608:
12604:
12603:
12599:
12598:
12592:
12581:
12576:
12574:
12567:
12558:
12547:
12546:
12545:
12541:
12539:
12536:
12535:
12515:
12504:
12503:
12502:
12500:
12497:
12496:
12475:
12469:
12468:
12457:
12456:
12453:
12450:
12449:
12447:sample variance
12422:
12418:
12416:
12413:
12412:
12401:
12395:
12390:
12364:
12360:
12359:
12353:
12342:
12337:
12331:
12327:
12325:
12313:
12307:
12304:
12303:
12278:
12273:
12267:
12263:
12248:
12247:
12243:
12241:
12238:
12237:
12220:
12215:
12209:
12206:
12205:
12180:
12176:
12171:
12166:
12160:
12149:
12134:
12129:
12126:
12125:
12100:
12096:
12091:
12086:
12080:
12069:
12059:
12054:
12041:
12030:
12029:
12023:
12020:
12019:
11999:
11994:
11981:
11976:
11970:
11967:
11966:
11936:
11935:
11931:
11929:
11926:
11925:
11904:
11899:
11888:
11884:
11879:
11874:
11868:
11857:
11844:
11833:
11832:
11826:
11823:
11822:
11802:
11797:
11791:
11788:
11787:
11784:
11768:
11742:
11738:
11732:
11721:
11720:
11719:
11710:
11706:
11697:
11692:
11676:
11672:
11661:
11660:
11638:
11633:
11618:
11607:
11606:
11600:
11599:
11597:
11594:
11593:
11567:
11563:
11559:
11557:
11543:
11542:
11540:
11537:
11536:
11511:
11507:
11496:
11495:
11486:
11482:
11470:
11465:
11454:
11453:
11440:
11429:
11428:
11427:
11416:
11415:
11406:
11402:
11396:
11392:
11375:
11374:
11365:
11361:
11349:
11338:
11337:
11336:
11324:
11320:
11314:
11303:
11302:
11301:
11290:
11289:
11280:
11276:
11270:
11266:
11259:
11255:
11246:
11242:
11231:
11230:
11208:
11203:
11188:
11181:
11170:
11169:
11168:
11167:
11161:
11160:
11158:
11155:
11154:
11140:
11123:
11100:
11096:
11090:
11079:
11078:
11077:
11068:
11064:
11055:
11050:
11040:
11029:
11016:
11012:
11006:
11002:
10996:
10985:
10977:
10972:
10963:
10950:
10946:
10939:
10928:
10927:
10926:
10917:
10913:
10912:
10910:
10901:
10897:
10887:
10883:
10882:
10876:
10865:
10853:
10842:
10841:
10840:
10835:
10816:
10805:
10804:
10803:
10796:
10794:
10793:
10791:
10788:
10787:
10761:
10757:
10746:
10743:
10742:
10716:
10712:
10703:
10699:
10681:
10677:
10660:
10657:
10656:
10627:
10623:
10616:
10605:
10604:
10603:
10594:
10590:
10589:
10587:
10579:
10575:
10568:
10557:
10556:
10555:
10546:
10542:
10541:
10539:
10530:
10519:
10518:
10517:
10516:
10512:
10506:
10495:
10485:
10474:
10462:
10451:
10450:
10449:
10444:
10425:
10414:
10413:
10412:
10405:
10403:
10402:
10378:
10377:
10370:
10368:
10367:
10365:
10362:
10361:
10336:
10332:
10330:
10327:
10326:
10306:
10302:
10296:
10292:
10280:
10274:
10271:
10270:
10253:
10249:
10243:
10239:
10227:
10221:
10218:
10217:
10183:
10179:
10170:
10166:
10148:
10144:
10127:
10124:
10123:
10094:
10093:
10079:
10078:
10064:
10063:
10061:
10058:
10057:
10024:
10023:
10009:
10008:
9994:
9993:
9982:
9981:
9954:
9953:
9946:
9944:
9943:
9932:
9931:
9907:
9906:
9899:
9897:
9896:
9895:
9891:
9883:
9872:
9871:
9870:
9865:
9849:
9845:
9838:
9834:
9823:
9822:
9813:
9809:
9808:
9806:
9798:
9794:
9787:
9783:
9772:
9771:
9762:
9758:
9757:
9755:
9746:
9735:
9734:
9733:
9732:
9728:
9722:
9711:
9701:
9690:
9678:
9667:
9666:
9665:
9660:
9636:
9635:
9628:
9626:
9625:
9623:
9620:
9619:
9591:
9587:
9578:
9572:
9558:
9554:
9545:
9539:
9538:
9534:
9528:
9517:
9503:
9482:
9472:
9468:
9462:
9451:
9446:
9436:
9426:
9422:
9416:
9405:
9400:
9398:
9383:
9373:
9371:
9357:
9356:
9354:
9351:
9350:
9325:
9315:
9313:
9299:
9298:
9296:
9293:
9292:
9264:
9260:
9258:
9255:
9254:
9235:
9231:
9226:
9217:
9213:
9211:
9208:
9207:
9188:
9184:
9179:
9170:
9166:
9164:
9161:
9160:
9140:
9136:
9127:
9123:
9121:
9118:
9117:
9096:
9092:
9086:
9075:
9074:
9073:
9064:
9060:
9051:
9046:
9036:
9025:
9012:
9008:
9002:
8998:
8992:
8981:
8973:
8968:
8949:
8938:
8937:
8936:
8929:
8927:
8926:
8924:
8921:
8920:
8893:Ratio estimator
8866:
8856:
8852:
8846:
8835:
8830:
8820:
8810:
8806:
8800:
8789:
8784:
8782:
8773:
8762:
8761:
8760:
8758:
8755:
8754:
8730:
8726:
8720:
8709:
8704:
8697:
8693:
8687:
8683:
8677:
8666:
8661:
8659:
8650:
8639:
8638:
8637:
8635:
8632:
8631:
8614:
8603:
8602:
8601:
8586:
8576:
8572:
8566:
8555:
8550:
8540:
8530:
8526:
8520:
8509:
8504:
8502:
8487:
8477:
8473:
8467:
8456:
8451:
8441:
8431:
8427:
8421:
8410:
8405:
8403:
8388:
8377:
8376:
8369:
8358:
8353:
8343:
8332:
8331:
8324:
8313:
8308:
8306:
8292:
8288:
8283:
8277:
8273:
8267:
8256:
8251:
8242:
8238:
8232:
8228:
8226:
8220:
8216:
8210:
8199:
8194:
8192:
8173:
8171:
8170:
8156:
8155:
8153:
8150:
8149:
8129:
8125:
8119:
8108:
8103:
8096:
8092:
8086:
8082:
8076:
8065:
8060:
8058:
8046:
8035:
8034:
8033:
8027:
8016:
8011:
8004:
7993:
7992:
7991:
7985:
7974:
7969:
7967:
7953:
7949:
7944:
7938:
7927:
7922:
7913:
7909:
7903:
7899:
7897:
7891:
7880:
7875:
7873:
7859:
7858:
7850:
7847:
7846:
7829:
7825:
7823:
7820:
7819:
7799:
7788:
7787:
7780:
7769:
7754:
7750:
7744:
7740:
7738:
7732:
7721:
7708:
7704:
7698:
7694:
7688:
7677:
7659:
7658:
7656:
7653:
7652:
7633:
7629:
7624:
7615:
7611:
7609:
7606:
7605:
7589:
7586:
7585:
7563:
7562:
7560:
7557:
7556:
7540:
7537:
7536:
7520:
7517:
7516:
7494:
7493:
7491:
7488:
7487:
7484:
7479:
7475:
7461:
7460:
7454:
7443:
7439:
7433:
7429:
7428:
7424:
7423:
7417:
7406:
7394:
7390:
7385:
7376:
7375:
7362:
7358:
7352:
7348:
7346:
7338:
7334:
7328:
7324:
7322:
7313:
7309:
7299:
7295:
7289:
7278:
7266:
7262:
7257:
7248:
7247:
7236:
7225:
7224:
7223:
7217:
7206:
7205:
7204:
7195:
7184:
7183:
7182:
7181:
7177:
7171:
7160:
7148:
7144:
7139:
7130:
7129:
7118:
7107:
7106:
7105:
7099:
7088:
7087:
7086:
7077:
7066:
7065:
7064:
7063:
7059:
7053:
7042:
7032:
7021:
7009:
7005:
7000:
6993:
6983:
6975:
6974:
6963:
6962:
6961:
6948:
6946:
6943:
6942:
6916:
6912:
6901:
6898:
6897:
6896:We assume that
6873:
6862:
6858:
6852:
6848:
6847:
6843:
6842:
6836:
6825:
6813:
6809:
6804:
6791:
6783:
6782:
6766:
6764:
6763:
6762:
6751:
6748:
6747:
6718:
6714:
6705:
6701:
6678:
6675:
6674:
6654:
6650:
6633:
6629:
6622:
6618:
6612:
6608:
6607:
6605:
6587:
6576:
6575:
6574:
6572:
6569:
6568:
6567:, and for i=j:
6546:
6542:
6535:
6531:
6525:
6521:
6520:
6518:
6500:
6489:
6488:
6487:
6485:
6482:
6481:
6461:
6457:
6455:
6452:
6451:
6431:
6427:
6418:
6414:
6408:
6404:
6392:
6388:
6376:
6372:
6363:
6359:
6351:
6348:
6347:
6328:
6324:
6318:
6314:
6312:
6303:
6292:
6291:
6290:
6288:
6285:
6284:
6260:
6249:
6248:
6247:
6241:
6230:
6229:
6228:
6219:
6208:
6207:
6206:
6205:
6201:
6195:
6184:
6174:
6163:
6151:
6147:
6142:
6129:
6121:
6120:
6104:
6102:
6101:
6100:
6089:
6086:
6085:
6076:
6059:
6058:
6052:
6034:
6032:
6023:
6019:
6013:
6009:
6008:
6004:
6003:
5997:
5986:
5969:
5964:
5955:
5954:
5948:
5930:
5928:
5919:
5915:
5909:
5905:
5904:
5900:
5899:
5893:
5882:
5865:
5860:
5849:
5845:
5843:
5834:
5833:
5827:
5810:
5806:
5800:
5796:
5790:
5779:
5774:
5772:
5758:
5754:
5748:
5744:
5742:
5738:
5734:
5733:
5727:
5716:
5699:
5694:
5684:
5675:
5664:
5660:
5654:
5650:
5644:
5633:
5619:
5608:
5604:
5598:
5594:
5592:
5582:
5581:
5577:
5576:
5570:
5559:
5542:
5537:
5527:
5518:
5517:
5511:
5494:
5483:
5482:
5481:
5470:
5466:
5460:
5456:
5454:
5453:
5449:
5448:
5442:
5431:
5414:
5409:
5399:
5392:
5383:
5372:
5371:
5370:
5357:
5355:
5352:
5351:
5322:
5318:
5312:
5308:
5307:
5305:
5299:
5288:
5268:
5266:
5264:
5261:
5260:
5243:
5225:
5223:
5214:
5210:
5204:
5200:
5199:
5195:
5194:
5188:
5177:
5160:
5155:
5137:
5126:
5125:
5124:
5113:
5110:
5109:
5082:
5072:
5068:
5062:
5051:
5036:
5032:
5023:
5017:
5011:
5000:
4984:
4980:
4976:
4967:
4961:
4955:
4944:
4929:
4925:
4916:
4910:
4904:
4893:
4879:
4864:
4853:
4852:
4851:
4849:
4846:
4845:
4829:
4826:
4825:
4804:
4801:
4800:
4778:
4774:
4768:
4757:
4745:
4742:
4741:
4709:
4699:
4695:
4689:
4678:
4673:
4671:
4654:
4650:
4641:
4635:
4629:
4618:
4613:
4611:
4594:
4590:
4581:
4575:
4569:
4558:
4544:
4543:
4541:
4521:
4510:
4509:
4508:
4506:
4493:
4492:
4476:
4474:
4473:
4472:
4470:
4467:
4466:
4439:
4421:
4419:
4410:
4406:
4400:
4396:
4395:
4391:
4390:
4384:
4373:
4356:
4351:
4343:
4339:
4334:
4317:
4316:
4300:
4298:
4297:
4296:
4292:
4284:
4281:
4280:
4266:sampling design
4236:
4226:
4222:
4216:
4205:
4200:
4198:
4178:
4167:
4166:
4165:
4163:
4150:
4149:
4133:
4131:
4130:
4129:
4127:
4124:
4123:
4116:
4087:
4083:
4076:
4071:
4060:
4056:
4051:
4042:
4038:
4036:
4033:
4032:
4007:
4003:
3996:
3992:
3986:
3982:
3981:
3979:
3970:
3959:
3958:
3957:
3951:
3947:
3935:
3924:
3923:
3920:
3917:
3916:
3899:
3888:
3887:
3886:
3872:
3868:
3862:
3858:
3856:
3854:
3851:
3850:
3830:
3827:
3826:
3807:
3803:
3797:
3793:
3791:
3782:
3771:
3770:
3769:
3767:
3764:
3763:
3743:
3740:
3739:
3716:
3712:
3697:
3693:
3687:
3682:
3666:
3662:
3650:
3645:
3626:
3614:
3611:
3610:
3593:
3589:
3583:
3579:
3567:
3563:
3551:
3547:
3532:
3520:
3517:
3516:
3499:
3495:
3489:
3485:
3473:
3467:
3464:
3463:
3446:
3442:
3440:
3437:
3436:
3419:
3415:
3413:
3410:
3409:
3384:
3380:
3378:
3375:
3374:
3352:
3348:
3346:
3337:
3333:
3324:one sample draw
3322:
3317:
3305:
3301:
3293:
3290:
3289:
3272:
3268:
3254:
3239:
3235:
3227:
3224:
3223:
3198:
3194:
3192:
3189:
3188:
3181:survey sampling
3154:
3153:
3151:
3148:
3147:
3131:
3128:
3127:
3097:sampling design
3075:
3071:
3069:
3066:
3065:
3058:
3024:
3019:
3009:
2998:
2993:
2991:
2977:
2973:
2971:
2969:
2966:
2965:
2942:
2938:
2932:
2921:
2916:
2914:
2900:
2899:
2897:
2894:
2893:
2869:
2862:
2858:
2847:
2846:
2837:
2833:
2824:
2813:
2808:
2806:
2797:
2792:
2781:
2780:
2777:
2774:
2773:
2751:
2740:
2739:
2738:
2728:
2724:
2721:
2715:
2710:
2699:
2698:
2685:
2674:
2673:
2672:
2661:
2658:
2657:
2620:
2615:
2577:
2576:
2568:
2565:
2564:
2541:
2537:
2535:
2532:
2531:
2510:
2506:
2497:
2492:
2486:
2475:
2454:
2453:
2445:
2442:
2441:
2420:
2416:
2415:
2403:
2399:
2391:
2388:
2387:
2362:
2361:
2359:
2356:
2355:
2352:
2347:
2309:
2304:
2294:
2290:
2288:
2282:
2271:
2261:
2250:
2249:
2231:
2230:
2228:
2225:
2224:
2196:
2182:
2177:
2167:
2156:
2151:
2147:
2146:
2138:
2133:
2120:
2115:
2110:
2104:
2093:
2088:
2086:
2076:
2071:
2060:
2056:
2051:
2046:
2040:
2029:
2016:
2005:
2004:
1998:
1995:
1994:
1974:
1970:
1961:
1957:
1955:
1952:
1951:
1932:
1926:
1921:
1908:
1897:
1896:
1890:
1887:
1886:
1859:
1855:
1849:
1838:
1833:
1827:
1811:
1806:
1796:
1785:
1780:
1774:
1759:
1758:
1754:
1752:
1749:
1748:
1715:
1711:
1705:
1694:
1689:
1677:
1673:
1664:
1660:
1659:
1655:
1649:
1638:
1633:
1631:
1616:
1611:
1601:
1595:
1584:
1579:
1565:
1560:
1550:
1546:
1543:
1539:
1533:
1522:
1517:
1515:
1501:
1500:
1498:
1495:
1494:
1466:
1461:
1452:
1443:
1439:
1437:
1434:
1433:
1413:
1408:
1402:
1399:
1398:
1374:
1370:
1368:
1365:
1364:
1361:
1355:
1349:
1324:
1320:
1315:
1305:
1294:
1288:
1270:
1269:
1265:
1263:
1260:
1259:
1229:
1228:
1224:
1222:
1219:
1218:
1197:
1193:
1191:
1188:
1187:
1162:
1158:
1157:
1151:
1140:
1126:
1124:
1121:
1120:
1092:
1088:
1087:
1081:
1070:
1065:
1059:
1055:
1053:
1041:
1035:
1032:
1031:
1006:
1002:
993:
988:
982:
971:
953:
952:
950:
947:
946:
916:
911:
905:
894:
888:
885:
884:
854:
850:
835:
831:
822:
818:
817:
810:
806:
800:
796:
781:
777:
771:
767:
758:
754:
748:
744:
743:
741:
727:
726:
724:
721:
720:
691:
687:
681:
670:
665:
658:
654:
648:
644:
638:
627:
622:
620:
606:
605:
603:
600:
599:
574:
570:
555:
551:
542:
538:
537:
533:
531:
528:
527:
502:
498:
483:
479:
470:
466:
465:
461:
459:
456:
455:
448:
379:
378:
376:
373:
372:
336:
331:
329:
326:
325:
292:
287:
285:
282:
281:
266:Since only the
264:
226:
191:
189:
175:
174:
172:
169:
168:
136:
122:
121:
119:
116:
115:
93:
89:
87:
81:
77:
75:
72:
67:
55:arithmetic mean
40:arithmetic mean
32:
21:Weighted median
17:
12:
11:
5:
20328:
20318:
20317:
20312:
20307:
20293:
20292:
20287:
20268:
20267:External links
20265:
20264:
20263:
20257:
20244:
20225:
20222:
20219:
20218:
20195:
20174:
20167:
20149:
20133:
20106:(4): 485–490.
20080:
20052:
20027:
19986:
19947:
19931:
19924:
19873:
19849:
19848:
19846:
19843:
19840:
19839:
19825:
19824:
19822:
19819:
19817:
19816:
19811:
19806:
19801:
19796:
19791:
19786:
19781:
19776:
19771:
19766:
19761:
19756:
19751:
19746:
19741:
19735:
19733:
19730:
19729:
19728:
19717:
19711:
19708:
19705:
19698:
19694:
19687:
19684:
19678:
19673:
19669:
19665:
19660:
19655:
19652:
19649:
19645:
19638:
19633:
19629:
19601:
19597:
19591:
19587:
19583:
19578:
19571:
19568:
19562:
19535:
19528:
19525:
19519:
19496:
19492:
19488:
19483:
19479:
19447:
19444:
19435:
19432:
19417:
19416:
19405:
19398:
19393:
19389:
19382:
19378:
19371:
19368:
19362:
19357:
19353:
19349:
19341:
19336:
19333:
19330:
19326:
19319:
19316:
19313:
19310:
19307:
19303:
19298:
19293:
19288:
19284:
19254:
19249:
19245:
19233:
19232:
19219:
19214:
19210:
19204:
19197:
19194:
19188:
19184:
19179:
19172:
19169:
19160:
19157:
19128:
19124:
19104:
19101:
19096:
19093:
19080:
19060:
19037:
19034:
19031:
19028:
19025:
19022:
19019:
19015:
19010:
18990:
18970:
18967:
18961:
18958:
18954:
18950:
18947:
18924:
18921:
18917:
18896:
18893:
18890:
18887:
18884:
18881:
18878:
18875:
18872:
18869:
18866:
18863:
18857:
18854:
18850:
18826:
18823:
18819:
18815:
18812:
18809:
18806:
18785:
18762:
18742:
18739:
18736:
18733:
18730:
18726:
18722:
18719:
18714:
18710:
18698:
18697:
18686:
18680:
18677:
18674:
18667:
18663:
18659:
18656:
18650:
18644:
18641:
18638:
18634:
18627:
18622:
18619:
18616:
18612:
18608:
18603:
18599:
18573:
18569:
18546:
18542:
18530:
18529:
18518:
18511:
18507:
18501:
18498:
18495:
18491:
18485:
18480:
18476:
18452:
18443:we can define
18432:
18429:
18426:
18423:
18420:
18400:
18397:
18394:
18391:
18388:
18369:
18366:
18365:
18364:
18353:
18348:
18345:
18342:
18339:
18336:
18332:
18326:
18322:
18316:
18311:
18308:
18305:
18301:
18297:
18292:
18288:
18264:
18244:
18221:
18217:
18194:
18190:
18169:
18147:
18143:
18122:
18102:
18082:
18070:
18067:
18066:
18065:
18054:
18049:
18046:
18041:
18036:
18032:
18017:
18016:
18005:
18002:
17998:
17993:
17987:
17982:
17977:
17972:
17965:
17962:
17956:
17952:
17946:
17943:
17926:
17925:
17914:
17909:
17906:
17902:
17897:
17892:
17886:
17881:
17876:
17873:
17868:
17861:
17858:
17852:
17824:
17802:
17798:
17794:
17791:
17788:
17785:
17782:
17779:
17769:vector of ones
17751:
17727:
17724:
17699:
17695:
17669:
17646:
17642:
17636:
17632:
17628:
17625:
17622:
17617:
17613:
17609:
17606:
17602:
17579:
17576:
17572:
17571:
17554:
17548:
17545:
17544:
17541:
17538:
17537:
17535:
17530:
17525:
17519:
17516:
17515:
17512:
17509:
17508:
17506:
17499:
17493:
17490:
17488:
17485:
17484:
17481:
17478:
17476:
17473:
17472:
17470:
17465:
17462:
17460:
17458:
17454:
17448:
17443:
17436:
17433:
17428:
17423:
17418:
17413:
17408:
17401:
17398:
17393:
17388:
17382:
17376:
17373:
17368:
17362:
17359:
17354:
17349:
17344:
17339:
17336:
17331:
17326:
17320:
17315:
17312:
17310:
17305:
17301:
17294:
17293:
17279:
17278:
17265:
17259:
17256:
17254:
17251:
17250:
17247:
17244:
17242:
17239:
17238:
17236:
17231:
17226:
17221:
17215:
17210:
17204:
17198:
17195:
17193:
17190:
17189:
17187:
17181:
17176:
17171:
17159:
17146:
17140:
17137:
17135:
17132:
17131:
17128:
17125:
17123:
17120:
17119:
17117:
17112:
17107:
17102:
17096:
17091:
17085:
17079:
17076:
17074:
17071:
17070:
17068:
17062:
17057:
17052:
17025:
17020:
17017:
17012:
17006:
17001:
16994:
16989:
16986:
16983:
16979:
16974:
16969:
16962:
16958:
16950:
16920:
16916:
16910:
16905:
16898:
16893:
16886:
16881:
16878:
16875:
16871:
16866:
16858:
16854:
16846:
16841:
16835:
16831:
16805:
16800:
16797:
16792:
16787:
16782:
16777:
16772:
16757:matrix inverse
16739:
16713:
16709:
16691:
16688:
16675:
16671:
16667:
16664:
16659:
16655:
16650:
16644:
16640:
16628:
16627:
16616:
16608:
16604:
16600:
16597:
16591:
16585:
16581:
16577:
16572:
16567:
16561:
16555:
16550:
16544:
16540:
16536:
16531:
16526:
16520:
16513:
16509:
16503:
16498:
16495:
16492:
16488:
16481:
16477:
16453:
16450:
16445:
16441:
16426:
16425:
16410:
16404:
16399:
16395:
16390:
16384:
16380:
16376:
16373:
16368:
16364:
16357:
16351:
16347:
16343:
16338:
16333:
16327:
16321:
16316:
16310:
16306:
16302:
16297:
16292:
16286:
16279:
16275:
16269:
16264:
16261:
16258:
16254:
16247:
16244:
16242:
16240:
16236:
16230:
16226:
16222:
16217:
16212:
16206:
16200:
16195:
16189:
16185:
16181:
16176:
16171:
16165:
16158:
16154:
16148:
16143:
16140:
16137:
16133:
16124:
16119:
16115:
16109:
16104:
16101:
16098:
16094:
16090:
16085:
16080:
16074:
16070:
16064:
16059:
16056:
16053:
16049:
16044:
16035:
16031:
16025:
16020:
16017:
16014:
16010:
16003:
16000:
15998:
15995:
15991:
15990:
15966:
15950:
15949:
15938:
15933:
15928:
15921:
15917:
15911:
15906:
15903:
15900:
15896:
15892:
15886:
15882:
15854:
15850:
15833:
15832:
15816:
15812:
15806:
15801:
15798:
15795:
15791:
15784:
15780:
15774:
15770:
15766:
15762:
15736:
15732:
15720:
15719:
15708:
15705:
15700:
15696:
15690:
15685:
15682:
15679:
15675:
15671:
15666:
15662:
15638:
15635:
15619:
15618:
15607:
15601:
15598:
15593:
15589:
15582:
15576:
15572:
15568:
15563:
15558:
15552:
15546:
15541:
15535:
15531:
15527:
15522:
15517:
15511:
15504:
15500:
15494:
15489:
15486:
15483:
15479:
15472:
15468:
15442:
15420:
15417:
15413:
15412:
15401:
15394:
15390:
15384:
15378:
15374:
15370:
15365:
15360:
15354:
15348:
15343:
15337:
15333:
15329:
15324:
15319:
15313:
15306:
15302:
15296:
15291:
15288:
15285:
15281:
15274:
15270:
15255:
15254:
15243:
15235:
15231:
15225:
15220:
15217:
15214:
15210:
15202:
15197:
15190:
15186:
15180:
15175:
15172:
15169:
15165:
15158:
15152:
15148:
15120:
15116:
15087:
15084:
15079:
15075:
15048:
15043:
15029:
15026:
14998:
14989:
14985:
14982:
14977:
14971:
14966:
14962:
14959:
14956:
14945:
14944:
14929:
14923:
14918:
14914:
14909:
14903:
14899:
14895:
14892:
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5708:
5705:
5702:
5698:
5691:
5688:
5683:
5678:
5673:
5667:
5663:
5657:
5653:
5647:
5642:
5639:
5636:
5632:
5626:
5623:
5618:
5611:
5607:
5601:
5597:
5589:
5586:
5580:
5573:
5568:
5565:
5562:
5558:
5551:
5548:
5545:
5541:
5534:
5531:
5526:
5523:
5521:
5519:
5514:
5509:
5503:
5500:
5497:
5490:
5487:
5480:
5473:
5469:
5463:
5459:
5452:
5445:
5440:
5437:
5434:
5430:
5423:
5420:
5417:
5413:
5406:
5403:
5398:
5395:
5393:
5391:
5379:
5376:
5369:
5366:
5363:
5360:
5359:
5331:
5325:
5321:
5315:
5311:
5302:
5297:
5294:
5291:
5287:
5283:
5278:
5274:
5271:
5246:
5241:
5235:
5231:
5228:
5222:
5217:
5213:
5207:
5203:
5198:
5191:
5186:
5183:
5180:
5176:
5169:
5166:
5163:
5159:
5154:
5151:
5146:
5143:
5140:
5133:
5130:
5123:
5120:
5117:
5089:
5085:
5081:
5075:
5071:
5065:
5060:
5057:
5054:
5050:
5046:
5039:
5035:
5030:
5026:
5022:
5014:
5009:
5006:
5003:
4999:
4995:
4987:
4983:
4979:
4974:
4970:
4966:
4958:
4953:
4950:
4947:
4943:
4939:
4932:
4928:
4923:
4919:
4915:
4907:
4902:
4899:
4896:
4892:
4886:
4883:
4878:
4873:
4870:
4867:
4860:
4857:
4833:
4808:
4781:
4777:
4771:
4766:
4763:
4760:
4756:
4752:
4749:
4738:
4737:
4726:
4721:
4716:
4712:
4708:
4702:
4698:
4692:
4687:
4684:
4681:
4677:
4670:
4665:
4657:
4653:
4648:
4644:
4640:
4632:
4627:
4624:
4621:
4617:
4610:
4605:
4597:
4593:
4588:
4584:
4580:
4572:
4567:
4564:
4561:
4557:
4551:
4548:
4540:
4535:
4530:
4527:
4524:
4517:
4514:
4505:
4500:
4488:
4483:
4480:
4457:
4456:
4455:
4442:
4437:
4431:
4427:
4424:
4418:
4413:
4409:
4403:
4399:
4394:
4387:
4382:
4379:
4376:
4372:
4365:
4362:
4359:
4355:
4346:
4342:
4338:
4333:
4329:
4324:
4312:
4307:
4304:
4295:
4291:
4288:
4248:
4243:
4239:
4235:
4229:
4225:
4219:
4214:
4211:
4208:
4204:
4197:
4192:
4187:
4184:
4181:
4174:
4171:
4162:
4157:
4145:
4140:
4137:
4115:
4108:
4090:
4086:
4082:
4079:
4075:
4070:
4063:
4059:
4055:
4050:
4045:
4041:
4010:
4006:
3999:
3995:
3989:
3985:
3978:
3973:
3966:
3963:
3954:
3950:
3946:
3942:
3938:
3931:
3928:
3902:
3895:
3892:
3885:
3882:
3875:
3871:
3865:
3861:
3834:
3810:
3806:
3800:
3796:
3790:
3785:
3778:
3775:
3762:values, i.e.:
3747:
3724:
3719:
3715:
3711:
3708:
3705:
3700:
3696:
3690:
3685:
3681:
3677:
3674:
3669:
3665:
3661:
3658:
3653:
3648:
3644:
3640:
3637:
3633:
3629:
3625:
3621:
3618:
3596:
3592:
3586:
3582:
3578:
3575:
3570:
3566:
3562:
3559:
3554:
3550:
3546:
3543:
3539:
3535:
3531:
3527:
3524:
3502:
3498:
3492:
3488:
3484:
3480:
3476:
3472:
3449:
3445:
3422:
3418:
3387:
3383:
3360:
3355:
3351:
3345:
3340:
3336:
3332:
3329:
3320:
3316:
3313:
3308:
3304:
3300:
3297:
3275:
3271:
3267:
3264:
3261:
3253:
3250:
3247:
3242:
3238:
3234:
3231:
3201:
3197:
3161:
3158:
3135:
3078:
3074:
3057:
3054:
3033:
3027:
3022:
3018:
3012:
3007:
3004:
3001:
2997:
2990:
2985:
2980:
2976:
2951:
2945:
2941:
2935:
2930:
2927:
2924:
2920:
2913:
2907:
2904:
2878:
2875:
2872:
2865:
2861:
2854:
2851:
2845:
2840:
2836:
2832:
2827:
2822:
2819:
2816:
2812:
2805:
2800:
2795:
2788:
2785:
2770:
2769:
2754:
2747:
2744:
2736:
2731:
2727:
2718:
2713:
2706:
2703:
2696:
2693:
2688:
2681:
2678:
2671:
2668:
2665:
2619:
2616:
2614:
2611:
2599:
2596:
2593:
2590:
2584:
2581:
2575:
2572:
2552:
2549:
2544:
2540:
2519:
2513:
2509:
2504:
2500:
2496:
2489:
2484:
2481:
2478:
2474:
2470:
2467:
2461:
2458:
2452:
2449:
2429:
2423:
2419:
2414:
2411:
2406:
2402:
2398:
2395:
2369:
2366:
2351:
2348:
2346:
2343:
2331:
2330:
2319:
2312:
2307:
2303:
2297:
2293:
2285:
2280:
2277:
2274:
2270:
2264:
2257:
2254:
2248:
2244:
2238:
2235:
2218:
2217:
2206:
2199:
2194:
2188:
2185:
2180:
2176:
2170:
2165:
2162:
2159:
2155:
2150:
2141:
2136:
2132:
2126:
2123:
2118:
2114:
2107:
2102:
2099:
2096:
2092:
2085:
2079:
2074:
2070:
2063:
2059:
2054:
2050:
2043:
2038:
2035:
2032:
2028:
2024:
2019:
2012:
2009:
2003:
1977:
1973:
1969:
1964:
1960:
1939:
1935:
1929:
1924:
1920:
1916:
1911:
1904:
1901:
1895:
1883:
1882:
1871:
1862:
1858:
1852:
1847:
1844:
1841:
1837:
1832:
1826:
1817:
1814:
1809:
1805:
1799:
1794:
1791:
1788:
1784:
1779:
1773:
1766:
1763:
1757:
1738:
1737:
1726:
1718:
1714:
1708:
1703:
1700:
1697:
1693:
1686:
1680:
1676:
1672:
1667:
1663:
1658:
1652:
1647:
1644:
1641:
1637:
1630:
1619:
1614:
1610:
1606:
1598:
1593:
1590:
1587:
1583:
1576:
1568:
1563:
1559:
1553:
1549:
1542:
1536:
1531:
1528:
1525:
1521:
1514:
1508:
1505:
1488:
1487:
1476:
1469:
1464:
1460:
1456:
1451:
1446:
1442:
1416:
1411:
1407:
1377:
1373:
1351:Main article:
1348:
1345:
1344:
1343:
1327:
1323:
1318:
1314:
1308:
1303:
1300:
1297:
1293:
1287:
1284:
1277:
1274:
1268:
1236:
1233:
1227:
1200:
1196:
1186:with variance
1165:
1161:
1154:
1149:
1146:
1143:
1139:
1133:
1130:
1114:
1113:
1095:
1091:
1084:
1079:
1076:
1073:
1069:
1062:
1058:
1052:
1048:
1044:
1040:
1025:
1024:
1009:
1005:
1000:
996:
992:
985:
980:
977:
974:
970:
966:
960:
957:
931:
928:
923:
919:
915:
908:
903:
900:
897:
893:
877:
876:
865:
857:
853:
849:
846:
843:
838:
834:
830:
825:
821:
813:
809:
803:
799:
795:
792:
789:
784:
780:
774:
770:
766:
761:
757:
751:
747:
740:
734:
731:
714:
713:
702:
694:
690:
684:
679:
676:
673:
669:
661:
657:
651:
647:
641:
636:
633:
630:
626:
619:
613:
610:
583:
577:
573:
569:
566:
563:
558:
554:
550:
545:
541:
536:
511:
505:
501:
497:
494:
491:
486:
482:
478:
473:
469:
464:
447:
444:
443:
442:
431:
428:
425:
422:
419:
416:
413:
410:
407:
404:
401:
398:
395:
392:
386:
383:
366:
365:
354:
351:
345:
342:
339:
335:
322:
321:
310:
307:
301:
298:
295:
291:
263:
260:
256:
255:
244:
241:
235:
232:
229:
224:
221:
218:
215:
212:
209:
206:
203:
200:
197:
194:
188:
182:
179:
151:
148:
143:
140:
135:
129:
126:
111:
110:
104:
103:
71:
68:
66:
63:
15:
9:
6:
4:
3:
2:
20327:
20316:
20313:
20311:
20308:
20306:
20303:
20302:
20300:
20291:
20288:
20283:
20282:
20277:
20271:
20270:
20260:
20254:
20250:
20245:
20241:
20237:
20233:
20228:
20227:
20215:
20214:0-9771170-1-4
20211:
20207:
20206:
20199:
20192:
20188:
20184:
20178:
20170:
20168:981-270-527-9
20164:
20160:
20153:
20147:
20143:
20137:
20129:
20125:
20121:
20117:
20113:
20109:
20105:
20101:
20094:
20087:
20085:
20069:
20062:
20056:
20041:
20037:
20031:
20022:
20017:
20013:
20009:
20005:
20001:
19997:
19990:
19984:
19978:
19974:
19970:
19966:
19962:
19958:
19951:
19945:
19941:
19935:
19927:
19921:
19917:
19910:
19908:
19906:
19904:
19902:
19900:
19898:
19896:
19894:
19892:
19890:
19888:
19886:
19884:
19882:
19880:
19878:
19871:
19867:
19861:
19859:
19857:
19855:
19850:
19836:
19830:
19826:
19815:
19812:
19810:
19807:
19805:
19802:
19800:
19797:
19795:
19792:
19790:
19787:
19785:
19782:
19780:
19777:
19775:
19772:
19770:
19767:
19765:
19762:
19760:
19757:
19755:
19752:
19750:
19747:
19745:
19742:
19740:
19737:
19736:
19715:
19709:
19706:
19703:
19696:
19682:
19676:
19671:
19667:
19658:
19653:
19650:
19647:
19643:
19636:
19631:
19627:
19619:
19618:
19617:
19615:
19599:
19595:
19589:
19585:
19581:
19576:
19566:
19560:
19551:
19533:
19523:
19517:
19494:
19490:
19486:
19481:
19477:
19467:
19465:
19442:
19430:
19403:
19396:
19391:
19387:
19380:
19366:
19360:
19355:
19351:
19339:
19334:
19331:
19328:
19324:
19314:
19311:
19308:
19301:
19296:
19291:
19286:
19282:
19274:
19273:
19272:
19270:
19252:
19247:
19243:
19217:
19212:
19208:
19202:
19192:
19186:
19182:
19177:
19167:
19155:
19144:
19143:
19142:
19126:
19122:
19110:
19100:
19092:
19078:
19058:
19032:
19029:
19026:
19020:
19017:
19013:
19008:
18988:
18968:
18965:
18959:
18956:
18952:
18948:
18945:
18922:
18919:
18915:
18891:
18888:
18885:
18879:
18876:
18870:
18867:
18864:
18855:
18852:
18848:
18824:
18821:
18813:
18810:
18807:
18783:
18774:
18760:
18737:
18734:
18731:
18724:
18720:
18717:
18712:
18708:
18684:
18678:
18675:
18672:
18665:
18661:
18657:
18654:
18648:
18642:
18639:
18636:
18632:
18625:
18620:
18617:
18614:
18610:
18606:
18601:
18597:
18589:
18588:
18587:
18571:
18567:
18544:
18540:
18516:
18509:
18505:
18499:
18496:
18493:
18489:
18483:
18478:
18474:
18466:
18465:
18464:
18450:
18427:
18424:
18421:
18418:
18398:
18395:
18389:
18386:
18375:
18351:
18346:
18343:
18340:
18337:
18334:
18330:
18324:
18320:
18314:
18309:
18306:
18303:
18299:
18295:
18290:
18286:
18278:
18277:
18276:
18262:
18242:
18219:
18215:
18192:
18188:
18167:
18145:
18141:
18120:
18100:
18080:
18052:
18047:
18044:
18034:
18022:
18021:
18020:
18003:
17985:
17970:
17960:
17954:
17950:
17941:
17931:
17930:
17929:
17912:
17907:
17904:
17884:
17871:
17866:
17856:
17850:
17842:
17841:
17840:
17838:
17822:
17800:
17792:
17789:
17786:
17783:
17780:
17770:
17766:
17765:design matrix
17722:
17697:
17693:
17684:
17644:
17634:
17630:
17626:
17623:
17620:
17615:
17611:
17604:
17589:
17585:
17575:
17552:
17546:
17539:
17533:
17528:
17523:
17517:
17510:
17504:
17497:
17491:
17486:
17479:
17474:
17468:
17463:
17461:
17452:
17446:
17434:
17431:
17426:
17416:
17411:
17399:
17396:
17391:
17380:
17374:
17371:
17366:
17360:
17357:
17352:
17342:
17337:
17334:
17329:
17318:
17313:
17311:
17284:
17283:
17282:
17263:
17257:
17252:
17245:
17240:
17234:
17229:
17224:
17213:
17202:
17196:
17191:
17185:
17179:
17174:
17160:
17144:
17138:
17133:
17126:
17121:
17115:
17110:
17105:
17094:
17083:
17077:
17072:
17066:
17060:
17055:
17041:
17040:
17039:
17036:
17023:
17018:
17015:
17010:
17004:
16992:
16987:
16984:
16981:
16977:
16972:
16967:
16938:
16934:
16918:
16914:
16908:
16896:
16884:
16879:
16876:
16873:
16869:
16864:
16839:
16816:
16803:
16798:
16795:
16790:
16780:
16775:
16760:
16758:
16754:
16729:
16711:
16707:
16698:
16687:
16673:
16669:
16665:
16662:
16657:
16653:
16648:
16642:
16638:
16614:
16606:
16602:
16598:
16595:
16589:
16583:
16579:
16575:
16570:
16559:
16553:
16548:
16542:
16538:
16534:
16529:
16518:
16511:
16507:
16501:
16496:
16493:
16490:
16486:
16479:
16467:
16466:
16465:
16451:
16448:
16443:
16439:
16429:
16408:
16397:
16393:
16388:
16382:
16378:
16371:
16366:
16362:
16355:
16349:
16345:
16341:
16336:
16325:
16319:
16314:
16308:
16304:
16300:
16295:
16284:
16277:
16273:
16267:
16262:
16259:
16256:
16252:
16245:
16243:
16234:
16228:
16224:
16220:
16215:
16204:
16198:
16193:
16187:
16183:
16179:
16174:
16163:
16156:
16152:
16146:
16141:
16138:
16135:
16131:
16122:
16117:
16113:
16107:
16102:
16099:
16096:
16092:
16088:
16083:
16078:
16072:
16068:
16062:
16057:
16054:
16051:
16047:
16042:
16033:
16029:
16023:
16018:
16015:
16012:
16008:
16001:
15999:
15981:
15980:
15979:
15955:
15936:
15931:
15919:
15915:
15909:
15904:
15901:
15898:
15894:
15890:
15880:
15871:
15870:
15869:
15848:
15838:
15837:weighted mean
15814:
15810:
15804:
15799:
15796:
15793:
15789:
15782:
15778:
15772:
15768:
15764:
15760:
15752:
15751:
15750:
15734:
15730:
15706:
15703:
15698:
15694:
15688:
15683:
15680:
15677:
15673:
15669:
15664:
15660:
15652:
15651:
15650:
15648:
15644:
15634:
15632:
15628:
15624:
15605:
15599:
15596:
15591:
15587:
15580:
15574:
15570:
15566:
15561:
15550:
15544:
15539:
15533:
15529:
15525:
15520:
15509:
15502:
15498:
15492:
15487:
15484:
15481:
15477:
15470:
15458:
15457:
15456:
15430:
15426:
15416:
15399:
15392:
15388:
15382:
15376:
15372:
15368:
15363:
15352:
15346:
15341:
15335:
15331:
15327:
15322:
15311:
15304:
15300:
15294:
15289:
15286:
15283:
15279:
15272:
15260:
15259:
15258:
15241:
15233:
15229:
15223:
15218:
15215:
15212:
15208:
15200:
15188:
15184:
15178:
15173:
15170:
15167:
15163:
15156:
15146:
15137:
15136:
15135:
15114:
15104:
15103:weighted mean
15099:
15085:
15082:
15077:
15073:
15064:
15046:
15025:
15022:
15019:
15017:
15012:
14996:
14987:
14983:
14975:
14964:
14957:
14927:
14916:
14912:
14907:
14901:
14897:
14890:
14885:
14881:
14873:
14863:
14859:
14855:
14850:
14846:
14837:
14833:
14827:
14822:
14819:
14816:
14805:
14803:
14787:
14782:
14778:
14773:
14767:
14763:
14756:
14753:
14747:
14730:
14721:
14719:
14709:
14698:
14686:
14685:
14684:
14681:
14666:
14660:
14655:
14651:
14646:
14640:
14636:
14631:
14627:
14624:
14616:
14598:
14595:
14592:
14588:
14584:
14579:
14575:
14570:
14564:
14559:
14555:
14530:
14525:
14521:
14518:
14515:
14509:
14487:
14479:
14474:
14470:
14464:
14460:
14454:
14451:
14447:
14424:
14419:
14415:
14409:
14404:
14401:
14398:
14394:
14390:
14385:
14381:
14358:
14354:
14348:
14343:
14340:
14337:
14333:
14329:
14324:
14320:
14290:
14281:
14276:
14268:
14263:
14259:
14253:
14249:
14243:
14240:
14236:
14232:
14230:
14217:
14206:
14200:
14194:
14191:
14182:
14172:
14167:
14163:
14157:
14153:
14147:
14139:
14128:
14122:
14116:
14113:
14104:
14098:
14096:
14084:
14080:
14070:
14060:
14056:
14052:
14047:
14043:
14033:
14025:
14021:
14015:
14010:
14007:
14004:
13993:
13991:
13981:
13964:
13954:
13942:
13933:
13928:
13923:
13919:
13916:
13913:
13907:
13903:
13901:
13888:
13877:
13871:
13865:
13862:
13853:
13845:
13842:
13837:
13829:
13818:
13812:
13806:
13803:
13794:
13788:
13786:
13776:
13767:
13759:
13756:
13751:
13747:
13737:
13729:
13724:
13721:
13718:
13707:
13705:
13695:
13685:
13675:
13661:
13660:
13659:
13643:
13634:
13613:
13605:
13595:
13593:
13589:
13585:
13580:
13564:
13560:
13539:
13517:
13513:
13507:
13502:
13499:
13496:
13492:
13466:
13461:
13455:
13451:
13447:
13442:
13438:
13433:
13426:
13422:
13416:
13411:
13408:
13405:
13401:
13394:
13391:
13386:
13382:
13376:
13371:
13368:
13365:
13361:
13353:
13349:
13343:
13338:
13335:
13332:
13328:
13321:
13313:
13309:
13301:
13300:
13299:
13280:
13276:
13263:
13246:
13243:
13240:
13237:
13234:
13208:
13205:
13202:
13199:
13196:
13170:
13167:
13164:
13161:
13158:
13155:
13152:
13149:
13146:
13143:
13140:
13128:
13108:
13105:
13100:
13096:
13090:
13085:
13082:
13079:
13075:
13067:
13062:
13056:
13052:
13048:
13043:
13039:
13034:
13027:
13023:
13017:
13012:
13009:
13006:
12995:
12987:
12983:
12975:
12974:
12973:
12971:
12961:
12959:
12955:
12951:
12947:
12943:
12939:
12934:
12932:
12914:
12910:
12887:
12870:
12858:
12842:
12839:
12834:
12830:
12824:
12819:
12816:
12813:
12809:
12776:
12772:
12766:
12761:
12758:
12755:
12751:
12743:
12738:
12732:
12728:
12724:
12719:
12715:
12710:
12703:
12699:
12693:
12688:
12685:
12682:
12671:
12669:
12662:
12645:
12632:
12626:
12621:
12617:
12614:
12609:
12605:
12600:
12593:
12588:
12585:
12582:
12571:
12569:
12559:
12549:
12534:
12533:
12532:
12516:
12506:
12494:
12476:
12459:
12448:
12444:
12439:
12423:
12419:
12410:
12406:
12400:
12385:
12365:
12361:
12354:
12349:
12346:
12343:
12339:
12332:
12328:
12322:
12318:
12314:
12310:
12300:
12298:
12280:
12274:
12268:
12264:
12260:
12250:
12244:
12221:
12216:
12212:
12191:
12188:
12181:
12177:
12172:
12168:
12161:
12156:
12153:
12150:
12146:
12142:
12139:
12135:
12131:
12108:
12101:
12097:
12092:
12088:
12081:
12076:
12073:
12070:
12066:
12060:
12055:
12051:
12047:
12042:
12032:
12026:
12018:
12017:
12016:
12000:
11995:
11991:
11987:
11982:
11977:
11973:
11963:
11961:
11938:
11932:
11905:
11900:
11896:
11889:
11885:
11880:
11876:
11869:
11864:
11861:
11858:
11854:
11850:
11845:
11835:
11829:
11821:
11820:
11819:
11803:
11798:
11794:
11779:
11777:
11776:Bootstrapping
11773:
11763:
11743:
11733:
11723:
11716:
11711:
11707:
11698:
11693:
11689:
11685:
11677:
11663:
11657:
11648:
11645:
11642:
11635:
11630:
11624:
11619:
11609:
11603:
11592:
11591:
11590:
11574:
11568:
11564:
11560:
11554:
11545:
11518:
11512:
11498:
11492:
11487:
11483:
11476:
11471:
11466:
11456:
11449:
11441:
11431:
11418:
11412:
11407:
11403:
11397:
11393:
11377:
11371:
11366:
11362:
11355:
11350:
11340:
11333:
11330:
11325:
11315:
11305:
11292:
11286:
11281:
11277:
11271:
11267:
11260:
11256:
11247:
11233:
11227:
11218:
11215:
11212:
11205:
11200:
11194:
11189:
11182:
11172:
11164:
11153:
11152:
11151:
11149:
11148:Taylor series
11145:
11144:bootstrapping
11135:
11131:
11129:
11122:
11119:
11106:
11101:
11091:
11081:
11074:
11069:
11065:
11056:
11051:
11047:
11041:
11036:
11033:
11030:
11026:
11017:
11007:
11003:
10997:
10992:
10989:
10986:
10982:
10974:
10969:
10964:
10959:
10951:
10947:
10940:
10930:
10923:
10918:
10914:
10902:
10898:
10894:
10891:
10884:
10877:
10872:
10869:
10866:
10862:
10854:
10844:
10837:
10832:
10826:
10817:
10807:
10797:
10773:
10770:
10762:
10758:
10754:
10751:
10728:
10725:
10717:
10713:
10709:
10704:
10700:
10693:
10690:
10685:
10682:
10674:
10671:
10668:
10665:
10653:
10640:
10636:
10628:
10624:
10617:
10607:
10600:
10595:
10591:
10580:
10576:
10569:
10559:
10552:
10547:
10543:
10534:
10531:
10513:
10507:
10502:
10499:
10496:
10492:
10486:
10481:
10478:
10475:
10471:
10463:
10453:
10446:
10441:
10435:
10426:
10416:
10406:
10399:
10393:
10380:
10371:
10359:
10345:
10342:
10337:
10333:
10323:
10307:
10303:
10297:
10293:
10289:
10285:
10281:
10277:
10254:
10250:
10244:
10240:
10236:
10232:
10228:
10224:
10215:
10211:
10195:
10192:
10184:
10180:
10176:
10171:
10167:
10160:
10157:
10152:
10149:
10141:
10138:
10135:
10132:
10121:
10096:
10090:
10081:
10066:
10054:
10039:
10026:
10020:
10011:
9996:
9984:
9978:
9975:
9969:
9956:
9947:
9934:
9928:
9922:
9909:
9900:
9892:
9884:
9874:
9867:
9862:
9858:
9850:
9846:
9839:
9835:
9825:
9819:
9814:
9810:
9799:
9795:
9788:
9784:
9774:
9768:
9763:
9759:
9750:
9747:
9729:
9723:
9718:
9715:
9712:
9708:
9702:
9697:
9694:
9691:
9687:
9679:
9669:
9662:
9657:
9651:
9638:
9629:
9617:
9614:
9600:
9592:
9588:
9583:
9579:
9575:
9569:
9566:
9559:
9555:
9550:
9546:
9542:
9535:
9529:
9524:
9521:
9518:
9514:
9508:
9505:
9500:
9497:
9494:
9487:
9483:
9479:
9473:
9469:
9463:
9458:
9455:
9452:
9448:
9441:
9437:
9433:
9427:
9423:
9417:
9412:
9409:
9406:
9402:
9395:
9385:
9375:
9368:
9359:
9348:
9327:
9317:
9310:
9301:
9285:
9283:
9265:
9261:
9236:
9232:
9228:
9223:
9218:
9214:
9189:
9185:
9181:
9176:
9171:
9167:
9146:
9141:
9137:
9133:
9128:
9124:
9097:
9087:
9077:
9070:
9065:
9061:
9052:
9047:
9043:
9037:
9032:
9029:
9026:
9022:
9013:
9003:
8999:
8993:
8988:
8985:
8982:
8978:
8970:
8965:
8959:
8950:
8940:
8930:
8919:
8918:
8917:
8915:
8910:
8909:Taylor series
8905:
8900:
8898:
8894:
8889:
8871:
8867:
8863:
8857:
8853:
8847:
8842:
8839:
8836:
8832:
8825:
8821:
8817:
8811:
8807:
8801:
8796:
8793:
8790:
8786:
8779:
8774:
8764:
8752:
8731:
8727:
8721:
8716:
8713:
8710:
8706:
8698:
8694:
8688:
8684:
8678:
8673:
8670:
8667:
8663:
8656:
8651:
8641:
8615:
8605:
8598:
8591:
8587:
8583:
8577:
8573:
8567:
8562:
8559:
8556:
8552:
8545:
8541:
8537:
8531:
8527:
8521:
8516:
8513:
8510:
8506:
8499:
8492:
8488:
8484:
8478:
8474:
8468:
8463:
8460:
8457:
8453:
8446:
8442:
8438:
8432:
8428:
8422:
8417:
8414:
8411:
8407:
8400:
8393:
8389:
8379:
8370:
8365:
8362:
8359:
8355:
8348:
8344:
8334:
8325:
8320:
8317:
8314:
8310:
8303:
8293:
8289:
8285:
8278:
8274:
8268:
8263:
8260:
8257:
8253:
8243:
8239:
8233:
8229:
8221:
8217:
8211:
8206:
8203:
8200:
8196:
8189:
8175:
8167:
8158:
8130:
8126:
8120:
8115:
8112:
8109:
8105:
8097:
8093:
8087:
8083:
8077:
8072:
8069:
8066:
8062:
8055:
8047:
8037:
8028:
8023:
8020:
8017:
8013:
8005:
7995:
7986:
7981:
7978:
7975:
7971:
7964:
7954:
7950:
7946:
7939:
7934:
7931:
7928:
7924:
7914:
7910:
7904:
7900:
7892:
7887:
7884:
7881:
7877:
7870:
7861:
7855:
7852:
7830:
7826:
7804:
7800:
7790:
7781:
7776:
7773:
7770:
7766:
7762:
7755:
7751:
7745:
7741:
7733:
7728:
7725:
7722:
7718:
7714:
7709:
7705:
7699:
7695:
7689:
7684:
7681:
7678:
7674:
7670:
7661:
7634:
7630:
7626:
7621:
7616:
7612:
7591:
7565:
7542:
7522:
7496:
7474:
7455:
7450:
7444:
7440:
7434:
7430:
7425:
7418:
7413:
7410:
7407:
7403:
7395:
7391:
7387:
7382:
7380:
7371:
7363:
7359:
7353:
7349:
7339:
7335:
7329:
7325:
7314:
7310:
7306:
7303:
7296:
7290:
7285:
7282:
7279:
7275:
7267:
7263:
7259:
7254:
7252:
7243:
7237:
7227:
7218:
7208:
7199:
7196:
7178:
7172:
7167:
7164:
7161:
7157:
7149:
7145:
7141:
7136:
7134:
7125:
7119:
7109:
7100:
7090:
7081:
7078:
7060:
7054:
7049:
7046:
7043:
7039:
7033:
7028:
7025:
7022:
7018:
7010:
7006:
7002:
6997:
6995:
6980:
6965:
6955:
6952:
6928:
6925:
6917:
6913:
6909:
6906:
6874:
6869:
6863:
6859:
6853:
6849:
6844:
6837:
6832:
6829:
6826:
6822:
6814:
6810:
6806:
6801:
6788:
6768:
6756:
6753:
6746:
6745:
6744:
6730:
6727:
6719:
6715:
6711:
6706:
6702:
6695:
6692:
6689:
6686:
6683:
6671:
6655:
6651:
6647:
6644:
6641:
6634:
6630:
6623:
6619:
6613:
6609:
6602:
6599:
6596:
6591:
6588:
6550:
6547:
6543:
6536:
6532:
6526:
6522:
6515:
6512:
6509:
6504:
6501:
6465:
6462:
6458:
6435:
6432:
6424:
6419:
6415:
6409:
6405:
6401:
6396:
6393:
6389:
6385:
6377:
6373:
6369:
6364:
6360:
6353:
6329:
6325:
6319:
6315:
6309:
6304:
6294:
6281:
6267:
6261:
6251:
6242:
6232:
6223:
6220:
6202:
6196:
6191:
6188:
6185:
6181:
6175:
6170:
6167:
6164:
6160:
6152:
6148:
6144:
6139:
6126:
6106:
6094:
6091:
6083:
6081:
6075:
6072:
6053:
6048:
6038:
6035:
6029:
6024:
6020:
6014:
6010:
6005:
5998:
5993:
5990:
5987:
5983:
5976:
5973:
5970:
5966:
5961:
5959:
5949:
5944:
5934:
5931:
5925:
5920:
5916:
5910:
5906:
5901:
5894:
5889:
5886:
5883:
5879:
5872:
5869:
5866:
5862:
5855:
5850:
5846:
5840:
5838:
5828:
5823:
5817:
5811:
5807:
5801:
5797:
5791:
5786:
5783:
5780:
5776:
5769:
5766:
5759:
5755:
5749:
5745:
5739:
5735:
5728:
5723:
5720:
5717:
5713:
5706:
5703:
5700:
5696:
5689:
5686:
5681:
5676:
5671:
5665:
5661:
5655:
5651:
5645:
5640:
5637:
5634:
5630:
5624:
5621:
5616:
5609:
5605:
5599:
5595:
5587:
5584:
5578:
5571:
5566:
5563:
5560:
5556:
5549:
5546:
5543:
5539:
5532:
5529:
5524:
5522:
5512:
5507:
5501:
5498:
5495:
5485:
5478:
5471:
5467:
5461:
5457:
5450:
5443:
5438:
5435:
5432:
5428:
5421:
5418:
5415:
5411:
5404:
5401:
5396:
5394:
5374:
5364:
5361:
5349:
5345:
5329:
5323:
5319:
5313:
5309:
5300:
5295:
5292:
5289:
5285:
5281:
5272:
5269:
5244:
5239:
5229:
5226:
5220:
5215:
5211:
5205:
5201:
5196:
5189:
5184:
5181:
5178:
5174:
5167:
5164:
5161:
5157:
5152:
5144:
5141:
5138:
5128:
5118:
5115:
5107:
5102:
5087:
5083:
5079:
5073:
5069:
5063:
5058:
5055:
5052:
5048:
5044:
5037:
5033:
5028:
5024:
5020:
5012:
5007:
5004:
5001:
4997:
4993:
4985:
4981:
4977:
4972:
4968:
4964:
4956:
4951:
4948:
4945:
4941:
4937:
4930:
4926:
4921:
4917:
4913:
4905:
4900:
4897:
4894:
4890:
4884:
4881:
4876:
4871:
4868:
4865:
4855:
4831:
4823:
4819:
4806:
4797:
4779:
4775:
4769:
4764:
4761:
4758:
4754:
4750:
4747:
4724:
4719:
4714:
4710:
4706:
4700:
4696:
4690:
4685:
4682:
4679:
4675:
4668:
4663:
4655:
4651:
4646:
4642:
4638:
4630:
4625:
4622:
4619:
4615:
4608:
4603:
4595:
4591:
4586:
4582:
4578:
4570:
4565:
4562:
4559:
4555:
4549:
4546:
4538:
4533:
4528:
4525:
4522:
4512:
4503:
4498:
4478:
4465:
4464:
4463:
4440:
4435:
4425:
4422:
4416:
4411:
4407:
4401:
4397:
4392:
4385:
4380:
4377:
4374:
4370:
4363:
4360:
4357:
4353:
4344:
4340:
4336:
4331:
4327:
4322:
4302:
4293:
4289:
4286:
4279:
4278:
4277:
4275:
4271:
4267:
4262:
4246:
4241:
4237:
4233:
4227:
4223:
4217:
4212:
4209:
4206:
4202:
4195:
4190:
4185:
4182:
4179:
4169:
4160:
4155:
4135:
4121:
4113:
4107:
4088:
4084:
4080:
4077:
4073:
4068:
4061:
4057:
4053:
4048:
4043:
4039:
4030:
4025:
4008:
4004:
3997:
3993:
3987:
3983:
3976:
3971:
3961:
3952:
3948:
3944:
3940:
3936:
3926:
3900:
3890:
3883:
3880:
3873:
3869:
3863:
3859:
3848:
3832:
3808:
3804:
3798:
3794:
3788:
3783:
3773:
3761:
3745:
3736:
3717:
3713:
3709:
3706:
3698:
3694:
3688:
3683:
3679:
3675:
3667:
3663:
3656:
3651:
3646:
3642:
3638:
3631:
3627:
3623:
3616:
3594:
3590:
3584:
3580:
3576:
3568:
3564:
3557:
3552:
3548:
3544:
3537:
3533:
3529:
3522:
3500:
3496:
3490:
3486:
3482:
3478:
3474:
3470:
3447:
3443:
3420:
3416:
3406:
3404:
3385:
3381:
3358:
3353:
3349:
3343:
3338:
3334:
3330:
3314:
3311:
3306:
3302:
3295:
3273:
3269:
3265:
3259:
3251:
3248:
3245:
3240:
3236:
3229:
3221:
3217:
3199:
3195:
3186:
3182:
3178:
3156:
3133:
3125:
3122:or sometimes
3121:
3117:
3113:
3109:
3104:
3102:
3098:
3094:
3076:
3072:
3063:
3053:
3051:
3046:
3031:
3025:
3020:
3016:
3010:
3005:
3002:
2999:
2995:
2988:
2978:
2974:
2949:
2943:
2939:
2933:
2928:
2925:
2922:
2918:
2911:
2902:
2876:
2873:
2870:
2863:
2849:
2843:
2838:
2834:
2825:
2820:
2817:
2814:
2810:
2803:
2798:
2793:
2783:
2752:
2742:
2729:
2725:
2716:
2711:
2701:
2694:
2686:
2676:
2666:
2663:
2656:
2655:
2654:
2652:
2648:
2644:
2640:
2636:
2632:
2629:
2625:
2610:
2597:
2594:
2591:
2579:
2570:
2550:
2547:
2542:
2538:
2517:
2511:
2507:
2502:
2498:
2494:
2487:
2482:
2479:
2476:
2472:
2468:
2456:
2447:
2427:
2421:
2417:
2412:
2404:
2400:
2393:
2384:
2364:
2342:
2340:
2336:
2317:
2310:
2305:
2301:
2295:
2291:
2283:
2278:
2275:
2272:
2268:
2262:
2252:
2246:
2242:
2233:
2223:
2222:
2221:
2204:
2197:
2192:
2186:
2183:
2178:
2174:
2168:
2163:
2160:
2157:
2153:
2148:
2139:
2134:
2130:
2124:
2121:
2116:
2112:
2105:
2100:
2097:
2094:
2090:
2083:
2077:
2072:
2068:
2061:
2057:
2052:
2048:
2041:
2036:
2033:
2030:
2026:
2022:
2017:
2007:
2001:
1993:
1992:
1991:
1975:
1971:
1967:
1962:
1958:
1937:
1933:
1927:
1922:
1918:
1914:
1909:
1899:
1893:
1869:
1860:
1856:
1850:
1845:
1842:
1839:
1835:
1830:
1824:
1815:
1812:
1807:
1803:
1797:
1792:
1789:
1786:
1782:
1777:
1771:
1761:
1755:
1747:
1746:
1745:
1743:
1724:
1716:
1712:
1706:
1701:
1698:
1695:
1691:
1684:
1678:
1674:
1670:
1665:
1661:
1656:
1650:
1645:
1642:
1639:
1635:
1628:
1617:
1612:
1608:
1604:
1596:
1591:
1588:
1585:
1581:
1574:
1566:
1561:
1557:
1551:
1547:
1540:
1534:
1529:
1526:
1523:
1519:
1512:
1503:
1493:
1492:
1491:
1474:
1467:
1462:
1458:
1454:
1449:
1444:
1440:
1432:
1431:
1430:
1414:
1409:
1405:
1397:
1393:
1375:
1371:
1360:
1354:
1325:
1321:
1316:
1312:
1306:
1301:
1298:
1295:
1285:
1282:
1272:
1266:
1258:
1257:
1256:
1254:
1231:
1225:
1216:
1198:
1194:
1185:
1180:
1163:
1159:
1152:
1147:
1144:
1141:
1131:
1128:
1119:
1118:ordinary mean
1093:
1089:
1082:
1077:
1074:
1071:
1060:
1056:
1050:
1046:
1042:
1038:
1030:
1029:
1028:
1007:
1003:
998:
994:
990:
983:
978:
975:
972:
964:
955:
945:
944:
943:
929:
926:
921:
917:
913:
906:
901:
898:
895:
881:
863:
855:
851:
847:
844:
841:
836:
832:
828:
823:
819:
811:
807:
801:
797:
793:
790:
787:
782:
778:
772:
768:
764:
759:
755:
749:
745:
738:
729:
719:
718:
717:
700:
692:
688:
682:
677:
674:
671:
659:
655:
649:
645:
639:
634:
631:
628:
617:
608:
598:
597:
596:
581:
575:
571:
567:
564:
561:
556:
552:
548:
543:
539:
534:
526:
509:
503:
499:
495:
492:
489:
484:
480:
476:
471:
467:
462:
453:
429:
426:
420:
417:
414:
408:
402:
399:
396:
390:
381:
371:
370:
369:
352:
349:
343:
340:
337:
333:
324:
323:
308:
305:
299:
296:
293:
289:
280:
279:
278:
275:
273:
269:
259:
242:
239:
233:
230:
227:
219:
216:
213:
207:
201:
198:
195:
186:
177:
167:
166:
165:
162:
149:
146:
141:
138:
133:
124:
108:
107:
106:
101:
100:
99:
70:Basic example
62:
60:
56:
51:
49:
45:
41:
37:
30:
26:
22:
20279:
20274:David Terr.
20248:
20231:
20204:
20198:
20183:Inequalities
20182:
20177:
20158:
20152:
20136:
20103:
20099:
20071:. Retrieved
20067:
20055:
20043:. Retrieved
20039:
20030:
20003:
19999:
19989:
19960:
19956:
19950:
19934:
19915:
19829:
19468:
19463:
19418:
19234:
19112:
19098:
18775:
18700:approaching
18699:
18531:
18377:
18072:
18018:
17927:
17591:
17573:
17280:
17037:
16817:
16761:
16693:
16629:
16430:
16427:
15953:
15951:
15834:
15721:
15642:
15640:
15623:standardized
15620:
15428:
15424:
15422:
15414:
15256:
15134:is given by
15100:
15062:
15031:
15023:
15020:
15015:
15013:
14946:
14682:
14311:
13603:
13601:
13584:standardized
13581:
13483:
13264:
13129:
13126:
12969:
12967:
12957:
12953:
12945:
12941:
12935:
12856:
12800:
12492:
12442:
12440:
12402:
12301:
12123:
11964:
11959:
11923:
11785:
11769:
11760:
11534:
11141:
11132:
11124:
11120:
10654:
10360:
10324:
10213:
10209:
10055:
9618:
9615:
9349:
9290:
9281:
9115:
8901:
8896:
8890:
7485:
6895:
6672:
6282:
6084:
6077:
6073:
5350:
5346:
5105:
5103:
4821:
4799:
4739:
4461:
4274:pps sampling
4272:(such as in
4269:
4263:
4119:
4117:
4111:
4029:design based
4028:
4026:
3846:
3759:
3737:
3407:
3215:
3176:
3123:
3119:
3115:
3111:
3105:
3061:
3059:
3049:
3047:
2771:
2628:uncorrelated
2623:
2621:
2385:
2353:
2332:
2219:
1884:
1741:
1739:
1489:
1362:
1214:
1181:
1115:
1026:
882:
878:
715:
449:
367:
276:
267:
265:
257:
163:
112:
105:
73:
52:
35:
33:
20073:22 December
20045:22 December
19616:(squared),
19552:(squared),
17815:(of length
17767:equal to a
16937:commutative
12299:, squared.
11782:Other notes
4495:known
4319:known
4152:known
3062:model based
2639:expectation
1394:with known
20299:Categories
19845:References
19107:See also:
18586:is simply
18372:See also:
17582:See also:
15647:normalized
15627:normalized
13588:normalized
12397:See also:
3845:-expanded
3758:-expanded
2350:Expectancy
1357:See also:
20281:MathWorld
20240:300283069
19804:Weighting
19707:−
19686:¯
19677:−
19644:∑
19628:σ
19586:σ
19570:¯
19561:σ
19527:¯
19518:σ
19491:σ
19478:σ
19446:¯
19434:^
19431:σ
19388:σ
19370:¯
19361:−
19325:∑
19312:−
19287:ν
19283:χ
19248:ν
19244:χ
19213:ν
19209:χ
19196:¯
19187:σ
19171:¯
19159:^
19156:σ
19123:χ
19030:−
19018:−
19009:≤
18957:−
18949:−
18920:−
18889:−
18868:−
18853:−
18822:−
18811:−
18735:−
18676:−
18658:−
18640:−
18611:∑
18497:−
18431:Δ
18428:−
18393:Δ
18344:−
18300:∑
18045:−
17964:¯
17955:σ
17945:¯
17905:−
17860:¯
17851:σ
17787:…
17726:¯
17624:…
17432:−
17397:−
17372:−
17358:−
17335:−
17304:¯
17209:⊤
17090:⊤
17016:−
16978:∑
16961:¯
16870:∑
16857:¯
16834:¯
16796:−
16708:σ
16599:−
16584:∗
16580:μ
16576:−
16543:∗
16539:μ
16535:−
16487:∑
16372:−
16350:∗
16346:μ
16342:−
16309:∗
16305:μ
16301:−
16253:∑
16229:∗
16225:μ
16221:−
16188:∗
16184:μ
16180:−
16132:∑
16093:∑
16089:−
16048:∑
16009:∑
15895:∑
15885:∗
15881:μ
15853:∗
15849:μ
15835:Then the
15790:∑
15674:∑
15597:−
15575:∗
15571:μ
15567:−
15534:∗
15530:μ
15526:−
15478:∑
15377:∗
15373:μ
15369:−
15336:∗
15332:μ
15328:−
15280:∑
15209:∑
15164:∑
15151:∗
15147:μ
15119:∗
15115:μ
15101:Then the
15083:≥
14988:σ
14958:
14891:−
14864:∗
14860:μ
14856:−
14813:∑
14757:−
14734:^
14731:σ
14628:−
14519:−
14455:−
14395:∑
14334:∑
14282:σ
14244:−
14201:
14195:−
14183:
14148:−
14123:
14117:−
14105:
14061:∗
14057:μ
14053:−
14034:
14001:∑
13968:^
13965:σ
13955:
13934:σ
13917:−
13872:
13866:−
13854:
13838:−
13813:
13807:−
13795:
13760:μ
13757:−
13738:
13715:∑
13689:^
13686:σ
13676:
13635:σ
13614:μ
13493:∑
13456:∗
13452:μ
13448:−
13402:∑
13392:−
13362:∑
13329:∑
13106:−
13076:∑
13057:∗
13053:μ
13049:−
13003:∑
12911:σ
12874:^
12871:σ
12810:∑
12752:∑
12733:∗
12729:μ
12725:−
12679:∑
12649:^
12646:σ
12618:μ
12615:−
12579:∑
12553:^
12550:σ
12510:^
12507:σ
12463:^
12460:σ
12445:weighted
12424:∗
12420:μ
12340:∑
12265:σ
12254:¯
12245:σ
12213:σ
12189:≤
12147:∑
12143:≤
12067:∑
12052:σ
12036:¯
12027:σ
11992:σ
11974:σ
11942:¯
11933:σ
11897:σ
11855:∑
11839:¯
11830:σ
11795:σ
11772:Jackknife
11727:¯
11717:−
11686:∑
11667:¯
11646:−
11625:^
11613:¯
11604:σ
11561:∑
11549:¯
11502:¯
11493:−
11477:∑
11460:¯
11435:¯
11422:¯
11413:−
11381:¯
11372:−
11356:∑
11344:¯
11331:−
11309:¯
11296:¯
11287:−
11261:∑
11237:¯
11216:−
11195:^
11176:¯
11165:σ
11085:¯
11075:−
11027:∑
10983:∑
10948:π
10934:¯
10924:−
10899:π
10895:−
10863:∑
10848:^
10827:^
10811:¯
10771:≈
10759:π
10755:−
10679:Δ
10669:≠
10663:∀
10625:π
10611:¯
10601:−
10577:π
10563:¯
10553:−
10525:ˇ
10522:Δ
10493:∑
10472:∑
10457:^
10436:^
10420:¯
10394:^
10384:^
10146:Δ
10136:≠
10130:∀
10100:^
10085:^
10070:^
10056:The term
10030:^
10015:^
10000:^
9988:^
9976:−
9970:^
9960:^
9938:^
9923:^
9913:^
9878:^
9847:π
9829:^
9820:−
9796:π
9778:^
9769:−
9741:ˇ
9738:Δ
9709:∑
9688:∑
9673:^
9652:^
9642:^
9589:π
9567:−
9556:π
9515:∑
9495:≈
9449:∑
9403:∑
9389:^
9379:^
9363:^
9331:^
9321:^
9305:^
9186:π
9134:≈
9125:π
9081:¯
9071:−
9023:∑
8979:∑
8960:^
8944:¯
8833:∑
8787:∑
8768:¯
8707:∑
8664:∑
8645:¯
8609:¯
8553:∑
8507:∑
8454:∑
8408:∑
8383:ˇ
8356:∑
8338:ˇ
8311:∑
8290:π
8254:∑
8240:π
8197:∑
8184:^
8179:¯
8162:^
8106:∑
8063:∑
8041:ˇ
8014:∑
7999:ˇ
7972:∑
7951:π
7925:∑
7911:π
7878:∑
7865:¯
7794:ˇ
7767:∑
7752:π
7719:∑
7675:∑
7665:^
7631:π
7569:^
7500:^
7404:∑
7360:π
7336:π
7311:π
7307:−
7276:∑
7231:ˇ
7212:ˇ
7190:ˇ
7187:Δ
7158:∑
7113:ˇ
7094:ˇ
7072:ˇ
7069:Δ
7040:∑
7019:∑
6969:^
6956:
6941:and that
6926:≈
6914:π
6910:−
6823:∑
6777:^
6772:¯
6757:
6687:≠
6681:∀
6652:π
6648:−
6631:π
6620:π
6610:π
6603:−
6582:ˇ
6579:Δ
6544:π
6533:π
6523:π
6516:−
6495:ˇ
6492:Δ
6459:π
6429:Δ
6416:π
6406:π
6402:−
6390:π
6326:π
6298:ˇ
6255:ˇ
6236:ˇ
6214:ˇ
6211:Δ
6182:∑
6161:∑
6115:^
6110:¯
6095:
6043:¯
6030:−
5984:∑
5974:−
5939:¯
5926:−
5880:∑
5870:−
5777:∑
5767:−
5756:π
5714:∑
5704:−
5631:∑
5617:−
5557:∑
5547:−
5489:^
5479:−
5429:∑
5419:−
5378:^
5365:
5286:∑
5277:¯
5234:¯
5221:−
5175:∑
5165:−
5132:^
5119:
5049:∑
5034:π
4998:∑
4994:≈
4942:∑
4891:∑
4859:^
4807:π
4755:∑
4676:∑
4652:π
4616:∑
4609:≈
4556:∑
4516:^
4487:^
4482:¯
4430:¯
4417:−
4371:∑
4361:−
4311:^
4306:¯
4290:
4203:∑
4196:≈
4173:^
4144:^
4139:¯
4081:×
4069:≈
4058:π
4005:π
3965:ˇ
3930:ˇ
3894:ˇ
3805:π
3777:ˇ
3746:π
3714:π
3710:−
3695:π
3591:π
3405:design).
3350:π
3344:≈
3270:π
3252:∣
3185:Bernoulli
3160:^
2996:∑
2984:¯
2919:∑
2906:¯
2874:−
2853:¯
2844:−
2811:∑
2787:^
2784:σ
2746:¯
2735:¯
2705:^
2702:σ
2680:¯
2667:
2595:μ
2583:¯
2551:μ
2539:μ
2508:μ
2473:∑
2460:¯
2418:μ
2368:¯
2302:σ
2269:∑
2256:¯
2247:σ
2237:¯
2184:−
2175:σ
2154:∑
2131:σ
2122:−
2113:σ
2091:∑
2069:σ
2027:∑
2011:¯
2002:σ
1972:σ
1959:σ
1950:when all
1919:σ
1903:¯
1894:σ
1836:∑
1813:−
1804:σ
1783:∑
1765:¯
1756:σ
1692:∑
1671:⋅
1636:∑
1609:σ
1582:∑
1558:σ
1520:∑
1507:¯
1459:σ
1406:σ
1292:∑
1286:σ
1276:¯
1267:σ
1235:¯
1226:σ
1195:σ
1138:∑
1068:∑
969:∑
959:¯
892:∑
845:⋯
791:⋯
733:¯
668:∑
625:∑
612:¯
565:…
493:…
418:×
400:×
385:¯
217:×
199:×
181:¯
128:¯
20144:, 2011.
20128:37828617
19983:pdf link
19732:See also
18180:at time
16751:and the
15954:unbiased
15952:and the
15769:′
15429:unbiased
12405:variance
12319:′
12178:′
12098:′
11886:′
10286:′
10233:′
10216:between
9584:′
9551:′
9488:′
9442:′
8872:′
8826:′
8751:estimand
8592:′
8546:′
8493:′
8447:′
8394:′
8349:′
7805:′
6346:. Also,
5088:′
5029:′
4973:′
4922:′
4715:′
4647:′
4587:′
4242:′
4027:In this
3941:′
3849:values:
3632:′
3538:′
3479:′
3091:are not
2635:variance
2613:Variance
2503:′
2058:′
1740:and the
1396:variance
1322:′
1047:′
999:′
922:′
454:of data
268:relative
88:students
65:Examples
20216:, 1980.
20193:, 1988.
20120:5073694
20040:Gnu.org
20008:Bibcode
19965:Bibcode
19739:Average
19267:is the
18113:, with
18019:where:
17835:). The
17681:is the
16935:is not
16755:by the
16726:by the
15839:vector
15105:vector
14613:is the
7651:we get
4264:If the
3060:From a
1255:to be:
525:weights
94:
90:
82:
78:
76:classes
44:average
20255:
20238:
20212:
20189:
20165:
20126:
20118:
19922:
19868:
19235:where
18532:where
17547:0.9901
17540:0.9901
17492:0.9901
17475:0.9901
15427:, the
14992:actual
14947:where
14715:
14552:
14312:where
14286:actual
13938:actual
13639:actual
13319:
12993:
12801:where
12565:
12493:biased
12443:biased
12124:where
11535:where
6450:where
5259:where
2964:, and
1213:, the
20305:Means
20124:S2CID
20096:(PDF)
20064:(PDF)
19821:Notes
17763:is a
13532:(not
9288:Proof
8904:ratio
6893:Proof
6283:With
4459:Proof
3093:i.i.d
2772:With
2651:proof
2649:(see
2643:i.i.d
452:tuple
27:, or
20253:ISBN
20236:OCLC
20210:ISBN
20187:ISBN
20163:ISBN
20116:PMID
20075:2017
20047:2017
19920:ISBN
19866:ISBN
19749:Mean
18969:0.61
18880:0.39
18396:<
18390:<
17928:and
17586:and
15978:is:
15625:nor
14373:and
13586:nor
12929:for
12441:The
12407:and
11774:and
10269:and
2637:and
1744:is:
1116:The
139:4300
34:The
20108:doi
20016:doi
19973:doi
19001:is
17241:100
17139:100
12931:iid
9206:or
9116:If
6953:Var
6754:Var
6092:Var
5385:pwr
5362:Var
5116:Var
5106:pwr
4822:pwr
4287:Var
4112:pwr
3106:In
2664:Var
2653:):
595:is
430:86.
415:0.6
397:0.4
353:0.6
309:0.4
243:86.
150:86.
96:and
84:one
20301::
20278:.
20208:,
20122:.
20114:.
20104:35
20102:.
20098:.
20083:^
20066:.
20038:.
20014:.
20004:27
20002:.
19998:.
19981:-
19971:.
19961:29
19959:.
19918:.
19876:^
19853:^
19466:.
19271::
18773:.
18275:.
17712:,
17659:,
17230::=
17180::=
17111::=
17061::=
15749::
15707:1.
15649::
15098:.
15011:.
12960:.
12531::
12384:.
11962:.
11778:.
10053:.
8899:.
8888:.
6670:.
5344:.
5101:.
4261:.
4106:.
3735:.
2892:,
1217:,
421:90
403:80
344:30
338:20
334:30
300:30
294:20
290:20
274:.
234:30
228:20
220:90
214:30
202:80
196:20
142:50
61:.
23:,
20284:.
20261:.
20242:.
20171:.
20130:.
20110::
20077:.
20049:.
20024:.
20018::
20010::
19979:.
19975::
19967::
19928:.
19837:.
19716:.
19710:1
19704:n
19697:2
19693:)
19683:x
19672:i
19668:x
19664:(
19659:n
19654:1
19651:=
19648:i
19637:=
19632:2
19600:n
19596:/
19590:2
19582:=
19577:2
19567:x
19534:2
19524:x
19495:0
19487:=
19482:i
19443:x
19404:;
19397:2
19392:i
19381:2
19377:)
19367:x
19356:i
19352:x
19348:(
19340:n
19335:1
19332:=
19329:i
19318:)
19315:1
19309:n
19306:(
19302:1
19297:=
19292:2
19253:2
19218:2
19203:2
19193:x
19183:=
19178:2
19168:x
19127:2
19079:w
19059:n
19036:)
19033:w
19027:1
19024:(
19021:n
19014:e
18989:n
18966:=
18960:1
18953:e
18946:1
18923:1
18916:e
18895:)
18892:w
18886:1
18883:(
18877:=
18874:)
18871:w
18865:1
18862:(
18856:1
18849:e
18825:1
18818:)
18814:w
18808:1
18805:(
18784:w
18761:m
18741:)
18738:w
18732:1
18729:(
18725:/
18721:1
18718:=
18713:1
18709:V
18685:,
18679:w
18673:1
18666:m
18662:w
18655:1
18649:=
18643:1
18637:i
18633:w
18626:m
18621:1
18618:=
18615:i
18607:=
18602:1
18598:V
18572:1
18568:V
18545:1
18541:V
18517:,
18510:1
18506:V
18500:1
18494:i
18490:w
18484:=
18479:i
18475:w
18451:m
18425:1
18422:=
18419:w
18399:1
18387:0
18352:.
18347:i
18341:1
18338:+
18335:k
18331:x
18325:i
18321:w
18315:m
18310:1
18307:=
18304:i
18296:=
18291:k
18287:z
18263:m
18243:z
18220:i
18216:x
18193:i
18189:t
18168:y
18146:i
18142:t
18121:n
18101:y
18081:x
18053:.
18048:1
18040:C
18035:=
18031:W
18004:,
18001:)
17997:X
17992:W
17986:T
17981:J
17976:(
17971:2
17961:x
17951:=
17942:x
17913:,
17908:1
17901:)
17896:J
17891:W
17885:T
17880:J
17875:(
17872:=
17867:2
17857:x
17823:n
17801:T
17797:]
17793:1
17790:,
17784:,
17781:1
17778:[
17750:J
17723:x
17698:i
17694:x
17668:C
17645:T
17641:]
17635:n
17631:x
17627:,
17621:,
17616:1
17612:x
17608:[
17605:=
17601:X
17553:]
17534:[
17529:=
17524:]
17518:1
17511:1
17505:[
17498:]
17487:0
17480:0
17469:[
17464:=
17453:)
17447:2
17442:x
17435:1
17427:2
17422:C
17417:+
17412:1
17407:x
17400:1
17392:1
17387:C
17381:(
17375:1
17367:)
17361:1
17353:2
17348:C
17343:+
17338:1
17330:1
17325:C
17319:(
17314:=
17300:x
17264:]
17258:1
17253:0
17246:0
17235:[
17225:2
17220:C
17214:,
17203:]
17197:1
17192:0
17186:[
17175:2
17170:x
17145:]
17134:0
17127:0
17122:1
17116:[
17106:1
17101:C
17095:,
17084:]
17078:0
17073:1
17067:[
17056:1
17051:x
17024:,
17019:1
17011:)
17005:i
17000:W
16993:n
16988:1
16985:=
16982:i
16973:(
16968:=
16957:x
16949:C
16919:,
16915:)
16909:i
16904:x
16897:i
16892:W
16885:n
16880:1
16877:=
16874:i
16865:(
16853:x
16845:C
16840:=
16830:x
16804:.
16799:1
16791:i
16786:C
16781:=
16776:i
16771:W
16738:C
16712:2
16674:N
16670:/
16666:1
16663:=
16658:1
16654:V
16649:/
16643:i
16639:w
16615:.
16607:2
16603:V
16596:1
16590:)
16571:i
16566:x
16560:(
16554:T
16549:)
16530:i
16525:x
16519:(
16512:i
16508:w
16502:N
16497:1
16494:=
16491:i
16480:=
16476:C
16452:1
16449:=
16444:1
16440:V
16409:.
16403:)
16398:1
16394:V
16389:/
16383:2
16379:V
16375:(
16367:1
16363:V
16356:)
16337:i
16332:x
16326:(
16320:T
16315:)
16296:i
16291:x
16285:(
16278:i
16274:w
16268:N
16263:1
16260:=
16257:i
16246:=
16235:)
16216:i
16211:x
16205:(
16199:T
16194:)
16175:i
16170:x
16164:(
16157:i
16153:w
16147:N
16142:1
16139:=
16136:i
16123:2
16118:i
16114:w
16108:N
16103:1
16100:=
16097:i
16084:2
16079:)
16073:i
16069:w
16063:N
16058:1
16055:=
16052:i
16043:(
16034:i
16030:w
16024:N
16019:1
16016:=
16013:i
16002:=
15994:C
15965:C
15937:.
15932:i
15927:x
15920:i
15916:w
15910:N
15905:1
15902:=
15899:i
15891:=
15815:i
15811:w
15805:N
15800:1
15797:=
15794:i
15783:i
15779:w
15773:=
15765:i
15761:w
15735:1
15731:V
15704:=
15699:i
15695:w
15689:N
15684:1
15681:=
15678:i
15670:=
15665:1
15661:V
15606:.
15600:1
15592:1
15588:V
15581:)
15562:i
15557:x
15551:(
15545:T
15540:)
15521:i
15516:x
15510:(
15503:i
15499:w
15493:N
15488:1
15485:=
15482:i
15471:=
15467:C
15441:C
15400:.
15393:1
15389:V
15383:)
15364:i
15359:x
15353:(
15347:T
15342:)
15323:i
15318:x
15312:(
15305:i
15301:w
15295:N
15290:1
15287:=
15284:i
15273:=
15269:C
15242:.
15234:i
15230:w
15224:N
15219:1
15216:=
15213:i
15201:i
15196:x
15189:i
15185:w
15179:N
15174:1
15171:=
15168:i
15157:=
15086:0
15078:i
15074:w
15063:K
15047:i
15042:x
15016:N
14997:2
14984:=
14981:]
14976:2
14970:w
14965:s
14961:[
14955:E
14928:,
14922:)
14917:1
14913:V
14908:/
14902:2
14898:V
14894:(
14886:1
14882:V
14874:2
14870:)
14851:i
14847:x
14843:(
14838:i
14834:w
14828:N
14823:1
14820:=
14817:i
14806:=
14793:)
14788:2
14783:1
14779:V
14774:/
14768:2
14764:V
14760:(
14754:1
14748:2
14742:w
14722:=
14710:2
14704:w
14699:s
14667:)
14661:2
14656:1
14652:V
14647:/
14641:2
14637:V
14632:(
14625:1
14599:f
14596:f
14593:e
14589:N
14585:=
14580:2
14576:V
14571:/
14565:2
14560:1
14556:V
14531:)
14526:N
14522:1
14516:N
14510:(
14488:)
14480:2
14475:1
14471:V
14465:2
14461:V
14452:1
14448:(
14425:2
14420:i
14416:w
14410:N
14405:1
14402:=
14399:i
14391:=
14386:2
14382:V
14359:i
14355:w
14349:N
14344:1
14341:=
14338:i
14330:=
14325:1
14321:V
14291:2
14277:)
14269:2
14264:1
14260:V
14254:2
14250:V
14241:1
14237:(
14233:=
14223:]
14218:2
14214:)
14210:]
14207:X
14204:[
14198:E
14192:X
14189:(
14186:[
14180:E
14173:2
14168:1
14164:V
14158:2
14154:V
14145:]
14140:2
14136:)
14132:]
14129:X
14126:[
14120:E
14114:X
14111:(
14108:[
14102:E
14099:=
14085:1
14081:V
14076:]
14071:2
14067:)
14048:i
14044:x
14040:(
14037:[
14031:E
14026:i
14022:w
14016:N
14011:1
14008:=
14005:i
13994:=
13987:]
13982:2
13976:w
13958:[
13952:E
13943:2
13929:)
13924:N
13920:1
13914:N
13908:(
13904:=
13894:]
13889:2
13885:)
13881:]
13878:X
13875:[
13869:E
13863:X
13860:(
13857:[
13851:E
13846:N
13843:1
13835:]
13830:2
13826:)
13822:]
13819:X
13816:[
13810:E
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13801:(
13798:[
13792:E
13789:=
13777:N
13773:]
13768:2
13764:)
13752:i
13748:x
13744:(
13741:[
13735:E
13730:N
13725:1
13722:=
13719:i
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13701:]
13696:2
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13673:E
13644:2
13565:i
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13540:N
13518:i
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13503:1
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13497:i
13467:2
13462:)
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13439:x
13434:(
13427:i
13423:w
13417:N
13412:1
13409:=
13406:i
13395:1
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13377:N
13372:1
13369:=
13366:i
13354:i
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13339:1
13336:=
13333:i
13322:=
13314:2
13310:s
13286:}
13281:i
13277:w
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13250:}
13247:3
13244:,
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13232:{
13212:}
13209:5
13206:,
13203:4
13200:,
13197:2
13194:{
13174:}
13171:5
13168:,
13165:5
13162:,
13159:5
13156:,
13153:4
13150:,
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13141:2
13138:{
13109:1
13101:i
13097:w
13091:N
13086:1
13083:=
13080:i
13068:2
13063:)
13044:i
13040:x
13035:(
13028:i
13024:w
13018:N
13013:1
13010:=
13007:i
12996:=
12988:2
12984:s
12946:N
12942:N
12915:2
12888:2
12882:w
12843:1
12840:=
12835:i
12831:w
12825:N
12820:1
12817:=
12814:i
12777:i
12773:w
12767:N
12762:1
12759:=
12756:i
12744:2
12739:)
12720:i
12716:x
12711:(
12704:i
12700:w
12694:N
12689:1
12686:=
12683:i
12672:=
12663:2
12657:w
12633:N
12627:2
12622:)
12610:i
12606:x
12601:(
12594:N
12589:1
12586:=
12583:i
12572:=
12560:2
12517:2
12477:2
12471:w
12366:i
12362:w
12355:n
12350:1
12347:=
12344:i
12333:i
12329:w
12323:=
12315:i
12311:w
12281:n
12275:/
12269:0
12261:=
12251:x
12222:2
12217:0
12192:1
12182:2
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12169:w
12162:n
12157:1
12154:=
12151:i
12140:n
12136:/
12132:1
12109:,
12102:2
12093:i
12089:w
12082:n
12077:1
12074:=
12071:i
12061:2
12056:0
12048:=
12043:2
12033:x
12001:2
11996:0
11988:=
11983:2
11978:i
11939:x
11906:2
11901:i
11890:2
11881:i
11877:w
11870:n
11865:1
11862:=
11859:i
11851:=
11846:2
11836:x
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11799:i
11744:2
11740:)
11734:w
11724:x
11712:i
11708:x
11704:(
11699:2
11694:i
11690:w
11678:2
11674:)
11664:w
11658:n
11655:(
11652:)
11649:1
11643:n
11640:(
11636:n
11631:=
11620:2
11610:x
11575:n
11569:i
11565:w
11555:=
11546:w
11519:]
11513:2
11509:)
11499:w
11488:i
11484:w
11480:(
11472:2
11467:w
11457:x
11450:+
11447:)
11442:w
11432:x
11419:w
11408:i
11404:x
11398:i
11394:w
11390:(
11387:)
11378:w
11367:i
11363:w
11359:(
11351:w
11341:x
11334:2
11326:2
11322:)
11316:w
11306:x
11293:w
11282:i
11278:x
11272:i
11268:w
11264:(
11257:[
11248:2
11244:)
11234:w
11228:n
11225:(
11222:)
11219:1
11213:n
11210:(
11206:n
11201:=
11190:2
11183:w
11173:x
11107:.
11102:2
11098:)
11092:w
11082:y
11070:i
11066:y
11062:(
11057:2
11052:i
11048:w
11042:n
11037:1
11034:=
11031:i
11018:2
11014:)
11008:i
11004:w
10998:n
10993:1
10990:=
10987:i
10979:(
10975:1
10970:=
10965:2
10960:)
10952:i
10941:w
10931:y
10919:i
10915:y
10908:)
10903:i
10892:1
10889:(
10885:(
10878:n
10873:1
10870:=
10867:i
10855:2
10845:N
10838:1
10833:=
10823:)
10818:w
10808:y
10801:(
10798:V
10774:1
10768:)
10763:i
10752:1
10749:(
10729:0
10726:=
10723:)
10718:j
10714:I
10710:,
10705:i
10701:I
10697:(
10694:C
10691:=
10686:j
10683:i
10675::
10672:j
10666:i
10641:.
10637:)
10629:j
10618:w
10608:y
10596:j
10592:y
10581:i
10570:w
10560:y
10548:i
10544:y
10535:j
10532:i
10514:(
10508:n
10503:1
10500:=
10497:j
10487:n
10482:1
10479:=
10476:i
10464:2
10454:N
10447:1
10442:=
10432:)
10427:w
10417:y
10410:(
10407:V
10400:=
10390:)
10381:R
10375:(
10372:V
10346:1
10343:=
10338:i
10334:z
10308:i
10304:z
10298:i
10294:I
10290:=
10282:i
10278:z
10255:i
10251:y
10245:i
10241:I
10237:=
10229:i
10225:y
10214:i
10210:n
10196:0
10193:=
10190:)
10185:j
10181:I
10177:,
10172:i
10168:I
10164:(
10161:C
10158:=
10153:j
10150:i
10142::
10139:j
10133:i
10106:)
10097:Z
10091:,
10082:Y
10076:(
10067:C
10040:]
10036:)
10027:Z
10021:,
10012:Y
10006:(
9997:C
9985:R
9979:2
9966:)
9957:Z
9951:(
9948:V
9935:R
9929:+
9919:)
9910:Y
9904:(
9901:V
9893:[
9885:2
9875:Z
9868:1
9863:=
9859:)
9851:j
9840:j
9836:z
9826:R
9815:j
9811:y
9800:i
9789:i
9785:z
9775:R
9764:i
9760:y
9751:j
9748:i
9730:(
9724:n
9719:1
9716:=
9713:j
9703:n
9698:1
9695:=
9692:i
9680:2
9670:Z
9663:1
9658:=
9648:)
9639:R
9633:(
9630:V
9601:)
9593:i
9580:i
9576:z
9570:R
9560:i
9547:i
9543:y
9536:(
9530:n
9525:1
9522:=
9519:i
9509:Z
9506:1
9501:+
9498:R
9484:i
9480:z
9474:i
9470:w
9464:n
9459:1
9456:=
9453:i
9438:i
9434:y
9428:i
9424:w
9418:n
9413:1
9410:=
9407:i
9396:=
9386:Z
9376:Y
9369:=
9360:R
9328:Z
9318:Y
9311:=
9302:R
9282:N
9266:i
9262:w
9237:i
9233:p
9229:1
9224:=
9219:i
9215:w
9190:i
9182:1
9177:=
9172:i
9168:w
9147:n
9142:i
9138:p
9129:i
9112:.
9098:2
9094:)
9088:w
9078:y
9066:i
9062:y
9058:(
9053:2
9048:i
9044:w
9038:n
9033:1
9030:=
9027:i
9014:2
9010:)
9004:i
9000:w
8994:n
8989:1
8986:=
8983:i
8975:(
8971:1
8966:=
8956:)
8951:w
8941:y
8934:(
8931:V
8897:R
8868:i
8864:1
8858:i
8854:w
8848:n
8843:1
8840:=
8837:i
8822:i
8818:y
8812:i
8808:w
8802:n
8797:1
8794:=
8791:i
8780:=
8775:w
8765:y
8732:i
8728:w
8722:n
8717:1
8714:=
8711:i
8699:i
8695:y
8689:i
8685:w
8679:n
8674:1
8671:=
8668:i
8657:=
8652:w
8642:y
8616:w
8606:y
8599:=
8588:i
8584:1
8578:i
8574:w
8568:n
8563:1
8560:=
8557:i
8542:i
8538:y
8532:i
8528:w
8522:n
8517:1
8514:=
8511:i
8500:=
8489:i
8485:1
8479:i
8475:w
8469:N
8464:1
8461:=
8458:i
8443:i
8439:y
8433:i
8429:w
8423:N
8418:1
8415:=
8412:i
8401:=
8390:i
8380:1
8371:N
8366:1
8363:=
8360:i
8345:i
8335:y
8326:N
8321:1
8318:=
8315:i
8304:=
8294:i
8286:1
8279:i
8275:I
8269:N
8264:1
8261:=
8258:i
8244:i
8234:i
8230:y
8222:i
8218:I
8212:N
8207:1
8204:=
8201:i
8190:=
8176:Y
8168:=
8159:R
8131:i
8127:w
8121:N
8116:1
8113:=
8110:i
8098:i
8094:y
8088:i
8084:w
8078:N
8073:1
8070:=
8067:i
8056:=
8048:i
8038:1
8029:N
8024:1
8021:=
8018:i
8006:i
7996:y
7987:N
7982:1
7979:=
7976:i
7965:=
7955:i
7947:1
7940:N
7935:1
7932:=
7929:i
7915:i
7905:i
7901:y
7893:N
7888:1
7885:=
7882:i
7871:=
7862:Y
7856:=
7853:R
7831:i
7827:y
7801:i
7791:1
7782:n
7777:1
7774:=
7771:i
7763:=
7756:i
7746:i
7742:I
7734:n
7729:1
7726:=
7723:i
7715:=
7710:i
7706:I
7700:i
7696:w
7690:n
7685:1
7682:=
7679:i
7671:=
7662:N
7635:i
7627:1
7622:=
7617:i
7613:w
7592:N
7566:N
7543:N
7523:N
7497:Y
7480:π
7456:2
7451:)
7445:i
7441:y
7435:i
7431:w
7426:(
7419:n
7414:1
7411:=
7408:i
7396:2
7392:N
7388:1
7383:=
7372:)
7364:i
7354:i
7350:y
7340:i
7330:i
7326:y
7320:)
7315:i
7304:1
7301:(
7297:(
7291:n
7286:1
7283:=
7280:i
7268:2
7264:N
7260:1
7255:=
7244:)
7238:i
7228:y
7219:i
7209:y
7200:i
7197:i
7179:(
7173:n
7168:1
7165:=
7162:i
7150:2
7146:N
7142:1
7137:=
7126:)
7120:j
7110:y
7101:i
7091:y
7082:j
7079:i
7061:(
7055:n
7050:1
7047:=
7044:j
7034:n
7029:1
7026:=
7023:i
7011:2
7007:N
7003:1
6998:=
6991:)
6985:)
6981:N
6966:Y
6959:(
6929:1
6923:)
6918:i
6907:1
6904:(
6875:2
6870:)
6864:i
6860:y
6854:i
6850:w
6845:(
6838:n
6833:1
6830:=
6827:i
6815:2
6811:N
6807:1
6802:=
6799:)
6793:)
6789:N
6769:Y
6760:(
6731:0
6728:=
6725:)
6720:j
6716:I
6712:,
6707:i
6703:I
6699:(
6696:C
6693::
6690:j
6684:i
6656:i
6645:1
6642:=
6635:i
6624:i
6614:i
6600:1
6597:=
6592:i
6589:i
6551:j
6548:i
6537:j
6527:i
6513:1
6510:=
6505:j
6502:i
6466:j
6463:i
6436:j
6433:i
6425:=
6420:j
6410:i
6397:j
6394:i
6386:=
6383:)
6378:j
6374:I
6370:,
6365:i
6361:I
6357:(
6354:C
6330:i
6320:i
6316:y
6310:=
6305:i
6295:y
6268:)
6262:j
6252:y
6243:i
6233:y
6224:j
6221:i
6203:(
6197:n
6192:1
6189:=
6186:j
6176:n
6171:1
6168:=
6165:i
6153:2
6149:N
6145:1
6140:=
6137:)
6131:)
6127:N
6107:Y
6098:(
6054:2
6049:)
6039:y
6036:w
6025:i
6021:y
6015:i
6011:w
6006:(
5999:n
5994:1
5991:=
5988:i
5977:1
5971:n
5967:n
5962:=
5950:2
5945:)
5935:y
5932:w
5921:i
5917:y
5911:i
5907:w
5902:(
5895:n
5890:1
5887:=
5884:i
5873:1
5867:n
5863:1
5856:n
5851:2
5847:n
5841:=
5829:2
5824:)
5818:n
5812:i
5808:y
5802:i
5798:w
5792:n
5787:1
5784:=
5781:i
5770:n
5760:i
5750:i
5746:y
5740:n
5736:(
5729:n
5724:1
5721:=
5718:i
5707:1
5701:n
5697:1
5690:n
5687:1
5682:=
5677:2
5672:)
5666:i
5662:y
5656:i
5652:w
5646:n
5641:1
5638:=
5635:i
5625:n
5622:n
5610:i
5606:p
5600:i
5596:y
5588:n
5585:n
5579:(
5572:n
5567:1
5564:=
5561:i
5550:1
5544:n
5540:1
5533:n
5530:1
5525:=
5513:2
5508:)
5502:r
5499:w
5496:p
5486:Y
5472:i
5468:p
5462:i
5458:y
5451:(
5444:n
5439:1
5436:=
5433:i
5422:1
5416:n
5412:1
5405:n
5402:1
5397:=
5390:)
5375:Y
5368:(
5330:n
5324:i
5320:y
5314:i
5310:w
5301:n
5296:1
5293:=
5290:i
5282:=
5273:y
5270:w
5245:2
5240:)
5230:y
5227:w
5216:i
5212:y
5206:i
5202:w
5197:(
5190:n
5185:1
5182:=
5179:i
5168:1
5162:n
5158:n
5153:=
5150:)
5145:r
5142:w
5139:p
5129:Y
5122:(
5084:i
5080:y
5074:i
5070:w
5064:n
5059:1
5056:=
5053:i
5045:=
5038:i
5025:i
5021:y
5013:n
5008:1
5005:=
5002:i
4986:i
4982:p
4978:n
4969:i
4965:y
4957:n
4952:1
4949:=
4946:i
4938:=
4931:i
4927:p
4918:i
4914:y
4906:n
4901:1
4898:=
4895:i
4885:n
4882:1
4877:=
4872:r
4869:w
4866:p
4856:Y
4832:p
4780:i
4776:y
4770:N
4765:1
4762:=
4759:i
4751:=
4748:Y
4725:.
4720:N
4711:i
4707:y
4701:i
4697:w
4691:n
4686:1
4683:=
4680:i
4669:=
4664:N
4656:i
4643:i
4639:y
4631:n
4626:1
4623:=
4620:i
4604:N
4596:i
4592:p
4583:i
4579:y
4571:n
4566:1
4563:=
4560:i
4550:n
4547:1
4539:=
4534:N
4529:r
4526:w
4523:p
4513:Y
4504:=
4499:N
4479:Y
4441:2
4436:)
4426:y
4423:w
4412:i
4408:y
4402:i
4398:w
4393:(
4386:n
4381:1
4378:=
4375:i
4364:1
4358:n
4354:n
4345:2
4341:N
4337:1
4332:=
4328:)
4323:N
4303:Y
4294:(
4270:n
4247:N
4238:i
4234:y
4228:i
4224:w
4218:n
4213:1
4210:=
4207:i
4191:N
4186:r
4183:w
4180:p
4170:Y
4161:=
4156:N
4136:Y
4120:N
4089:i
4085:p
4078:n
4074:1
4062:i
4054:1
4049:=
4044:i
4040:w
4009:i
3998:i
3994:y
3988:i
3984:I
3977:=
3972:i
3962:y
3953:i
3949:I
3945:=
3937:i
3927:y
3901:i
3891:y
3884:n
3881:=
3874:i
3870:p
3864:i
3860:y
3847:y
3833:p
3809:i
3799:i
3795:y
3789:=
3784:i
3774:y
3760:y
3723:)
3718:i
3707:1
3704:(
3699:i
3689:2
3684:i
3680:y
3676:=
3673:]
3668:i
3664:I
3660:[
3657:V
3652:2
3647:i
3643:y
3639:=
3636:]
3628:i
3624:y
3620:[
3617:V
3595:i
3585:i
3581:y
3577:=
3574:]
3569:i
3565:I
3561:[
3558:E
3553:i
3549:y
3545:=
3542:]
3534:i
3530:y
3526:[
3523:E
3501:i
3497:I
3491:i
3487:y
3483:=
3475:i
3471:y
3448:i
3444:I
3421:i
3417:y
3386:i
3382:p
3359:n
3354:i
3339:i
3335:p
3331:=
3328:)
3319:|
3315:1
3312:=
3307:i
3303:I
3299:(
3296:P
3274:i
3266:=
3263:)
3260:n
3249:1
3246:=
3241:i
3237:I
3233:(
3230:P
3216:i
3200:i
3196:I
3177:y
3157:N
3134:N
3124:T
3120:Y
3116:y
3112:y
3077:i
3073:y
3050:y
3032:n
3026:2
3021:i
3017:w
3011:n
3006:1
3003:=
3000:i
2989:=
2979:2
2975:w
2950:n
2944:i
2940:w
2934:n
2929:1
2926:=
2923:i
2912:=
2903:w
2877:1
2871:n
2864:2
2860:)
2850:y
2839:i
2835:y
2831:(
2826:n
2821:1
2818:=
2815:i
2804:=
2799:2
2794:y
2753:2
2743:w
2730:2
2726:w
2717:2
2712:y
2695:=
2692:)
2687:w
2677:y
2670:(
2624:n
2598:.
2592:=
2589:)
2580:x
2574:(
2571:E
2548:=
2543:i
2518:.
2512:i
2499:i
2495:w
2488:n
2483:1
2480:=
2477:i
2469:=
2466:)
2457:x
2451:(
2448:E
2428:,
2422:i
2413:=
2410:)
2405:i
2401:x
2397:(
2394:E
2365:x
2318:.
2311:2
2306:i
2296:i
2292:x
2284:n
2279:1
2276:=
2273:i
2263:2
2253:x
2243:=
2234:x
2205:.
2198:2
2193:)
2187:2
2179:i
2169:n
2164:1
2161:=
2158:i
2149:(
2140:2
2135:i
2125:4
2117:i
2106:n
2101:1
2098:=
2095:i
2084:=
2078:2
2073:i
2062:2
2053:i
2049:w
2042:n
2037:1
2034:=
2031:i
2023:=
2018:2
2008:x
1976:0
1968:=
1963:i
1938:n
1934:/
1928:2
1923:0
1915:=
1910:2
1900:x
1870:,
1861:i
1857:w
1851:n
1846:1
1843:=
1840:i
1831:1
1825:=
1816:2
1808:i
1798:n
1793:1
1790:=
1787:i
1778:1
1772:=
1762:x
1725:,
1717:i
1713:w
1707:n
1702:1
1699:=
1696:i
1685:)
1679:i
1675:w
1666:i
1662:x
1657:(
1651:n
1646:1
1643:=
1640:i
1629:=
1618:2
1613:i
1605:1
1597:n
1592:1
1589:=
1586:i
1575:)
1567:2
1562:i
1552:i
1548:x
1541:(
1535:n
1530:1
1527:=
1524:i
1513:=
1504:x
1475:.
1468:2
1463:i
1455:1
1450:=
1445:i
1441:w
1415:2
1410:i
1376:i
1372:x
1326:2
1317:i
1313:w
1307:n
1302:1
1299:=
1296:i
1283:=
1273:x
1232:x
1199:2
1164:i
1160:x
1153:n
1148:1
1145:=
1142:i
1132:n
1129:1
1112:.
1094:j
1090:w
1083:n
1078:1
1075:=
1072:j
1061:i
1057:w
1051:=
1043:i
1039:w
1023:.
1008:i
1004:x
995:i
991:w
984:n
979:1
976:=
973:i
965:=
956:x
930:1
927:=
918:i
914:w
907:n
902:1
899:=
896:i
864:.
856:n
852:w
848:+
842:+
837:2
833:w
829:+
824:1
820:w
812:n
808:x
802:n
798:w
794:+
788:+
783:2
779:x
773:2
769:w
765:+
760:1
756:x
750:1
746:w
739:=
730:x
701:,
693:i
689:w
683:n
678:1
675:=
672:i
660:i
656:x
650:i
646:w
640:n
635:1
632:=
629:i
618:=
609:x
582:)
576:n
572:w
568:,
562:,
557:2
553:w
549:,
544:1
540:w
535:(
510:)
504:n
500:x
496:,
490:,
485:2
481:x
477:,
472:1
468:x
463:(
427:=
424:)
412:(
409:+
406:)
394:(
391:=
382:x
350:=
341:+
306:=
297:+
240:=
231:+
223:)
211:(
208:+
205:)
193:(
187:=
178:x
147:=
134:=
125:x
92:—
80:—
31:.
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