Standard deviation

5636:(pp) and Stock B, over the same period, had average returns of 12 percent but a higher standard deviation of 30 pp. On the basis of risk and return, an investor may decide that Stock A is the safer choice, because Stock B's additional two percentage points of return is not worth the additional 10 pp standard deviation (greater risk or uncertainty of the expected return). Stock B is likely to fall short of the initial investment (but also to exceed the initial investment) more often than Stock A under the same circumstances, and is estimated to return only two percent more on average. In this example, Stock A is expected to earn about 10 percent, plus or minus 20 pp (a range of 30 percent to −10 percent), about two-thirds of the future year returns. When considering more extreme possible returns or outcomes in future, an investor should expect results of as much as 10 percent plus or minus 60 pp, or a range from 70 percent to −50 percent, which includes outcomes for three standard deviations from the average return (about 99.7 percent of probable returns). 56: 7454: 7442: 14423: 11270: 5580:

be found, which will always be slightly different from the long-term average. By using standard deviations, a minimum and maximum value can be calculated that the averaged weight will be within some very high percentage of the time (99.9% or more). If it falls outside the range then the production process may need to be corrected. Statistical tests such as these are particularly important when the testing is relatively expensive. For example, if the product needs to be opened and drained and weighed, or if the product was otherwise used up by the test.

5525: 9202: 5629:). The fundamental concept of risk is that as it increases, the expected return on an investment should increase as well, an increase known as the risk premium. In other words, investors should expect a higher return on an investment when that investment carries a higher level of risk or uncertainty. When evaluating investments, investors should estimate both the expected return and the uncertainty of future returns. Standard deviation provides a quantified estimate of the uncertainty of future returns. 6930: 40: 14409: 8898: 6458: 11245: 10915: 5499: 14447: 14435: 5536:

deviations have the same units as the data points themselves. If, for instance, the data set {0, 6, 8, 14} represents the ages of a population of four siblings in years, the standard deviation is 5 years. As another example, the population {1000, 1006, 1008, 1014} may represent the distances traveled by four athletes, measured in meters. It has a mean of 1007 meters, and a standard deviation of 5 meters.

6063: 835: 9197:{\displaystyle {\begin{aligned}\operatorname {var} ({\text{mean}})&=\operatorname {var} \left({\frac {1}{N}}\sum _{i=1}^{N}X_{i}\right)={\frac {1}{N^{2}}}\operatorname {var} \left(\sum _{i=1}^{N}X_{i}\right)\\&={\frac {1}{N^{2}}}\sum _{i=1}^{N}\operatorname {var} (X_{i})={\frac {N}{N^{2}}}\operatorname {var} (X)={\frac {1}{N}}\operatorname {var} (X).\end{aligned}}} 10516: 5198: 8811: 427: 10018: 5612:

same average maximum temperature, the standard deviation of the daily maximum temperature for the coastal city will be less than that of the inland city as, on any particular day, the actual maximum temperature is more likely to be farther from the average maximum temperature for the inland city than for the coastal one.

6453:{\displaystyle {\begin{aligned}L\cdot (P-M)&=0\\(r,r,r)\cdot (x_{1}-\ell ,x_{2}-\ell ,x_{3}-\ell )&=0\\r(x_{1}-\ell +x_{2}-\ell +x_{3}-\ell )&=0\\r\left(\sum _{i}x_{i}-3\ell \right)&=0\\\sum _{i}x_{i}-3\ell &=0\\{\frac {1}{3}}\sum _{i}x_{i}&=\ell \\{\bar {x}}&=\ell \end{aligned}}} 5624:

associated with price-fluctuations of a given asset (stocks, bonds, property, etc.), or the risk of a portfolio of assets (actively managed mutual funds, index mutual funds, or ETFs). Risk is an important factor in determining how to efficiently manage a portfolio of investments because it determines

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number of repeated samples from the population and computing a mean for each sample. The mean's standard error turns out to equal the population standard deviation divided by the square root of the sample size, and is estimated by using the sample standard deviation divided by the square root of the

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An observation is rarely more than a few standard deviations away from the mean. Chebyshev's inequality ensures that, for all distributions for which the standard deviation is defined, the amount of data within a number of standard deviations of the mean is at least as much as given in the following

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Standard deviation is often used to compare real-world data against a model to test the model. For example, in industrial applications the weight of products coming off a production line may need to comply with a legally required value. By weighing some fraction of the products an average weight can

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has 9 degrees of freedom for estimating the standard deviation. The same computations as above give us in this case a 95% CI running from 0.69 × SD to 1.83 × SD. So even with a sample population of 10, the actual SD can still be almost a factor 2 higher than the sampled SD. For a

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Dark blue is one standard deviation on either side of the mean. For the normal distribution, this accounts for 68.27 percent of the set; while two standard deviations from the mean (medium and dark blue) account for 95.45 percent; three standard deviations (light, medium, and dark blue) account for

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This formula is valid only if the eight values with which we began form the complete population. If the values instead were a random sample drawn from some large parent population (for example, they were 8 students randomly and independently chosen from a class of 2 million), then one divides

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of the poll), is the expected standard deviation of the estimated mean if the same poll were to be conducted multiple times. Thus, the standard error estimates the standard deviation of an estimate, which itself measures how much the estimate depends on the particular sample that was taken from the

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beforehand. However, in most applications this parameter is unknown. For example, if a series of 10 measurements of a previously unknown quantity is performed in a laboratory, it is possible to calculate the resulting sample mean and sample standard deviation, but it is impossible to calculate the

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Financial time series are known to be non-stationary series, whereas the statistical calculations above, such as standard deviation, apply only to stationary series. To apply the above statistical tools to non-stationary series, the series first must be transformed to a stationary series, enabling

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As a simple example, consider the average daily maximum temperatures for two cities, one inland and one on the coast. It is helpful to understand that the range of daily maximum temperatures for cities near the coast is smaller than for cities inland. Thus, while these two cities may each have the

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Calculating the average (or arithmetic mean) of the return of a security over a given period will generate the expected return of the asset. For each period, subtracting the expected return from the actual return results in the difference from the mean. Squaring the difference in each period and

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Often, we want some information about the precision of the mean we obtained. We can obtain this by determining the standard deviation of the sampled mean. Assuming statistical independence of the values in the sample, the standard deviation of the mean is related to the standard deviation of the

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of those measurements. When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements is of crucial importance: if the mean of the measurements is too far away from the prediction (with the distance measured in standard deviations), then the

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For example, each of the three populations {0, 0, 14, 14}, {0, 6, 8, 14} and {6, 6, 8, 8} has a mean of 7. Their standard deviations are 7, 5, and 1, respectively. The third population has a much smaller standard deviation than the other two because its values are all close to 7. These standard

2393: 4405: 7080: 8661: 10910:{\displaystyle {\begin{aligned}A_{0}&=0\\A_{k}&=A_{k-1}+{\frac {w_{k}}{W_{k}}}\left(x_{k}-A_{k-1}\right)\\Q_{0}&=0\\Q_{k}&=Q_{k-1}+{\frac {w_{k}W_{k-1}}{W_{k}}}\left(x_{k}-A_{k-1}\right)^{2}=Q_{k-1}+w_{k}\left(x_{k}-A_{k-1}\right)\left(x_{k}-A_{k}\right)\end{aligned}}} 3691: 3452:

is still a biased estimator for the population standard deviation, though markedly less biased than the uncorrected sample standard deviation. This estimator is commonly used and generally known simply as the "sample standard deviation". The bias may still be large for small samples

185:, it is common to report both the standard deviation of the data (as a summary statistic) and the standard error of the estimate (as a measure of potential error in the findings). By convention, only effects more than two standard errors away from a null expectation are considered 5494:{\displaystyle {\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}}={\sqrt {{\frac {1}{N}}\left(\sum _{i=1}^{N}x_{i}^{2}\right)-{\bar {x}}^{2}}}={\sqrt {\left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}^{2}\right)-\left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)^{2}}},} 11235: 4091:

The standard deviation we obtain by sampling a distribution is itself not absolutely accurate, both for mathematical reasons (explained here by the confidence interval) and for practical reasons of measurement (measurement error). The mathematical effect can be described by the

4052: 2025: 1471: 5078: 4226: 9780: 4734: 4849: 2228: 1194:) – two standard deviations. If the standard deviation were zero, then all men would share an identical height of 69 inches. Three standard deviations account for 99.73% of the sample population being studied, assuming the distribution is 1150:. Roughly, the reason for it is that the formula for the sample variance relies on computing differences of observations from the sample mean, and the sample mean itself was constructed to be as close as possible to the observations, so just dividing by 1867: 3834: 5640:

taking the average gives the overall variance of the return of the asset. The larger the variance, the greater risk the security carries. Finding the square root of this variance will give the standard deviation of the investment tool in question.

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standard deviation (of a finite population) can be applied to the sample, using the size of the sample as the size of the population (though the actual population size from which the sample is drawn may be much larger). This estimator, denoted by

830:{\displaystyle {\begin{array}{lll}(2-5)^{2}=(-3)^{2}=9&&(5-5)^{2}=0^{2}=0\\(4-5)^{2}=(-1)^{2}=1&&(5-5)^{2}=0^{2}=0\\(4-5)^{2}=(-1)^{2}=1&&(7-5)^{2}=2^{2}=4\\(4-5)^{2}=(-1)^{2}=1&&(9-5)^{2}=4^{2}=16.\\\end{array}}} 5548:

theory being tested probably needs to be revised. This makes sense since they fall outside the range of values that could reasonably be expected to occur if the prediction were correct and the standard deviation appropriately quantified. See

5591:" for the declaration of a discovery. A five-sigma level translates to one chance in 3.5 million that a random fluctuation would yield the result. This level of certainty was required in order to assert that a particle consistent with the 3014: 2729: 2261: 8512: 3440: 1324: 7168: 5189: 4277: 6068: 6955: 4509:

data taken to be approximately normal follows from the heuristic that 95% of the area under the normal curve lies roughly two standard deviations to either side of the mean, so that, with 95% probability the total range of values

3165: 8891: 3571: 10460: 7362: 11193: 3914: 1885: 2511: 1337: 944: 4932: 3264: 4115: 1550: 11132: 4594: 416: 9249: 8646: 6558: 2583: − 1), defined below, and this is often referred to as the "sample standard deviation", without qualifiers. However, other estimators are better in other respects: the uncorrected estimator (using 5924: 5839: 4888: 9594: 9489: 4747: 2088: 8806:{\displaystyle {\begin{aligned}\operatorname {var} (X)&\equiv \sigma _{X}^{2}\\\operatorname {var} (X_{1}+X_{2})&\equiv \operatorname {var} (X_{1})+\operatorname {var} (X_{2})\\\end{aligned}}} 10977: 10336: 1664: 6896: 3719: 1162:

If the population of interest is approximately normally distributed, the standard deviation provides information on the proportion of observations above or below certain values. For example, the

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if the center of the data is measured about the mean. This is because the standard deviation from the mean is smaller than from any other point. The precise statement is the following: suppose

9396: 9651: 91:) of the set, while a high standard deviation indicates that the values are spread out over a wider range. The standard deviation is commonly used in the determination of what constitutes an 7192: 4910: 10116: 314: 168:

of a statistic (e.g., of the sample mean) are quite different, but related. The sample mean's standard error is the standard deviation of the set of means that would be found by drawing an

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standard deviation of the mean. However, one can estimate the standard deviation of the entire population from the sample, and thus obtain an estimate for the standard error of the mean.

10013:{\displaystyle {\begin{aligned}Q_{0}&=0\\Q_{k}&=Q_{k-1}+{\frac {k-1}{k}}\left(x_{k}-A_{k-1}\right)^{2}=Q_{k-1}+\left(x_{k}-A_{k-1}\right)\left(x_{k}-A_{k}\right)\end{aligned}}} 1049: 2909: 2624: 986: 6041: 6013: 9625:

samples without the need to store prior data during the calculation. Applying this method to a time series will result in successive values of standard deviation corresponding to

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A large standard deviation indicates that the data points can spread far from the mean and a small standard deviation indicates that they are clustered closely around the mean.

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Suppose that the entire population of interest is eight students in a particular class. For a finite set of numbers, the population standard deviation is found by taking the

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Example of samples from two populations with the same mean but different standard deviations. Red population has mean 100 and SD 10; blue population has mean 100 and SD 50.

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which means that the standard deviation is equal to the square root of the difference between the average of the squares of the values and the square of the average value.

8818: 4454:, this is down to 0.88 × SD to 1.16 × SD. To be more certain that the sampled SD is close to the actual SD we need to sample a large number of points. 10511: 10370: 7106:

is the number of random variables. The standard deviation therefore is simply a scaling variable that adjusts how broad the curve will be, though it also appears in the

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states that the distribution of an average of many independent, identically distributed random variables tends toward the famous bell-shaped normal distribution with a

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the bias is below 1%. Thus for very large sample sizes, the uncorrected sample standard deviation is generally acceptable. This estimator also has a uniformly smaller

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For example, assume an investor had to choose between two stocks. Stock A over the past 20 years had an average return of 10 percent, with a standard deviation of 20

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of the sample, which is a downward-biased estimate of the population variance) is used to compute an estimate of the population's standard deviation, the result is

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has only one degree of freedom for estimating the standard deviation. The result is that a 95% CI of the SD runs from 0.45 × SD to 31.9 × SD;

846: 11586:, National Center for Health Statistics: Vital and Health Statistics, vol. 3, Centers for Disease Control and Prevention, January 2021, p. 16, Table 12 5969: 5949: 6934:

99.73 percent; and four standard deviations account for 99.994 percent. The two points of the curve that are one standard deviation from the mean are also the

1122: 3026:. The bias in the variance is easily corrected, but the bias from the square root is more difficult to correct, and depends on the distribution in question. 1142: 1073: 325: 9209: 8606: 3535:

is used as a basis, and is scaled by a correction factor to produce an unbiased estimate. For the normal distribution, an unbiased estimator is given by

2388:{\displaystyle \sigma ={\sqrt {\int _{\mathbf {X} }(x-\mu )^{2}\,p(x)\,\mathrm {d} x}},{\text{ where }}\mu =\int _{\mathbf {X} }x\,p(x)\,\mathrm {d} x,} 7366:

If a data distribution is approximately normal then about 68 percent of the data values are within one standard deviation of the mean (mathematically,

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The practical value of understanding the standard deviation of a set of values is in appreciating how much variation there is from the average (mean).

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For other distributions, the correct formula depends on the distribution, but a rule of thumb is to use the further refinement of the approximation:

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Standard deviation may serve as a measure of uncertainty. In physical science, for example, the reported standard deviation of a group of repeated

9514: 9431: 4400:{\displaystyle \Pr \left(k{\frac {s^{2}}{q_{1-{\frac {\alpha }{2}}}}}<\sigma ^{2}<k{\frac {s^{2}}{q_{\frac {\alpha }{2}}}}\right)=1-\alpha .} 11056: 7075:{\displaystyle f\left(x,\mu ,\sigma ^{2}\right)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}} 5736:. This is the "main diagonal" going through the origin. If our three given values were all equal, then the standard deviation would be zero and 3180: 13544: 9621:. The method below calculates the running sums method with reduced rounding errors. This is a "one pass" algorithm for calculating variance of 10922: 3686:{\displaystyle c_{4}(N)\,=\,{\sqrt {\frac {2}{N-1}}}\,\,\,{\frac {\Gamma \left({\frac {N}{2}}\right)}{\Gamma \left({\frac {N-1}{2}}\right)}}.} 14476: 14049: 1476: 11254: 10261: 5555:

While the standard deviation does measure how far typical values tend to be from the mean, other measures are available. An example is the

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can refer to either the above-mentioned quantity as applied to those data, or to a modified quantity that is an unbiased estimate of the

11230:{\displaystyle {\text{SDI}}={\frac {{\text{Laboratory mean}}-{\text{Consensus group mean}}}{\text{Consensus group standard deviation}}}} 10123: 4047:{\displaystyle {\hat {\sigma }}={\sqrt {{\frac {1}{N-1.5-{\frac {1}{4}}\gamma _{2}}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}},} 2020:{\displaystyle \sigma ={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}(x_{i}-\mu )^{2}}},{\text{ where }}\mu ={\frac {1}{N}}\sum _{i=1}^{N}x_{i}.} 13823: 12464: 9330: 11166:

in 1894, following his use of it in lectures. This was as a replacement for earlier alternative names for the same idea: for example,

6490: 5854: 4856: 3528: 2570: 2529: 1466:{\displaystyle \sigma \equiv {\sqrt {\operatorname {E} \left}}={\sqrt {\int _{-\infty }^{+\infty }(x-\mu )^{2}f(x)\,\mathrm {d} x}},} 10061: 5073:{\displaystyle \sigma (X)={\sqrt {\operatorname {E} \left)^{2}\right]}}={\sqrt {\operatorname {E} \left-(\operatorname {E} )^{2}}}.} 4086: 236: 13597: 5600: 2550:, and the estimator (or the value of the estimator, namely the estimate) is called a sample standard deviation, and is denoted by 14036: 11804:

LIGO Scientific Collaboration, Virgo Collaboration (2016), "Observation of Gravitational Waves from a Binary Black Hole Merger",

4221:{\displaystyle \Pr \left(q_{\frac {\alpha }{2}}<k{\frac {s^{2}}{\sigma ^{2}}}<q_{1-{\frac {\alpha }{2}}}\right)=1-\alpha ,} 4099:

To show how a larger sample will make the confidence interval narrower, consider the following examples: A small population of

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of the squared deviations of the values subtracted from their average value. The marks of a class of eight students (that is, a

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the variation in returns on the asset or portfolio and gives investors a mathematical basis for investment decisions (known as

4729:{\displaystyle {\begin{aligned}\sigma (c)&=0\\\sigma (X+c)&=\sigma (X),\\\sigma (cX)&=|c|\sigma (X).\end{aligned}}} 1576:

has tails going out to infinity, but its mean and standard deviation do exist, because the tails diminish quickly enough. The

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less than 10). As sample size increases, the amount of bias decreases. We obtain more information and the difference between

955: 12459: 12159: 6862: 11782: 161:. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same unit as the data. 13063: 12211: 11294: 9276: 4738:

The standard deviation of the sum of two random variables can be related to their individual standard deviations and the

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The standard deviation of a probability distribution is the same as that of a random variable having that distribution.

31: 17: 4844:{\displaystyle \sigma (X+Y)={\sqrt {\operatorname {var} (X)+\operatorname {var} (Y)+2\,\operatorname {cov} (X,Y)}}.\,} 2223:{\displaystyle \sigma ={\sqrt {\sum _{i=1}^{N}p_{i}(x_{i}-\mu )^{2}}},{\text{ where }}\mu =\sum _{i=1}^{N}p_{i}x_{i}.} 13846: 13738: 11903: 8540: 7175: 5506:

See computational formula for the variance for proof, and for an analogous result for the sample standard deviation.

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Gurland, John; Tripathi, Ram C. (1971), "A Simple Approximation for Unbiased Estimation of the Standard Deviation",

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The following two formulas can represent a running (repeatedly updated) standard deviation. A set of two power sums

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Browne, Richard H. (2001). "Using the Sample Range as a Basis for Calculating Sample Size in Power Calculations".

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This estimator is unbiased if the variance exists and the sample values are drawn independently with replacement.

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of the sample, which is used as an estimate of the population standard deviation. Such a statistic is called an

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stands for the size of the sample: this is the square root of the sample variance, which is the average of the

1862:{\displaystyle \sigma ={\sqrt {{\frac {1}{N}}\left}},{\text{ where }}\mu ={\frac {1}{N}}(x_{1}+\cdots +x_{N}),} 1186:) – one standard deviation – and almost all men (about 95%) have a height within 13073: 4529:

estimation, as the range of possible values is easier to estimate than the standard deviation. Other divisors

3829:{\displaystyle {\hat {\sigma }}={\sqrt {{\frac {1}{N-1.5}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}},} 2569:, maximum likelihood), there is no single estimator for the standard deviation with all these properties, and 1010: 14376: 13335: 12238: 12123: 11372: 11708:

Shiffler, Ronald E.; Harsha, Phillip D. (1980). "Upper and Lower Bounds for the Sample Standard Deviation".

4107: 4069:. The excess kurtosis may be either known beforehand for certain distributions, or estimated from the data. 2845:(it converges in probability to the population value as the number of samples goes to infinity), and is the 2538:) where every member of a population is sampled. In cases where that cannot be done, the standard deviation 1572:

going out to infinity, the standard deviation might not exist, because the integral might not converge. The

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is the number of observations in the sample used to estimate the mean. This can easily be proven with (see

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The above formulas become equal to the simpler formulas given above if weights are taken as equal to one.

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The incremental method with reduced rounding errors can also be applied, with some additional complexity.

6819: 6616: 4436:. The reciprocals of the square roots of these two numbers give us the factors 0.45 and 31.9 given above. 14481: 14439: 14271: 14072: 13996: 13297: 13051: 12720: 12184: 12118: 12113: 11946:

Welford, B. P. (August 1962). "Note on a Method for Calculating Corrected Sums of Squares and Products".

11183: 9764:{\displaystyle {\begin{aligned}A_{0}&=0\\A_{k}&=A_{k-1}+{\frac {x_{k}-A_{k-1}}{k}}\end{aligned}}} 14156: 14128: 14123: 13871: 13630: 13536: 13516: 13424: 13135: 12953: 12436: 12308: 11357: 8595: 5560: 4082: 158: 11252:

In two dimensions, the standard deviation can be illustrated with the standard deviation ellipse (see

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Taking square roots reintroduces bias (because the square root is a nonlinear function which does not

2416:, the standard deviation can be expressed in terms of the parameters. For example, in the case of the 1583: 13888: 13656: 13377: 13302: 13231: 13160: 13080: 13068: 12938: 12926: 12919: 12627: 12348: 11392: 11377: 11347: 11299: 6594: 2853:, as the estimates are generally too low. The bias decreases as sample size grows, dropping off as 1/ 1618:

has a mean, but not a standard deviation (loosely speaking, the standard deviation is infinite). The

11960: 11779:"CERN experiments observe particle consistent with long-sought Higgs boson | CERN press office" 5652:

To gain some geometric insights and clarification, we will start with a population of three values,

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These same formulae can be used to obtain confidence intervals on the variance of residuals from a

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This arises because the sampling distribution of the sample standard deviation follows a (scaled)

3531:, there is no formula that works across all distributions, unlike for mean and variance. Instead, 2573:

is a very technically involved problem. Most often, the standard deviation is estimated using the

55: 14280: 13893: 13833: 13770: 13408: 13392: 13130: 12992: 12982: 12832: 12746: 11284: 10474: 7402: 5626: 5556: 2417: 2238: 1199: 48: 8379:

usually reported together. In a certain sense, the standard deviation is a "natural" measure of

2804: 14318: 14248: 14041: 13978: 13733: 13620: 12617: 12514: 12421: 12300: 12199: 11955: 11412: 11315: 11289: 8380: 8376: 5544: 3034: 3019: 3009:{\displaystyle s_{N}={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}}.} 2724:{\displaystyle s_{N}={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}},} 1870: 1146: 230: 134: 80: 11578: 6795: 6766: 6737: 6708: 6679: 6650: 1078: 59:

Cumulative probability of a normal distribution with expected value 0 and standard deviation 1

14343: 14285: 14228: 14054: 13947: 13856: 13582: 13466: 13325: 13317: 13207: 13199: 13014: 12910: 12888: 12847: 12812: 12779: 12725: 12700: 12655: 12594: 12554: 12356: 12179: 11864: 8521: 8507:{\displaystyle \sigma (r)={\sqrt {{\frac {1}{N-1}}\sum _{i=1}^{N}\left(x_{i}-r\right)^{2}}}.} 6943: 3435:{\displaystyle s={\sqrt {{\frac {1}{N-1}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}}.} 2860: 2557:

Unlike in the case of estimating the population mean of a normal distribution, for which the

1319:{\displaystyle \mu \equiv \operatorname {E} =\int _{-\infty }^{+\infty }xf(x)\,\mathrm {d} x} 1144:

gives an unbiased estimate of the variance of the larger parent population. This is known as

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grows larger with each new sample, rather than a constant-width sliding window calculation.

7453: 7441: 6585:, multiplied by the square root of the number of dimensions of the vector (3 in this case). 47:(or bell-shaped curve) where each band has a width of 1 standard deviation – See also: 14266: 13841: 13790: 13766: 13728: 13646: 13625: 13577: 13456: 13434: 13403: 13312: 13189: 13140: 13058: 13031: 12987: 12943: 12705: 12481: 12361: 12033: 11823: 11488: 11336: 11320: 11167: 10209: 10182: 9618: 8583: 7163:{\displaystyle {\text{Proportion}}=\operatorname {erf} \left({\frac {z}{\sqrt {2}}}\right)} 7107: 6902: 5184:{\displaystyle s(X)={\sqrt {\frac {N}{N-1}}}{\sqrt {\operatorname {E} \left)^{2}\right]}}.} 4920: 4233: 3269: 2842: 1163: 7113:

If a data distribution is approximately normal, then the proportion of data values within

8: 14413: 14338: 14261: 13942: 13706: 13699: 13661: 13569: 13549: 13521: 13254: 13120: 13115: 13105: 13097: 12915: 12876: 12766: 12756: 12665: 12444: 12400: 12318: 12243: 12145: 12062: 11920: 11888: 11623: 11342: 9614: 6476: 5549: 5519: 5515: 4093: 2566: 1619: 1577: 1573: 1195: 1179: 189:, a safeguard against spurious conclusion that is really due to random sampling error. 130: 44: 12037: 11827: 2029:

If, instead of having equal probabilities, the values have different probabilities, let

102:, and is most commonly represented in mathematical texts and equations by the lowercase 14427: 14238: 14092: 13988: 13937: 13813: 13710: 13694: 13671: 13448: 13182: 13165: 13125: 13036: 12931: 12893: 12864: 12824: 12784: 12730: 12647: 12328: 11988: 11847: 11813: 11760: 11752: 11721: 11665: 11598: 11532: 11465: 11440: 11275: 11187: 5954: 5934: 5559:, which might be considered a more direct measure of average distance, compared to the 4572: 3160:{\displaystyle s^{2}={\frac {1}{N-1}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}.} 2884: 2846: 2835: 2562: 1191: 1183: 193: 14422: 14333: 14303: 14295: 14115: 14106: 14031: 13962: 13818: 13803: 13778: 13666: 13607: 13473: 13461: 13087: 13004: 12948: 12871: 12715: 12637: 12416: 12290: 12091: 12081: 11995: 11899: 11851: 11839: 11764: 11529: 11470: 11269: 8886:{\displaystyle \operatorname {var} (cX_{1})\equiv c^{2}\,\operatorname {var} (X_{1})} 2398: 1101: 154: 10455:{\displaystyle {\begin{aligned}W_{0}&=0\\W_{k}&=W_{k-1}+w_{k}\end{aligned}}} 8655: 5744:. So it is not unreasonable to assume that the standard deviation is related to the 2542:

is estimated by examining a random sample taken from the population and computing a

14358: 14313: 14077: 14064: 13957: 13932: 13866: 13798: 13676: 13284: 13177: 13110: 13023: 12970: 12789: 12660: 12454: 12338: 12253: 12220: 12041: 11969: 11965: 11835: 11831: 11744: 11717: 11657: 11460: 11452: 11417: 11382: 9497:, as mentioned above, is the size of the set of values (or can also be regarded as 6935: 5633: 5584: 3696: 3023: 2850: 2413: 1127: 1058: 432: 421: 8119: 7417:, the percentage of values expected to lie in and outside the symmetric interval, 7357:{\displaystyle {\text{Proportion}}\leq x={\frac {1}{2}}\left={\frac {1}{2}}\left.} 2591: − 1.5 (for the normal distribution) almost completely eliminates bias. 14275: 14019: 13881: 13808: 13483: 13357: 13330: 13307: 13276: 12903: 12898: 12852: 12582: 12233: 11510: 11352: 9610: 5524: 4576: 4078: 4066: 2898: 2523: 1223: 174: 126: 13765: 11803: 14224: 14219: 12682: 12612: 12258: 11748: 11580:

Anthropometric Reference Data for Children and Adults: United States, 2015–2018

11407: 11402: 11248:

The standard deviation ellipse (green) of a two-dimensional normal distribution

7195: 7089: 3858:), and it is suited for all but the smallest samples or highest precision: for 3565: 2903: 1219: 165: 103: 88: 76: 11456: 7380:

is the arithmetic mean), about 95 percent are within two standard deviations (

2534:

One can find the standard deviation of an entire population in cases (such as

2506:{\displaystyle {\sqrt {\left(e^{\sigma ^{2}}-1\right)e^{2\mu +\sigma ^{2}}}}.} 14470: 14381: 14348: 14211: 14172: 13983: 13952: 13416: 13370: 12975: 12677: 12504: 12268: 12263: 12131: 12095: 11983: 6929: 4458: 1167: 39: 1568:

Not all random variables have a standard deviation. If the distribution has

939:{\displaystyle \sigma ^{2}={\frac {9+1+1+1+0+0+4+16}{8}}={\frac {32}{8}}=4.} 216: 14323: 14256: 14233: 14148: 13478: 12774: 12672: 12607: 12549: 12534: 12471: 12426: 12046: 12019: 12015: 11843: 11163: 173:

sample size. For example, a poll's standard error (what is reported as the

112: 11474: 3259:{\displaystyle \textstyle (x_{1}-{\bar {x}},\;\dots ,\;x_{n}-{\bar {x}}).} 14366: 14328: 14011: 13912: 13774: 13587: 13554: 13046: 12963: 12958: 12602: 12559: 12539: 12519: 12509: 12278: 11387: 5644:

use of statistical tools that now have a valid basis from which to work.

5592: 5540: 4923:

calculated directly from the data. In the following formula, the letter

4526: 2558: 1659:, with each value having the same probability, the standard deviation is 222: 146: 9262:

it is necessary to know the standard deviation of the entire population

5193:

For a finite population with equal probabilities at all points, we have

1545:{\textstyle {\sqrt {\operatorname {E} \left-(\operatorname {E} )^{2}}}.} 13212: 12692: 12392: 12323: 12273: 12248: 12168: 11756: 11669: 11362: 4913: 4739: 2857:, and thus is most significant for small or moderate sample sizes; for 64: 4072: 1869:

Note: The above expression has a built-in bias. See the discussion on

420:

First, calculate the deviations of each data point from the mean, and

13365: 13217: 12837: 12632: 12544: 12529: 12524: 12489: 11537: 11397: 11326: 11244: 8582:, which is the ratio of the standard deviation to the mean. It is a 2547: 2543: 1877: 1569: 1157: 1051:

In that case, the result of the original formula would be called the

11661: 2830:

is the mean value of these observations, while the denominator

2405:

ranging over the set of possible values of the random variable

411:{\displaystyle \mu ={\frac {2+4+4+4+5+5+7+9}{8}}={\frac {40}{8}}=5.} 12881: 12499: 12376: 12371: 12366: 11818: 11367: 11310: 9244:{\displaystyle \sigma _{\text{mean}}={\frac {\sigma }{\sqrt {N}}}.} 8641:{\displaystyle \sigma _{\text{mean}}={\frac {1}{\sqrt {N}}}\sigma } 8517: 8370: 5843:

whose coordinates are the mean of the values we started out with.

5621: 4919:

The calculation of the sum of squared deviations can be related to

1555: 840: 169: 138: 72: 2621:(considered as the entire population), and is defined as follows: 14386: 14087: 6553:{\textstyle {\sqrt {\sum _{i}\left(x_{i}-{\bar {x}}\right)^{2}}}} 5620:

In finance, standard deviation is often used as a measure of the

5583:

In experimental science, a theoretical model of reality is used.

3699:, and the correction factor is the mean of the chi distribution. 3018:

Here taking the square root introduces further downward bias, by

226: 182: 150: 92: 11527: 7390:), and about 99.7 percent lie within three standard deviations ( 5574: 2849:

when the population is normally distributed. However, this is a

952:

standard deviation is equal to the square root of the variance:

14308: 13289: 13263: 13243: 12494: 12285: 5919:{\displaystyle M=\left({\bar {x}},{\bar {x}},{\bar {x}}\right)} 5834:{\displaystyle M=\left({\bar {x}},{\bar {x}},{\bar {x}}\right)} 4883:{\displaystyle \textstyle \operatorname {var} \,=\,\sigma ^{2}} 4272:

is the confidence level. This is equivalent to the following:

1554:

Using words, the standard deviation is the square root of the

12137: 9589:{\displaystyle s={\sqrt {\frac {Ns_{2}-s_{1}^{2}}{N(N-1)}}}.} 9484:{\displaystyle \sigma ={\frac {\sqrt {Ns_{2}-s_{1}^{2}}}{N}}} 107: 11127:{\displaystyle s_{n}^{2}={\frac {n'}{n'-1}}\sigma _{n}^{2},} 10349:

is now the sum of the weights and not the number of samples

12228: 11517:. Baltimore, MD: Williams & Wilkins Co. pp. 24–25. 11255:

Multivariate normal distribution § Geometric interpretation

5596: 319: 84: 9400:

Given the results of these running summations, the values

7096:

equals their distribution's standard deviation divided by

4087:

Student's t-distribution § Robust parametric modeling

1007:

in the denominator of the last formula, and the result is

12063:"Earliest Known Uses of Some of the Words of Mathematics" 8375:

The mean and the standard deviation of a set of data are

7198:. The proportion that is less than or equal to a number, 4566: 3838:

The error in this approximation decays quadratically (as

217:

Population standard deviation of grades of eight students

164:

The standard deviation of a population or sample and the

11491:(1816). "Bestimmung der Genauigkeit der Beobachtungen". 10972:{\displaystyle \sigma _{n}^{2}={\frac {Q_{n}}{W_{n}}}\,} 9253:

In order to estimate the standard deviation of the mean

2594: 2575: 1171: 11493:

Zeitschrift für Astronomie und Verwandte Wissenschaften

10340:

And the standard deviation equations remain unchanged.

111:(sigma), for the population standard deviation, or the 11339:

generalizing number of standard deviations to the mean

10331:{\displaystyle s_{j}=\sum _{k=1}^{N}w_{k}x_{k}^{j}.\,} 7179: 6924: 6493: 5595:

had been discovered in two independent experiments at

4897: 4860: 3448:

is an unbiased estimator for the population variance,

3278: 3184: 2890: 2561:

is a simple estimator with many desirable properties (

1479: 1130: 1104: 1061: 1013: 11196: 11059: 10987: 10925: 10519: 10477: 10373: 10264: 10212: 10185: 10126: 10064: 9783: 9654: 9517: 9434: 9333: 9212: 8901: 8821: 8664: 8609: 8543: 8409: 7214: 7178: 7125: 6958: 6905: 6891:{\displaystyle {\frac {1}{\sqrt {1-\ell }}}\,\sigma } 6865: 6822: 6798: 6769: 6740: 6711: 6682: 6653: 6619: 6066: 6021: 5977: 5957: 5937: 5857: 5772: 5756:. That is indeed the case. To move orthogonally from 5201: 5088: 4935: 4896: 4859: 4750: 4597: 4571:

The standard deviation is invariant under changes in

4280: 4236: 4118: 3917: 3722: 3574: 3522: 3490: 3463: 3337: 3183: 3059: 2912: 2863: 2807: 2739: 2627: 2436: 2264: 2091: 1888: 1667: 1586: 1340: 1246: 1081: 958: 849: 430: 328: 239: 117: 14050:

Autoregressive conditional heteroskedasticity (ARCH)

12020:"On the dissection of asymmetrical frequency curves" 11420:

for calculating standard deviation of wind direction

11265: 11894:(2nd ed.). New Jersey: Prentice Hall. p. 11044:{\displaystyle s_{n}^{2}={\frac {Q_{n}}{W_{n}-1}},} 10359:A running sum of weights must be computed for each 4927:is interpreted to mean expected value, i.e., mean. 4083:

Variance § Distribution of the sample variance

4073:

Confidence interval of a sampled standard deviation

2794:{\displaystyle \{x_{1},\,x_{2},\,\ldots ,\,x_{N}\}} 208:(the standard deviation of the entire population). 13512: 11987: 11229: 11126: 11043: 10971: 10909: 10505: 10454: 10330: 10225: 10198: 10165:{\displaystyle \sigma _{n}^{2}={\frac {Q_{n}}{n}}} 10164: 10110: 10012: 9763: 9588: 9483: 9390: 9243: 9196: 8885: 8805: 8640: 8568: 8506: 7356: 7186: 7162: 7074: 6911: 6890: 6848: 6807: 6778: 6749: 6720: 6691: 6662: 6633: 6552: 6452: 6035: 6007: 5963: 5943: 5918: 5833: 5493: 5183: 5082:The sample standard deviation can be computed as: 5072: 4904: 4882: 4843: 4728: 4399: 4249: 4220: 4046: 3828: 3685: 3511: 3476: 3434: 3316: 3258: 3159: 3008: 2875: 2822: 2793: 2723: 2505: 2387: 2222: 2019: 1861: 1610: 1544: 1465: 1318: 1209: 1182:) have a height within 3 inches of the mean ( 1178:. This means that most men (about 68%, assuming a 1158:Standard deviation of average height for adult men 1136: 1116: 1090: 1067: 1043: 980: 938: 829: 410: 308: 83:indicates that the values tend to be close to the 12025:Philosophical Transactions of the Royal Society A 11147:is the number of elements with non-zero weights. 9391:{\displaystyle s_{j}=\sum _{k=1}^{N}{x_{k}^{j}}.} 8589: 6560:is equal to the standard deviation of the vector 6467:A little algebra shows that the distance between 5509: 4261:-th quantile of the chi-square distribution with 2801:are the observed values of the sample items, and 196:of data from a population is available, the term 14468: 11781:. Press.web.cern.ch. 4 July 2012. Archived from 11515:Studies in the History of the Statistical Method 8371:Relationship between standard deviation and mean 4281: 4119: 3560:, where the correction factor (which depends on 13598:Multivariate adaptive regression splines (MARS) 11647: 7187:{\displaystyle \textstyle \operatorname {erf} } 7117:standard deviations of the mean is defined by: 4905:{\displaystyle \textstyle \operatorname {cov} } 4472: 2587:) yields lower mean squared error, while using 2085:. In this case, the standard deviation will be 11707: 11302:An inequality on location and scale parameters 11182:The standard deviation index (SDI) is used in 10111:{\displaystyle s_{n}^{2}={\frac {Q_{n}}{n-1}}} 2887:than the corrected sample standard deviation. 309:{\displaystyle 2,\ 4,\ 4,\ 4,\ 5,\ 5,\ 7,\ 9.} 12153: 12080:. Philadelphia: F. A. Davis Co. p. 236. 11918: 5575:Experiment, industrial and hypothesis testing 4579:of the random variable. Thus, for a constant 4502:. An estimate of the standard deviation for 2232: 1622:has neither a mean nor a standard deviation. 12069: 9270: 4514:represents four standard deviations so that 3875:A more accurate approximation is to replace 3173: − 1 corresponds to the number of 2788: 2740: 11438: 11177: 5647: 4488:, an upper bound on the standard deviation 3702:An approximation can be given by replacing 3177:in the vector of deviations from the mean, 1634:takes random values from a finite data set 1625: 12198: 12160: 12146: 11990:The Oxford Dictionary of Statistical Terms 11885: 8401:are real numbers and define the function: 3223: 3216: 12811: 12045: 11959: 11817: 11464: 10968: 10327: 9598:In a computer implementation, as the two 9509:Similarly for sample standard deviation, 9426:value of the running standard deviation: 8860: 8565: 6884: 6627: 6588: 6029: 5599:, also leading to the declaration of the 4868: 4864: 4840: 4813: 4525:. This so-called range rule is useful in 3621: 3620: 3619: 3598: 3594: 3529:unbiased estimation of standard deviation 2777: 2770: 2756: 2571:unbiased estimation of standard deviation 2530:Unbiased estimation of standard deviation 2373: 2360: 2320: 2307: 1449: 1307: 1044:{\textstyle s={\sqrt {32/7}}\approx 2.1.} 11432: 11243: 8578:Variability can also be measured by the 7452: 7440: 6928: 5601:first observation of gravitational waves 5523: 4461:fit under standard normal theory, where 54: 38: 12014: 11945: 10174: 9609:sums become large, we need to consider 9422:can be used at any time to compute the 8815:(Statistical independence is assumed.) 6050:is to be orthogonal to the vector from 5566: 14: 14469: 14124:Kaplan–Meier estimator (product limit) 11734: 11509: 4567:Identities and mathematical properties 2239:continuous real-valued random variable 1174:, with a standard deviation of around 981:{\displaystyle \sigma ={\sqrt {4}}=2.} 98:Standard deviation may be abbreviated 75:of the values of a variable about its 14197: 13764: 13511: 12810: 12580: 12197: 12141: 11982: 11596: 11528: 11487: 11441:"Statistics notes: measurement error" 11140:is the total number of elements, and 6036:{\displaystyle \ell \in \mathbb {R} } 6008:{\displaystyle M=(\ell ,\ell ,\ell )} 2615:uncorrected sample standard deviation 2595:Uncorrected sample standard deviation 1154:would underestimate the variability. 122:, for the sample standard deviation. 27:In statistics, a measure of variation 14477:Statistical deviation and dispersion 14434: 14134:Accelerated failure time (AFT) model 12075: 11239: 6849:{\displaystyle 1-{\frac {1}{k^{2}}}} 6634:{\displaystyle {\sqrt {2}}\,\sigma } 5563:inherent in the standard deviation. 3872:the bias is already less than 0.1%. 3326:corrected sample standard deviation, 14446: 13729:Analysis of variance (ANOVA, anova) 12581: 11295:Algorithms for calculating variance 9277:Algorithms for calculating variance 6925:Rules for normally distributed data 5587:conventionally uses a standard of " 3865:the bias is equal to 1.3%, and for 3022:, due to the square root's being a 2891:Corrected sample standard deviation 2576:corrected sample standard deviation 1175: 32:Standard deviation (disambiguation) 24: 13824:Cochran–Mantel–Haenszel statistics 12450:Pearson product-moment correlation 12060: 11722:10.1111/j.1467-9639.1980.tb00398.x 11439:Bland, J.M.; Altman, D.G. (1996). 11223:Consensus group standard deviation 10471:is used above must be replaced by 10206:are weighted with unequal weights 8535:has a unique minimum at the mean: 5146: 5126: 5040: 5010: 4973: 4953: 4563:and for non-normal distributions. 4559:are available for other values of 3648: 3625: 3523:Unbiased sample standard deviation 2414:parametric family of distributions 2375: 2322: 1512: 1482: 1451: 1410: 1402: 1349: 1309: 1287: 1279: 1253: 1187: 1055:standard deviation and denoted by 233:) are the following eight values: 25: 14493: 12106: 12078:Medical laboratory science review 11590: 3272:with the expectation, i.e. often 318:These eight data points have the 211: 153:simpler, though in practice less 14445: 14433: 14421: 14408: 14407: 14198: 11268: 8656:basic properties of the variance 7204:cumulative distribution function 4484:data spanning a range of values 3512:{\displaystyle {\frac {1}{N-1}}} 2619:standard deviation of the sample 2351: 2279: 1611:{\displaystyle \alpha \in (1,2]} 198:standard deviation of the sample 14083:Least-squares spectral analysis 12054: 12008: 11976: 11939: 11912: 11879: 11857: 11797: 11771: 11728: 11701: 11683:"Standard Deviation Calculator" 11675: 8569:{\displaystyle r={\bar {x}}.\,} 4575:, and scales directly with the 4108:the factors here are as follows 1210:Definition of population values 13064:Mean-unbiased minimum-variance 12167: 11970:10.1080/00401706.1962.10490022 11836:10.1103/PhysRevLett.116.061102 11641: 11616: 11571: 11546: 11521: 11503: 11481: 9576: 9564: 9184: 9178: 9156: 9150: 9121: 9108: 8920: 8912: 8880: 8867: 8844: 8828: 8796: 8783: 8771: 8758: 8742: 8716: 8681: 8675: 8590:Standard deviation of the mean 8556: 8524:, it is possible to show that 8419: 8413: 6531: 6430: 6264: 6207: 6187: 6130: 6124: 6106: 6089: 6077: 6002: 5984: 5905: 5890: 5875: 5820: 5805: 5790: 5510:Interpretation and application 5350: 5260: 5162: 5158: 5152: 5137: 5098: 5092: 5056: 5052: 5046: 5037: 4989: 4985: 4979: 4964: 4945: 4939: 4832: 4820: 4804: 4798: 4786: 4780: 4766: 4754: 4716: 4710: 4703: 4695: 4684: 4675: 4662: 4656: 4643: 4631: 4611: 4605: 4022: 3924: 3804: 3729: 3591: 3585: 3477:{\displaystyle {\frac {1}{N}}} 3410: 3317:{\textstyle E\neq {\sqrt {E}}} 3309: 3303: 3292: 3282: 3249: 3243: 3207: 3185: 3137: 3029:An unbiased estimator for the 2984: 2814: 2699: 2370: 2364: 2317: 2311: 2298: 2285: 2151: 2131: 1948: 1928: 1853: 1821: 1781: 1761: 1743: 1723: 1711: 1691: 1605: 1593: 1528: 1524: 1518: 1509: 1446: 1440: 1428: 1415: 1373: 1360: 1304: 1298: 1265: 1259: 795: 782: 764: 754: 742: 729: 697: 684: 666: 656: 644: 631: 599: 586: 568: 558: 546: 533: 501: 488: 470: 460: 448: 435: 71:is a measure of the amount of 13: 1: 14377:Geographic information system 13593:Simultaneous equations models 12132:Standard Deviation Calculator 11554:"Standard Deviation Formulas" 11425: 11373:Reduced chi-squared statistic 11162:was first used in writing by 2515: 843:is the mean of these values: 206:population standard deviation 13560:Coefficient of determination 13171:Uniformly most powerful test 11865:"What is Standard Deviation" 11332:Geometric standard deviation 11184:external quality assessments 6948:probability density function 4473:Bounds on standard deviation 2430:, the standard deviation is 2397:and where the integrals are 2246:probability density function 2237:The standard deviation of a 1473:which can be shown to equal 1164:average height for adult men 125:The standard deviation of a 7: 14129:Proportional hazards models 14073:Spectral density estimation 14055:Vector autoregression (VAR) 13489:Maximum posterior estimator 12721:Randomized controlled trial 12119:Encyclopedia of Mathematics 11994:. Oxford University Press. 11890:Fundamentals of Probability 11323:Distance standard deviation 11261: 10506:{\displaystyle w_{k}/W_{k}} 9299:are computed over a set of 5764:, one begins at the point: 3564:) is given in terms of the 2847:maximum-likelihood estimate 2554:(possibly with modifiers). 187:"statistically significant" 10: 14498: 13889:Multivariate distributions 12309:Average absolute deviation 11886:Ghahramani, Saeed (2000). 11749:10.1198/000313001753272420 11358:Propagation of uncertainty 11153: 9274: 8596:Standard error of the mean 8593: 6592: 6475:(which is the same as the 5627:mean-variance optimization 5615: 5606: 5513: 4076: 3444:As explained above, while 3041: − 1 instead of 2823:{\displaystyle {\bar {x}}} 2527: 2521: 2233:Continuous random variable 159:average absolute deviation 29: 14403: 14357: 14294: 14247: 14210: 14206: 14193: 14165: 14147: 14114: 14105: 14063: 14010: 13971: 13920: 13911: 13877:Structural equation model 13832: 13789: 13785: 13760: 13719: 13685: 13639: 13606: 13568: 13535: 13531: 13507: 13447: 13356: 13275: 13239: 13230: 13213:Score/Lagrange multiplier 13198: 13151: 13096: 13022: 13013: 12823: 12819: 12806: 12765: 12739: 12691: 12646: 12628:Sample size determination 12593: 12589: 12576: 12480: 12435: 12409: 12391: 12347: 12299: 12219: 12210: 12206: 12193: 12175: 11737:The American Statistician 11650:The American Statistician 11457:10.1136/bmj.312.7047.1654 11378:Robust standard deviation 11348:Median absolute deviation 9271:Rapid calculation methods 7476: 7468: 7092:of the random variables, 5561:root mean square distance 3047:unbiased sample variance, 202:sample standard deviation 14372:Environmental statistics 13894:Elliptical distributions 13687:Generalized linear model 13616:Simple linear regression 13386:Hodges–Lehmann estimator 12843:Probability distribution 12752:Stochastic approximation 12314:Coefficient of variation 12076:Harr, Robert R. (2012). 11306:Coefficient of variation 11178:Standard deviation index 8580:coefficient of variation 7400:). This is known as the 6808:{\displaystyle k\sigma } 6779:{\displaystyle 6\sigma } 6750:{\displaystyle 5\sigma } 6721:{\displaystyle 4\sigma } 6692:{\displaystyle 3\sigma } 6663:{\displaystyle 2\sigma } 5648:Geometric interpretation 4265:degrees of freedom, and 1626:Discrete random variable 1198:or bell-shaped (see the 1091:{\displaystyle \sigma .} 143:probability distribution 14032:Cross-correlation (XCF) 13640:Non-standard predictors 13074:Lehmann–Scheffé theorem 12747:Adaptive clinical trial 11921:"Distribution Function" 11806:Physical Review Letters 11190:. It is calculated as: 10919:In the final division, 5675:. This defines a point 5557:mean absolute deviation 4912:stand for variance and 4540:of the range such that 4439:A larger population of 4065:denotes the population 2876:{\displaystyle N>75} 2838:about the sample mean. 2418:log-normal distribution 1326:The standard deviation 1206:for more information). 149:of its variance. It is 14428:Mathematics portal 14249:Engineering statistics 14157:Nelson–Aalen estimator 13734:Analysis of covariance 13621:Ordinary least squares 13545:Pearson product-moment 12949:Statistical functional 12860:Empirical distribution 12693:Controlled experiments 12422:Frequency distribution 12200:Descriptive statistics 12047:10.1098/rsta.1894.0003 11624:"Consistent estimator" 11413:Statistical dispersion 11393:Samuelson's inequality 11316:Deviation (statistics) 11300:Chebyshev's inequality 11290:Accuracy and precision 11249: 11231: 11128: 11045: 10973: 10911: 10507: 10456: 10332: 10298: 10256:are each computed as: 10227: 10200: 10166: 10112: 10014: 9765: 9590: 9485: 9392: 9367: 9245: 9198: 9101: 9038: 8971: 8887: 8807: 8642: 8570: 8508: 8465: 8381:statistical dispersion 8377:descriptive statistics 7772:1 / 370.398 7461: 7450: 7413:For various values of 7358: 7188: 7164: 7076: 6939: 6913: 6892: 6850: 6809: 6780: 6751: 6722: 6693: 6664: 6635: 6595:Chebyshev's inequality 6589:Chebyshev's inequality 6554: 6454: 6037: 6009: 5965: 5945: 5920: 5835: 5529: 5495: 5464: 5404: 5319: 5234: 5185: 5074: 4906: 4884: 4845: 4730: 4401: 4251: 4222: 4048: 3996: 3830: 3778: 3687: 3513: 3478: 3436: 3384: 3318: 3260: 3161: 3111: 3010: 2958: 2877: 2824: 2795: 2725: 2673: 2507: 2389: 2224: 2196: 2120: 2021: 2003: 1927: 1863: 1612: 1546: 1467: 1320: 1138: 1118: 1092: 1069: 1045: 982: 940: 831: 412: 310: 231:statistical population 135:statistical population 60: 52: 14344:Population statistics 14286:System identification 14020:Autocorrelation (ACF) 13948:Exponential smoothing 13862:Discriminant analysis 13857:Canonical correlation 13721:Partition of variance 13583:Regression validation 13427:(Jonckheere–Terpstra) 13326:Likelihood-ratio test 13015:Frequentist inference 12927:Location–scale family 12848:Sampling distribution 12813:Statistical inference 12780:Cross-sectional study 12767:Observational studies 12726:Randomized experiment 12555:Stem-and-leaf display 12357:Central limit theorem 12114:"Quadratic deviation" 11603:mathworld.wolfram.com 11533:"Bessel's Correction" 11489:Gauss, Carl Friedrich 11247: 11232: 11129: 11046: 10974: 10912: 10508: 10457: 10333: 10278: 10228: 10226:{\displaystyle w_{k}} 10201: 10199:{\displaystyle x_{k}} 10167: 10120:Population variance: 10113: 10015: 9766: 9591: 9486: 9393: 9347: 9246: 9199: 9081: 9018: 8951: 8888: 8808: 8643: 8571: 8522:completing the square 8509: 8445: 7456: 7444: 7359: 7189: 7165: 7077: 6944:central limit theorem 6932: 6914: 6912:{\displaystyle \ell } 6893: 6851: 6810: 6781: 6752: 6723: 6694: 6665: 6636: 6555: 6455: 6038: 6010: 5966: 5946: 5921: 5836: 5527: 5514:Further information: 5496: 5444: 5384: 5299: 5214: 5186: 5075: 4907: 4885: 4846: 4731: 4583:and random variables 4465:is now the number of 4402: 4252: 4250:{\displaystyle q_{p}} 4223: 4049: 3976: 3831: 3758: 3688: 3514: 3479: 3437: 3364: 3319: 3261: 3162: 3091: 3033:is given by applying 3011: 2938: 2878: 2825: 2796: 2726: 2653: 2508: 2390: 2225: 2176: 2100: 2022: 1983: 1907: 1864: 1613: 1547: 1468: 1321: 1139: 1119: 1093: 1070: 1046: 983: 941: 832: 413: 311: 58: 42: 14267:Probabilistic design 13852:Principal components 13695:Exponential families 13647:Nonlinear regression 13626:General linear model 13588:Mixed effects models 13578:Errors and residuals 13555:Confounding variable 13457:Bayesian probability 13435:Van der Waerden test 13425:Ordered alternative 13190:Multiple comparisons 13069:Rao–Blackwellization 13032:Estimating equations 12988:Statistical distance 12706:Factorial experiment 12239:Arithmetic-Geometric 11599:"Standard Deviation" 11337:Mahalanobis distance 11321:Distance correlation 11218:Consensus group mean 11194: 11188:medical laboratories 11057: 10985: 10923: 10517: 10475: 10371: 10262: 10210: 10183: 10175:Weighted calculation 10124: 10062: 9781: 9652: 9619:arithmetic underflow 9515: 9432: 9331: 9210: 8899: 8819: 8662: 8607: 8584:dimensionless number 8541: 8407: 7585:1 / 3.125 7212: 7176: 7123: 7108:normalizing constant 6956: 6903: 6863: 6820: 6796: 6767: 6738: 6709: 6680: 6651: 6617: 6491: 6064: 6019: 5975: 5955: 5935: 5855: 5770: 5709:. Consider the line 5567:Application examples 5199: 5086: 4933: 4894: 4857: 4748: 4595: 4278: 4234: 4116: 3915: 3720: 3572: 3488: 3461: 3335: 3276: 3181: 3057: 2910: 2861: 2843:consistent estimator 2805: 2737: 2625: 2599:The formula for the 2536:standardized testing 2434: 2262: 2089: 1886: 1873:further down below. 1665: 1584: 1477: 1338: 1244: 1128: 1102: 1079: 1059: 1011: 956: 847: 428: 424:the result of each: 326: 237: 30:For other uses, see 14339:Official statistics 14262:Methods engineering 13943:Seasonal adjustment 13711:Poisson regressions 13631:Bayesian regression 13570:Regression analysis 13550:Partial correlation 13522:Regression analysis 13121:Prediction interval 13116:Likelihood interval 13106:Confidence interval 13098:Interval estimation 13059:Unbiased estimators 12877:Model specification 12757:Up-and-down designs 12445:Partial correlation 12401:Index of dispersion 12319:Interquartile range 12038:1894RSPTA.185...71P 11919:Eric W. Weisstein. 11828:2016PhRvL.116f1102A 11710:Teaching Statistics 11597:Weisstein, Eric W. 11343:Mean absolute error 11186:, particularly for 11120: 11074: 11002: 10940: 10323: 10141: 10079: 9777:is the mean value. 9615:arithmetic overflow 9558: 9474: 9383: 8705: 7477:Proportion without 7460:(Percentage within) 6609:Minimum population 6477:orthogonal distance 5550:prediction interval 5520:Confidence interval 5516:Prediction interval 5419: 5334: 4094:confidence interval 3035:Bessel's correction 3020:Jensen's inequality 2617:, or sometimes the 1871:Bessel's correction 1620:Cauchy distribution 1578:Pareto distribution 1574:normal distribution 1414: 1291: 1180:normal distribution 1147:Bessel's correction 95:and what does not. 45:normal distribution 18:Standard deviations 14482:Summary statistics 14359:Spatial statistics 14239:Medical statistics 14139:First hitting time 14093:Whittle likelihood 13744:Degrees of freedom 13739:Multivariate ANOVA 13672:Heteroscedasticity 13484:Bayesian estimator 13449:Bayesian inference 13298:Kolmogorov–Smirnov 13183:Randomization test 13153:Testing hypotheses 13126:Tolerance interval 13037:Maximum likelihood 12932:Exponential family 12865:Density estimation 12825:Statistical theory 12785:Natural experiment 12731:Scientific control 12648:Survey methodology 12334:Standard deviation 11558:www.mathsisfun.com 11530:Weisstein, Eric W. 11276:Mathematics portal 11250: 11227: 11160:standard deviation 11124: 11106: 11060: 11041: 10988: 10969: 10926: 10907: 10905: 10503: 10452: 10450: 10328: 10309: 10223: 10196: 10162: 10127: 10108: 10065: 10010: 10008: 9761: 9759: 9586: 9544: 9481: 9460: 9388: 9369: 9241: 9194: 9192: 8883: 8803: 8801: 8691: 8638: 8566: 8504: 7743:1 / 100 7474:Proportion within 7462: 7451: 7445:Percentage within( 7438:, are as follows: 7408:the empirical rule 7354: 7202:, is given by the 7184: 7183: 7160: 7072: 6940: 6909: 6888: 6846: 6805: 6776: 6747: 6718: 6689: 6660: 6631: 6606:Distance from mean 6550: 6505: 6450: 6448: 6399: 6346: 6298: 6033: 6005: 5961: 5941: 5916: 5831: 5530: 5491: 5405: 5320: 5181: 5070: 4902: 4901: 4880: 4879: 4841: 4726: 4724: 4467:degrees of freedom 4447:sample population 4397: 4247: 4218: 4044: 3826: 3683: 3509: 3474: 3432: 3314: 3256: 3255: 3175:degrees of freedom 3157: 3006: 2885:mean squared error 2873: 2836:squared deviations 2820: 2791: 2721: 2613:, is known as the 2503: 2399:definite integrals 2385: 2220: 2017: 1859: 1630:In the case where 1608: 1542: 1463: 1394: 1316: 1271: 1134: 1114: 1088: 1065: 1041: 978: 936: 827: 825: 408: 306: 69:standard deviation 61: 53: 14461: 14460: 14399: 14398: 14395: 14394: 14334:National accounts 14304:Actuarial science 14296:Social statistics 14189: 14188: 14185: 14184: 14181: 14180: 14116:Survival function 14101: 14100: 13963:Granger causality 13804:Contingency table 13779:Survival analysis 13756: 13755: 13752: 13751: 13608:Linear regression 13503: 13502: 13499: 13498: 13474:Credible interval 13443: 13442: 13226: 13225: 13042:Method of moments 12911:Parametric family 12872:Statistical model 12802: 12801: 12798: 12797: 12716:Random assignment 12638:Statistical power 12572: 12571: 12568: 12567: 12417:Contingency table 12387: 12386: 12254:Generalized/power 12087:978-0-8036-3796-2 12001:978-0-19-920613-1 11240:Higher dimensions 11225: 11224: 11219: 11211: 11200: 11104: 11036: 10966: 10751: 10606: 10464:and places where 10233:, the power sums 10160: 10106: 10058:Sample variance: 9864: 9755: 9581: 9580: 9479: 9475: 9236: 9235: 9220: 9170: 9142: 9079: 9005: 8949: 8918: 8633: 8632: 8617: 8601:distribution by: 8559: 8499: 8443: 8368: 8367: 7686:1 / 20 7664:1 / 10 7340: 7339: 7307: 7285: 7282: 7237: 7218: 7154: 7153: 7129: 7058: 7035: 7017: 7014: 6936:inflection points 6922: 6921: 6882: 6881: 6844: 6625: 6548: 6534: 6496: 6464: 6463: 6433: 6390: 6388: 6337: 6289: 5964:{\displaystyle L} 5944:{\displaystyle M} 5908: 5893: 5878: 5823: 5808: 5793: 5634:percentage points 5486: 5442: 5382: 5362: 5353: 5292: 5277: 5263: 5212: 5176: 5122: 5121: 5065: 5003: 4835: 4375: 4372: 4327: 4323: 4194: 4170: 4139: 4039: 4025: 3974: 3961: 3927: 3821: 3807: 3756: 3732: 3678: 3671: 3640: 3617: 3616: 3519:becomes smaller. 3507: 3472: 3427: 3413: 3362: 3312: 3290: 3246: 3210: 3140: 3089: 3001: 2987: 2936: 2817: 2716: 2702: 2651: 2498: 2412:In the case of a 2337: 2336: where 2329: 2168: 2167: where 2160: 2074:have probability 2056:have probability 2038:have probability 1981: 1965: 1964: where 1957: 1905: 1819: 1803: 1802: where 1795: 1684: 1537: 1458: 1387: 1222:(the average) of 1192:63–75 inches 1184:66–72 inches 1033: 970: 928: 915: 400: 387: 302: 293: 284: 275: 266: 257: 248: 87:(also called the 79:. A low standard 16:(Redirected from 14489: 14449: 14448: 14437: 14436: 14426: 14425: 14411: 14410: 14314:Crime statistics 14208: 14207: 14195: 14194: 14112: 14111: 14078:Fourier analysis 14065:Frequency domain 14045: 13992: 13958:Structural break 13918: 13917: 13867:Cluster analysis 13814:Log-linear model 13787: 13786: 13762: 13761: 13703: 13677:Homoscedasticity 13533: 13532: 13509: 13508: 13428: 13420: 13412: 13411:(Kruskal–Wallis) 13396: 13381: 13336:Cross validation 13321: 13303:Anderson–Darling 13250: 13237: 13236: 13208:Likelihood-ratio 13200:Parametric tests 13178:Permutation test 13161:1- & 2-tails 13052:Minimum distance 13024:Point estimation 13020: 13019: 12971:Optimal decision 12922: 12821: 12820: 12808: 12807: 12790:Quasi-experiment 12740:Adaptive designs 12591: 12590: 12578: 12577: 12455:Rank correlation 12217: 12216: 12208: 12207: 12195: 12194: 12162: 12155: 12148: 12139: 12138: 12127: 12100: 12099: 12073: 12067: 12066: 12058: 12052: 12051: 12049: 12012: 12006: 12005: 11993: 11980: 11974: 11973: 11963: 11943: 11937: 11936: 11934: 11932: 11916: 11910: 11909: 11893: 11883: 11877: 11876: 11874: 11872: 11861: 11855: 11854: 11821: 11801: 11795: 11794: 11792: 11790: 11785:on 25 March 2016 11775: 11769: 11768: 11732: 11726: 11725: 11705: 11699: 11698: 11696: 11694: 11679: 11673: 11672: 11645: 11639: 11638: 11636: 11634: 11628:www.statlect.com 11620: 11614: 11613: 11611: 11609: 11594: 11588: 11587: 11585: 11575: 11569: 11568: 11566: 11564: 11550: 11544: 11543: 11542: 11525: 11519: 11518: 11507: 11501: 11500: 11485: 11479: 11478: 11468: 11436: 11418:Yamartino method 11383:Root mean square 11278: 11273: 11272: 11236: 11234: 11233: 11228: 11226: 11222: 11221: 11220: 11217: 11212: 11209: 11206: 11201: 11198: 11146: 11145: 11139: 11133: 11131: 11130: 11125: 11119: 11114: 11105: 11103: 11096: 11087: 11079: 11073: 11068: 11050: 11048: 11047: 11042: 11037: 11035: 11028: 11027: 11017: 11016: 11007: 11001: 10996: 10978: 10976: 10975: 10970: 10967: 10965: 10964: 10955: 10954: 10945: 10939: 10934: 10916: 10914: 10913: 10908: 10906: 10902: 10898: 10897: 10896: 10884: 10883: 10869: 10865: 10864: 10863: 10845: 10844: 10830: 10829: 10817: 10816: 10798: 10797: 10792: 10788: 10787: 10786: 10768: 10767: 10752: 10750: 10749: 10740: 10739: 10738: 10723: 10722: 10712: 10707: 10706: 10684: 10683: 10660: 10659: 10646: 10642: 10641: 10640: 10622: 10621: 10607: 10605: 10604: 10595: 10594: 10585: 10580: 10579: 10557: 10556: 10533: 10532: 10512: 10510: 10509: 10504: 10502: 10501: 10492: 10487: 10486: 10470: 10461: 10459: 10458: 10453: 10451: 10447: 10446: 10434: 10433: 10411: 10410: 10387: 10386: 10366: 10362: 10352: 10348: 10337: 10335: 10334: 10329: 10322: 10317: 10308: 10307: 10297: 10292: 10274: 10273: 10255: 10244: 10232: 10230: 10229: 10224: 10222: 10221: 10205: 10203: 10202: 10197: 10195: 10194: 10179:When the values 10171: 10169: 10168: 10163: 10161: 10156: 10155: 10146: 10140: 10135: 10117: 10115: 10114: 10109: 10107: 10105: 10094: 10093: 10084: 10078: 10073: 10054: 10038: 10031: 10019: 10017: 10016: 10011: 10009: 10005: 10001: 10000: 9999: 9987: 9986: 9972: 9968: 9967: 9966: 9948: 9947: 9930: 9929: 9911: 9910: 9905: 9901: 9900: 9899: 9881: 9880: 9865: 9860: 9849: 9844: 9843: 9821: 9820: 9797: 9796: 9776: 9770: 9768: 9767: 9762: 9760: 9756: 9751: 9750: 9749: 9731: 9730: 9720: 9715: 9714: 9692: 9691: 9668: 9667: 9645: 9632: 9628: 9624: 9608: 9595: 9593: 9592: 9587: 9582: 9579: 9559: 9557: 9552: 9540: 9539: 9526: 9525: 9505: 9496: 9490: 9488: 9487: 9482: 9480: 9473: 9468: 9456: 9455: 9443: 9442: 9421: 9412: 9403: 9397: 9395: 9394: 9389: 9384: 9382: 9377: 9366: 9361: 9343: 9342: 9324: 9306: 9302: 9298: 9289: 9265: 9261: 9250: 9248: 9247: 9242: 9237: 9231: 9227: 9222: 9221: 9218: 9203: 9201: 9200: 9195: 9193: 9171: 9163: 9143: 9141: 9140: 9128: 9120: 9119: 9100: 9095: 9080: 9078: 9077: 9065: 9057: 9053: 9049: 9048: 9047: 9037: 9032: 9006: 9004: 9003: 8991: 8986: 8982: 8981: 8980: 8970: 8965: 8950: 8942: 8919: 8916: 8892: 8890: 8889: 8884: 8879: 8878: 8859: 8858: 8843: 8842: 8812: 8810: 8809: 8804: 8802: 8795: 8794: 8770: 8769: 8741: 8740: 8728: 8727: 8704: 8699: 8653: 8647: 8645: 8644: 8639: 8634: 8628: 8624: 8619: 8618: 8615: 8575: 8573: 8572: 8567: 8561: 8560: 8552: 8534: 8513: 8511: 8510: 8505: 8500: 8498: 8497: 8492: 8488: 8481: 8480: 8464: 8459: 8444: 8442: 8428: 8426: 8400: 8364: 8363: 8360: 8357: 8348: 8347: 8344: 8341: 8334: 8333: 8330: 8327: 8320: 8312: 8311: 8308: 8305: 8296: 8295: 8292: 8284: 8283: 8280: 8273: 8270: 8269: 8260: 8259: 8256: 8253: 8244: 8243: 8240: 8232: 8231: 8228: 8221: 8218: 8217: 8208: 8207: 8204: 8201: 8192: 8191: 8183: 8182: 8175: 8172: 8171: 8162: 8161: 8158: 8149: 8148: 8145: 8137: 8136: 8133: 8125: 8122: 8114: 8113: 8110: 8101: 8100: 8092: 8091: 8084: 8081: 8080: 8071: 8070: 8067: 8058: 8057: 8049: 8048: 8041: 8038: 8037: 8028: 8027: 8024: 8015: 8014: 8011: 8003: 8002: 7999: 7992: 7984: 7983: 7980: 7971: 7965: 7960: 7957: 7956: 7947: 7946: 7943: 7938:6.8 / 7936: 7934: 7926: 7925: 7922: 7919: 7912: 7911: 7908: 7905: 7898: 7895: 7888: 7887: 7873: 7870: 7869: 7860: 7859: 7850: 7849: 7841: 7840: 7833: 7825: 7824: 7810: 7807: 7806: 7797: 7785: 7782: 7781: 7768: 7767: 7759: 7758: 7751: 7734: 7731: 7730: 7721: 7720: 7711: 7710: 7702: 7701: 7694: 7677: 7674: 7673: 7655: 7652: 7651: 7642:1 / 5 7633: 7630: 7629: 7620: 7619: 7610: 7609: 7601: 7600: 7593: 7576: 7573: 7572: 7563:1 / 3 7554: 7551: 7550: 7541: 7534: 7528: 7523: 7520: 7519: 7501: 7498: 7497: 7466: 7465: 7437: 7416: 7399: 7389: 7379: 7375: 7363: 7361: 7360: 7355: 7350: 7346: 7345: 7341: 7335: 7331: 7308: 7300: 7295: 7291: 7290: 7286: 7284: 7283: 7278: 7272: 7261: 7238: 7230: 7219: 7216: 7201: 7193: 7191: 7190: 7185: 7169: 7167: 7166: 7161: 7159: 7155: 7149: 7145: 7130: 7127: 7116: 7105: 7101: 7095: 7087: 7081: 7079: 7078: 7073: 7071: 7070: 7069: 7068: 7063: 7059: 7054: 7043: 7036: 7028: 7018: 7016: 7015: 7007: 6998: 6993: 6989: 6988: 6987: 6918: 6916: 6915: 6910: 6897: 6895: 6894: 6889: 6883: 6871: 6867: 6855: 6853: 6852: 6847: 6845: 6843: 6842: 6830: 6814: 6812: 6811: 6806: 6785: 6783: 6782: 6777: 6756: 6754: 6753: 6748: 6727: 6725: 6724: 6719: 6698: 6696: 6695: 6690: 6669: 6667: 6666: 6661: 6640: 6638: 6637: 6632: 6626: 6621: 6603: 6602: 6584: 6559: 6557: 6556: 6551: 6549: 6547: 6546: 6541: 6537: 6536: 6535: 6527: 6521: 6520: 6504: 6495: 6486: 6482: 6474: 6470: 6459: 6457: 6456: 6451: 6449: 6435: 6434: 6426: 6409: 6408: 6398: 6389: 6381: 6356: 6355: 6345: 6322: 6318: 6308: 6307: 6297: 6257: 6256: 6238: 6237: 6219: 6218: 6180: 6179: 6161: 6160: 6142: 6141: 6057: 6053: 6049: 6042: 6040: 6039: 6034: 6032: 6014: 6012: 6011: 6006: 5970: 5968: 5967: 5962: 5950: 5948: 5947: 5942: 5925: 5923: 5922: 5917: 5915: 5911: 5910: 5909: 5901: 5895: 5894: 5886: 5880: 5879: 5871: 5847: 5846: 5840: 5838: 5837: 5832: 5830: 5826: 5825: 5824: 5816: 5810: 5809: 5801: 5795: 5794: 5786: 5763: 5759: 5755: 5751: 5743: 5739: 5735: 5708: 5702: 5674: 5585:Particle physics 5500: 5498: 5497: 5492: 5487: 5485: 5484: 5479: 5475: 5474: 5473: 5463: 5458: 5443: 5435: 5424: 5420: 5418: 5413: 5403: 5398: 5383: 5375: 5368: 5363: 5361: 5360: 5355: 5354: 5346: 5339: 5335: 5333: 5328: 5318: 5313: 5293: 5285: 5283: 5278: 5276: 5275: 5270: 5266: 5265: 5264: 5256: 5250: 5249: 5233: 5228: 5213: 5205: 5203: 5190: 5188: 5187: 5182: 5177: 5175: 5171: 5170: 5169: 5125: 5123: 5120: 5106: 5105: 5079: 5077: 5076: 5071: 5066: 5064: 5063: 5033: 5029: 5028: 5009: 5004: 5002: 4998: 4997: 4996: 4952: 4926: 4916:, respectively. 4911: 4909: 4908: 4903: 4889: 4887: 4886: 4881: 4878: 4877: 4850: 4848: 4847: 4842: 4836: 4773: 4735: 4733: 4732: 4727: 4725: 4706: 4698: 4590: 4586: 4582: 4562: 4558: 4539: 4524: 4513: 4508: 4501: 4491: 4487: 4483: 4464: 4453: 4445: 4435: 4425: 4415: 4406: 4404: 4403: 4398: 4381: 4377: 4376: 4374: 4373: 4365: 4359: 4358: 4349: 4341: 4340: 4328: 4326: 4325: 4324: 4316: 4303: 4302: 4293: 4271: 4264: 4260: 4256: 4254: 4253: 4248: 4246: 4245: 4227: 4225: 4224: 4219: 4202: 4198: 4197: 4196: 4195: 4187: 4171: 4169: 4168: 4159: 4158: 4149: 4141: 4140: 4132: 4105: 4064: 4053: 4051: 4050: 4045: 4040: 4038: 4037: 4032: 4028: 4027: 4026: 4018: 4012: 4011: 3995: 3990: 3975: 3973: 3972: 3971: 3962: 3954: 3936: 3934: 3929: 3928: 3920: 3907: 3906: 3904: 3903: 3896: 3893: 3881: 3871: 3864: 3857: 3856: 3854: 3853: 3848: 3845: 3835: 3833: 3832: 3827: 3822: 3820: 3819: 3814: 3810: 3809: 3808: 3800: 3794: 3793: 3777: 3772: 3757: 3755: 3741: 3739: 3734: 3733: 3725: 3715: 3714: − 1.5 3708: 3697:chi distribution 3692: 3690: 3689: 3684: 3679: 3677: 3676: 3672: 3667: 3656: 3646: 3645: 3641: 3633: 3623: 3618: 3615: 3601: 3600: 3584: 3583: 3563: 3559: 3558: 3556: 3555: 3547: 3544: 3534: 3518: 3516: 3515: 3510: 3508: 3506: 3492: 3483: 3481: 3480: 3475: 3473: 3465: 3441: 3439: 3438: 3433: 3428: 3426: 3425: 3420: 3416: 3415: 3414: 3406: 3400: 3399: 3383: 3378: 3363: 3361: 3347: 3345: 3324:), yielding the 3323: 3321: 3320: 3315: 3313: 3299: 3291: 3286: 3265: 3263: 3262: 3257: 3248: 3247: 3239: 3233: 3232: 3212: 3211: 3203: 3197: 3196: 3166: 3164: 3163: 3158: 3153: 3152: 3147: 3143: 3142: 3141: 3133: 3127: 3126: 3110: 3105: 3090: 3088: 3074: 3069: 3068: 3024:concave function 3015: 3013: 3012: 3007: 3002: 3000: 2999: 2994: 2990: 2989: 2988: 2980: 2974: 2973: 2957: 2952: 2937: 2929: 2927: 2922: 2921: 2882: 2880: 2879: 2874: 2851:biased estimator 2829: 2827: 2826: 2821: 2819: 2818: 2810: 2800: 2798: 2797: 2792: 2787: 2786: 2766: 2765: 2752: 2751: 2730: 2728: 2727: 2722: 2717: 2715: 2714: 2709: 2705: 2704: 2703: 2695: 2689: 2688: 2672: 2667: 2652: 2644: 2642: 2637: 2636: 2512: 2510: 2509: 2504: 2499: 2497: 2496: 2495: 2494: 2471: 2467: 2460: 2459: 2458: 2457: 2438: 2429: 2423: 2420:with parameters 2408: 2404: 2394: 2392: 2391: 2386: 2378: 2356: 2355: 2354: 2338: 2335: 2330: 2325: 2306: 2305: 2284: 2283: 2282: 2272: 2257: 2243: 2229: 2227: 2226: 2221: 2216: 2215: 2206: 2205: 2195: 2190: 2169: 2166: 2161: 2159: 2158: 2143: 2142: 2130: 2129: 2119: 2114: 2099: 2084: 2073: 2055: 2046: 2037: 2026: 2024: 2023: 2018: 2013: 2012: 2002: 1997: 1982: 1974: 1966: 1963: 1958: 1956: 1955: 1940: 1939: 1926: 1921: 1906: 1898: 1896: 1868: 1866: 1865: 1860: 1852: 1851: 1833: 1832: 1820: 1812: 1804: 1801: 1796: 1794: 1790: 1789: 1788: 1773: 1772: 1751: 1750: 1735: 1734: 1719: 1718: 1703: 1702: 1685: 1677: 1675: 1658: 1633: 1617: 1615: 1614: 1609: 1561: 1551: 1549: 1548: 1543: 1538: 1536: 1535: 1505: 1501: 1500: 1481: 1472: 1470: 1469: 1464: 1459: 1454: 1436: 1435: 1413: 1405: 1393: 1388: 1386: 1382: 1381: 1380: 1348: 1334:is defined as 1333: 1329: 1325: 1323: 1322: 1317: 1312: 1290: 1282: 1239: 1237: 1228: 1193: 1189: 1185: 1177: 1173: 1143: 1141: 1140: 1135: 1123: 1121: 1120: 1117:{\textstyle n-1} 1115: 1097: 1095: 1094: 1089: 1074: 1072: 1071: 1066: 1050: 1048: 1047: 1042: 1034: 1029: 1021: 1006: 998: 987: 985: 984: 979: 971: 966: 945: 943: 942: 937: 929: 921: 916: 911: 864: 859: 858: 836: 834: 833: 828: 826: 816: 815: 803: 802: 780: 772: 771: 750: 749: 718: 717: 705: 704: 682: 674: 673: 652: 651: 620: 619: 607: 606: 584: 576: 575: 554: 553: 522: 521: 509: 508: 486: 478: 477: 456: 455: 417: 415: 414: 409: 401: 393: 388: 383: 336: 322:(average) of 5: 315: 313: 312: 307: 300: 291: 282: 273: 264: 255: 246: 21: 14497: 14496: 14492: 14491: 14490: 14488: 14487: 14486: 14467: 14466: 14464: 14462: 14457: 14420: 14391: 14353: 14290: 14276:quality control 14243: 14225:Clinical trials 14202: 14177: 14161: 14149:Hazard function 14143: 14097: 14059: 14043: 14006: 14002:Breusch–Godfrey 13990: 13967: 13907: 13882:Factor analysis 13828: 13809:Graphical model 13781: 13748: 13715: 13701: 13681: 13635: 13602: 13564: 13527: 13526: 13495: 13439: 13426: 13418: 13410: 13394: 13379: 13358:Rank statistics 13352: 13331:Model selection 13319: 13277:Goodness of fit 13271: 13248: 13222: 13194: 13147: 13092: 13081:Median unbiased 13009: 12920: 12853:Order statistic 12815: 12794: 12761: 12735: 12687: 12642: 12585: 12583:Data collection 12564: 12476: 12431: 12405: 12383: 12343: 12295: 12212:Continuous data 12202: 12189: 12171: 12166: 12112: 12109: 12104: 12103: 12088: 12074: 12070: 12059: 12055: 12013: 12009: 12002: 11981: 11977: 11961:10.1.1.302.7503 11944: 11940: 11930: 11928: 11917: 11913: 11906: 11884: 11880: 11870: 11868: 11863: 11862: 11858: 11802: 11798: 11788: 11786: 11777: 11776: 11772: 11733: 11729: 11706: 11702: 11692: 11690: 11687:PureCalculators 11681: 11680: 11676: 11662:10.2307/2682923 11646: 11642: 11632: 11630: 11622: 11621: 11617: 11607: 11605: 11595: 11591: 11583: 11577: 11576: 11572: 11562: 11560: 11552: 11551: 11547: 11526: 11522: 11508: 11504: 11486: 11482: 11437: 11433: 11428: 11423: 11353:Pooled variance 11285:68–95–99.7 rule 11274: 11267: 11264: 11242: 11216: 11210:Laboratory mean 11208: 11207: 11205: 11197: 11195: 11192: 11191: 11180: 11156: 11143: 11141: 11137: 11115: 11110: 11089: 11088: 11080: 11078: 11069: 11064: 11058: 11055: 11054: 11023: 11019: 11018: 11012: 11008: 11006: 10997: 10992: 10986: 10983: 10982: 10960: 10956: 10950: 10946: 10944: 10935: 10930: 10924: 10921: 10920: 10904: 10903: 10892: 10888: 10879: 10875: 10874: 10870: 10853: 10849: 10840: 10836: 10835: 10831: 10825: 10821: 10806: 10802: 10793: 10776: 10772: 10763: 10759: 10758: 10754: 10753: 10745: 10741: 10728: 10724: 10718: 10714: 10713: 10711: 10696: 10692: 10685: 10679: 10675: 10672: 10671: 10661: 10655: 10651: 10648: 10647: 10630: 10626: 10617: 10613: 10612: 10608: 10600: 10596: 10590: 10586: 10584: 10569: 10565: 10558: 10552: 10548: 10545: 10544: 10534: 10528: 10524: 10520: 10518: 10515: 10514: 10497: 10493: 10488: 10482: 10478: 10476: 10473: 10472: 10469: 10465: 10449: 10448: 10442: 10438: 10423: 10419: 10412: 10406: 10402: 10399: 10398: 10388: 10382: 10378: 10374: 10372: 10369: 10368: 10364: 10360: 10350: 10347: 10344: 10341: 10318: 10313: 10303: 10299: 10293: 10282: 10269: 10265: 10263: 10260: 10259: 10254: 10251: 10247: 10242: 10240: 10237: 10234: 10217: 10213: 10211: 10208: 10207: 10190: 10186: 10184: 10181: 10180: 10177: 10151: 10147: 10145: 10136: 10131: 10125: 10122: 10121: 10095: 10089: 10085: 10083: 10074: 10069: 10063: 10060: 10059: 10053: 10050: 10046: 10043: 10040: 10036: 10033: 10029: 10026: 10023: 10007: 10006: 9995: 9991: 9982: 9978: 9977: 9973: 9956: 9952: 9943: 9939: 9938: 9934: 9919: 9915: 9906: 9889: 9885: 9876: 9872: 9871: 9867: 9866: 9850: 9848: 9833: 9829: 9822: 9816: 9812: 9809: 9808: 9798: 9792: 9788: 9784: 9782: 9779: 9778: 9774: 9758: 9757: 9739: 9735: 9726: 9722: 9721: 9719: 9704: 9700: 9693: 9687: 9683: 9680: 9679: 9669: 9663: 9659: 9655: 9653: 9650: 9649: 9644: 9640: 9637: 9630: 9629:data points as 9626: 9622: 9611:round-off error 9607: 9606: 9602: 9599: 9560: 9553: 9548: 9535: 9531: 9527: 9524: 9516: 9513: 9512: 9504: 9501: 9498: 9494: 9469: 9464: 9451: 9447: 9441: 9433: 9430: 9429: 9420: 9417: 9414: 9411: 9408: 9405: 9401: 9378: 9373: 9368: 9362: 9351: 9338: 9334: 9332: 9329: 9328: 9323: 9322: 9318: 9314: 9311: 9308: 9304: 9300: 9297: 9294: 9291: 9288: 9285: 9282: 9279: 9273: 9263: 9260: 9257: 9254: 9226: 9217: 9213: 9211: 9208: 9207: 9191: 9190: 9162: 9136: 9132: 9127: 9115: 9111: 9096: 9085: 9073: 9069: 9064: 9055: 9054: 9043: 9039: 9033: 9022: 9017: 9013: 8999: 8995: 8990: 8976: 8972: 8966: 8955: 8941: 8940: 8936: 8923: 8915: 8902: 8900: 8897: 8896: 8874: 8870: 8854: 8850: 8838: 8834: 8820: 8817: 8816: 8800: 8799: 8790: 8786: 8765: 8761: 8745: 8736: 8732: 8723: 8719: 8707: 8706: 8700: 8695: 8684: 8665: 8663: 8660: 8659: 8651: 8623: 8614: 8610: 8608: 8605: 8604: 8598: 8592: 8551: 8550: 8542: 8539: 8538: 8532: 8528: 8525: 8493: 8476: 8472: 8471: 8467: 8466: 8460: 8449: 8432: 8427: 8425: 8408: 8405: 8404: 8399: 8398: 8394: 8390: 8387: 8384: 8373: 8361: 8358: 8355: 8353: 8345: 8342: 8339: 8337: 8331: 8328: 8325: 8323: 8318: 8309: 8306: 8303: 8301: 8293: 8290: 8288: 8281: 8278: 8276: 8271: 8267: 8265: 8257: 8254: 8251: 8249: 8241: 8238: 8236: 8229: 8226: 8224: 8219: 8215: 8213: 8205: 8202: 8199: 8197: 8189: 8187: 8180: 8178: 8173: 8169: 8167: 8159: 8156: 8154: 8146: 8143: 8141: 8134: 8131: 8129: 8123: 8120: 8111: 8108: 8106: 8098: 8096: 8089: 8087: 8082: 8078: 8076: 8068: 8065: 8063: 8055: 8053: 8046: 8044: 8039: 8035: 8033: 8025: 8022: 8020: 8012: 8009: 8007: 8000: 7997: 7995: 7990: 7981: 7978: 7976: 7969: 7963: 7958: 7954: 7952: 7944: 7941: 7939: 7937: 7932: 7930: 7923: 7920: 7917: 7915: 7909: 7906: 7903: 7901: 7896: 7893: 7885: 7883: 7871: 7867: 7865: 7857: 7855: 7847: 7845: 7838: 7836: 7831: 7822: 7820: 7808: 7804: 7802: 7795: 7783: 7779: 7777: 7765: 7763: 7756: 7754: 7749: 7732: 7728: 7726: 7718: 7716: 7708: 7706: 7699: 7697: 7692: 7675: 7671: 7669: 7653: 7649: 7647: 7631: 7627: 7625: 7617: 7615: 7607: 7605: 7598: 7596: 7591: 7574: 7570: 7568: 7552: 7548: 7546: 7539: 7532: 7526: 7521: 7517: 7515: 7499: 7495: 7493: 7470: 7435: 7432: 7428: 7425: 7421: 7418: 7414: 7403:68–95–99.7 rule 7398: 7394: 7391: 7388: 7384: 7381: 7377: 7374: 7370: 7367: 7330: 7326: 7313: 7309: 7299: 7277: 7273: 7262: 7260: 7256: 7243: 7239: 7229: 7215: 7213: 7210: 7209: 7199: 7177: 7174: 7173: 7144: 7140: 7126: 7124: 7121: 7120: 7114: 7103: 7100: 7097: 7093: 7085: 7064: 7044: 7042: 7038: 7037: 7027: 7023: 7019: 7006: 7002: 6997: 6983: 6979: 6966: 6962: 6957: 6954: 6953: 6927: 6904: 6901: 6900: 6866: 6864: 6861: 6860: 6838: 6834: 6829: 6821: 6818: 6817: 6797: 6794: 6793: 6768: 6765: 6764: 6739: 6736: 6735: 6710: 6707: 6706: 6681: 6678: 6677: 6652: 6649: 6648: 6620: 6618: 6615: 6614: 6597: 6591: 6582: 6579: 6575: 6572: 6568: 6565: 6561: 6542: 6526: 6525: 6516: 6512: 6511: 6507: 6506: 6500: 6494: 6492: 6489: 6488: 6484: 6480: 6472: 6468: 6465: 6447: 6446: 6436: 6425: 6424: 6421: 6420: 6410: 6404: 6400: 6394: 6380: 6377: 6376: 6366: 6351: 6347: 6341: 6334: 6333: 6323: 6303: 6299: 6293: 6288: 6284: 6278: 6277: 6267: 6252: 6248: 6233: 6229: 6214: 6210: 6201: 6200: 6190: 6175: 6171: 6156: 6152: 6137: 6133: 6103: 6102: 6092: 6067: 6065: 6062: 6061: 6055: 6051: 6047: 6028: 6020: 6017: 6016: 5976: 5973: 5972: 5956: 5953: 5952: 5936: 5933: 5932: 5926: 5900: 5899: 5885: 5884: 5870: 5869: 5868: 5864: 5856: 5853: 5852: 5815: 5814: 5800: 5799: 5785: 5784: 5783: 5779: 5771: 5768: 5767: 5761: 5757: 5753: 5749: 5741: 5737: 5729: 5725: 5721: 5717: 5713: 5710: 5704: 5700: 5697: 5693: 5690: 5686: 5683: 5679: 5676: 5673: 5670: 5666: 5663: 5659: 5656: 5653: 5650: 5618: 5609: 5577: 5569: 5522: 5512: 5480: 5469: 5465: 5459: 5448: 5434: 5433: 5429: 5428: 5414: 5409: 5399: 5388: 5374: 5373: 5369: 5367: 5356: 5345: 5344: 5343: 5329: 5324: 5314: 5303: 5298: 5294: 5284: 5282: 5271: 5255: 5254: 5245: 5241: 5240: 5236: 5235: 5229: 5218: 5204: 5202: 5200: 5197: 5196: 5165: 5161: 5136: 5132: 5124: 5110: 5104: 5087: 5084: 5083: 5059: 5055: 5024: 5020: 5016: 5008: 4992: 4988: 4963: 4959: 4951: 4934: 4931: 4930: 4924: 4895: 4892: 4891: 4873: 4869: 4858: 4855: 4854: 4772: 4749: 4746: 4745: 4723: 4722: 4702: 4694: 4687: 4669: 4668: 4646: 4625: 4624: 4614: 4598: 4596: 4593: 4592: 4588: 4584: 4580: 4569: 4560: 4556: 4552: 4548: 4544: 4541: 4537: 4533: 4530: 4522: 4518: 4515: 4511: 4506: 4503: 4500: 4496: 4493: 4489: 4485: 4481: 4478: 4475: 4462: 4451: 4448: 4443: 4440: 4433: 4430: 4427: 4423: 4420: 4417: 4413: 4410: 4364: 4360: 4354: 4350: 4348: 4336: 4332: 4315: 4308: 4304: 4298: 4294: 4292: 4288: 4284: 4279: 4276: 4275: 4270: 4266: 4262: 4258: 4241: 4237: 4235: 4232: 4231: 4186: 4179: 4175: 4164: 4160: 4154: 4150: 4148: 4131: 4127: 4126: 4122: 4117: 4114: 4113: 4103: 4100: 4089: 4079:Margin of error 4075: 4067:excess kurtosis 4063: 4060: 4057: 4033: 4017: 4016: 4007: 4003: 4002: 3998: 3997: 3991: 3980: 3967: 3963: 3953: 3940: 3935: 3933: 3919: 3918: 3916: 3913: 3912: 3901: 3897: 3894: 3891: 3890: 3888: 3886: 3883: 3879: 3876: 3869: 3866: 3862: 3859: 3852: 3849: 3846: 3843: 3842: 3840: 3839: 3815: 3799: 3798: 3789: 3785: 3784: 3780: 3779: 3773: 3762: 3745: 3740: 3738: 3724: 3723: 3721: 3718: 3717: 3713: 3710: 3706: 3703: 3657: 3655: 3651: 3647: 3632: 3628: 3624: 3622: 3605: 3599: 3579: 3575: 3573: 3570: 3569: 3561: 3554: 3551: 3548: 3545: 3543: 3540: 3539: 3537: 3536: 3532: 3525: 3496: 3491: 3489: 3486: 3485: 3464: 3462: 3459: 3458: 3421: 3405: 3404: 3395: 3391: 3390: 3386: 3385: 3379: 3368: 3351: 3346: 3344: 3336: 3333: 3332: 3298: 3285: 3277: 3274: 3273: 3238: 3237: 3228: 3224: 3202: 3201: 3192: 3188: 3182: 3179: 3178: 3148: 3132: 3131: 3122: 3118: 3117: 3113: 3112: 3106: 3095: 3078: 3073: 3064: 3060: 3058: 3055: 3054: 2995: 2979: 2978: 2969: 2965: 2964: 2960: 2959: 2953: 2942: 2928: 2926: 2917: 2913: 2911: 2908: 2907: 2899:sample variance 2893: 2862: 2859: 2858: 2809: 2808: 2806: 2803: 2802: 2782: 2778: 2761: 2757: 2747: 2743: 2738: 2735: 2734: 2710: 2694: 2693: 2684: 2680: 2679: 2675: 2674: 2668: 2657: 2643: 2641: 2632: 2628: 2626: 2623: 2622: 2612: 2597: 2532: 2526: 2524:Sample variance 2518: 2490: 2486: 2476: 2472: 2453: 2449: 2448: 2444: 2443: 2439: 2437: 2435: 2432: 2431: 2428: 2425: 2421: 2406: 2402: 2374: 2350: 2349: 2345: 2334: 2321: 2301: 2297: 2278: 2277: 2273: 2271: 2263: 2260: 2259: 2255: 2251: 2248: 2241: 2235: 2211: 2207: 2201: 2197: 2191: 2180: 2165: 2154: 2150: 2138: 2134: 2125: 2121: 2115: 2104: 2098: 2090: 2087: 2086: 2083: 2082: 2078: 2075: 2072: 2071: 2067: 2063: 2060: 2057: 2054: 2051: 2048: 2045: 2042: 2039: 2036: 2033: 2030: 2008: 2004: 1998: 1987: 1973: 1962: 1951: 1947: 1935: 1931: 1922: 1911: 1897: 1895: 1887: 1884: 1883: 1847: 1843: 1828: 1824: 1811: 1800: 1784: 1780: 1768: 1764: 1746: 1742: 1730: 1726: 1714: 1710: 1698: 1694: 1690: 1686: 1676: 1674: 1666: 1663: 1662: 1657: 1656: 1652: 1648: 1645: 1641: 1638: 1635: 1631: 1628: 1585: 1582: 1581: 1580:with parameter 1559: 1531: 1527: 1496: 1492: 1488: 1480: 1478: 1475: 1474: 1450: 1431: 1427: 1406: 1398: 1392: 1376: 1372: 1359: 1355: 1347: 1339: 1336: 1335: 1331: 1327: 1308: 1283: 1275: 1245: 1242: 1241: 1235: 1233: 1230: 1226: 1224:random variable 1212: 1204:empirical rule, 1200:68–95–99.7 rule 1160: 1129: 1126: 1125: 1124:rather than by 1103: 1100: 1099: 1080: 1077: 1076: 1060: 1057: 1056: 1025: 1020: 1012: 1009: 1008: 1000: 992: 965: 957: 954: 953: 920: 865: 863: 854: 850: 848: 845: 844: 824: 823: 811: 807: 798: 794: 779: 767: 763: 745: 741: 726: 725: 713: 709: 700: 696: 681: 669: 665: 647: 643: 628: 627: 615: 611: 602: 598: 583: 571: 567: 549: 545: 530: 529: 517: 513: 504: 500: 485: 473: 469: 451: 447: 431: 429: 426: 425: 392: 337: 335: 327: 324: 323: 238: 235: 234: 219: 214: 175:margin of error 127:random variable 49:68–95–99.7 rule 35: 28: 23: 22: 15: 12: 11: 5: 14495: 14485: 14484: 14479: 14459: 14458: 14456: 14455: 14443: 14431: 14417: 14404: 14401: 14400: 14397: 14396: 14393: 14392: 14390: 14389: 14384: 14379: 14374: 14369: 14363: 14361: 14355: 14354: 14352: 14351: 14346: 14341: 14336: 14331: 14326: 14321: 14316: 14311: 14306: 14300: 14298: 14292: 14291: 14289: 14288: 14283: 14278: 14269: 14264: 14259: 14253: 14251: 14245: 14244: 14242: 14241: 14236: 14231: 14222: 14220:Bioinformatics 14216: 14214: 14204: 14203: 14191: 14190: 14187: 14186: 14183: 14182: 14179: 14178: 14176: 14175: 14169: 14167: 14163: 14162: 14160: 14159: 14153: 14151: 14145: 14144: 14142: 14141: 14136: 14131: 14126: 14120: 14118: 14109: 14103: 14102: 14099: 14098: 14096: 14095: 14090: 14085: 14080: 14075: 14069: 14067: 14061: 14060: 14058: 14057: 14052: 14047: 14039: 14034: 14029: 14028: 14027: 14025:partial (PACF) 14016: 14014: 14008: 14007: 14005: 14004: 13999: 13994: 13986: 13981: 13975: 13973: 13972:Specific tests 13969: 13968: 13966: 13965: 13960: 13955: 13950: 13945: 13940: 13935: 13930: 13924: 13922: 13915: 13909: 13908: 13906: 13905: 13904: 13903: 13902: 13901: 13886: 13885: 13884: 13874: 13872:Classification 13869: 13864: 13859: 13854: 13849: 13844: 13838: 13836: 13830: 13829: 13827: 13826: 13821: 13819:McNemar's test 13816: 13811: 13806: 13801: 13795: 13793: 13783: 13782: 13758: 13757: 13754: 13753: 13750: 13749: 13747: 13746: 13741: 13736: 13731: 13725: 13723: 13717: 13716: 13714: 13713: 13697: 13691: 13689: 13683: 13682: 13680: 13679: 13674: 13669: 13664: 13659: 13657:Semiparametric 13654: 13649: 13643: 13641: 13637: 13636: 13634: 13633: 13628: 13623: 13618: 13612: 13610: 13604: 13603: 13601: 13600: 13595: 13590: 13585: 13580: 13574: 13572: 13566: 13565: 13563: 13562: 13557: 13552: 13547: 13541: 13539: 13529: 13528: 13525: 13524: 13519: 13513: 13505: 13504: 13501: 13500: 13497: 13496: 13494: 13493: 13492: 13491: 13481: 13476: 13471: 13470: 13469: 13464: 13453: 13451: 13445: 13444: 13441: 13440: 13438: 13437: 13432: 13431: 13430: 13422: 13414: 13398: 13395:(Mann–Whitney) 13390: 13389: 13388: 13375: 13374: 13373: 13362: 13360: 13354: 13353: 13351: 13350: 13349: 13348: 13343: 13338: 13328: 13323: 13320:(Shapiro–Wilk) 13315: 13310: 13305: 13300: 13295: 13287: 13281: 13279: 13273: 13272: 13270: 13269: 13261: 13252: 13240: 13234: 13232:Specific tests 13228: 13227: 13224: 13223: 13221: 13220: 13215: 13210: 13204: 13202: 13196: 13195: 13193: 13192: 13187: 13186: 13185: 13175: 13174: 13173: 13163: 13157: 13155: 13149: 13148: 13146: 13145: 13144: 13143: 13138: 13128: 13123: 13118: 13113: 13108: 13102: 13100: 13094: 13093: 13091: 13090: 13085: 13084: 13083: 13078: 13077: 13076: 13071: 13056: 13055: 13054: 13049: 13044: 13039: 13028: 13026: 13017: 13011: 13010: 13008: 13007: 13002: 12997: 12996: 12995: 12985: 12980: 12979: 12978: 12968: 12967: 12966: 12961: 12956: 12946: 12941: 12936: 12935: 12934: 12929: 12924: 12908: 12907: 12906: 12901: 12896: 12886: 12885: 12884: 12879: 12869: 12868: 12867: 12857: 12856: 12855: 12845: 12840: 12835: 12829: 12827: 12817: 12816: 12804: 12803: 12800: 12799: 12796: 12795: 12793: 12792: 12787: 12782: 12777: 12771: 12769: 12763: 12762: 12760: 12759: 12754: 12749: 12743: 12741: 12737: 12736: 12734: 12733: 12728: 12723: 12718: 12713: 12708: 12703: 12697: 12695: 12689: 12688: 12686: 12685: 12683:Standard error 12680: 12675: 12670: 12669: 12668: 12663: 12652: 12650: 12644: 12643: 12641: 12640: 12635: 12630: 12625: 12620: 12615: 12613:Optimal design 12610: 12605: 12599: 12597: 12587: 12586: 12574: 12573: 12570: 12569: 12566: 12565: 12563: 12562: 12557: 12552: 12547: 12542: 12537: 12532: 12527: 12522: 12517: 12512: 12507: 12502: 12497: 12492: 12486: 12484: 12478: 12477: 12475: 12474: 12469: 12468: 12467: 12462: 12452: 12447: 12441: 12439: 12433: 12432: 12430: 12429: 12424: 12419: 12413: 12411: 12410:Summary tables 12407: 12406: 12404: 12403: 12397: 12395: 12389: 12388: 12385: 12384: 12382: 12381: 12380: 12379: 12374: 12369: 12359: 12353: 12351: 12345: 12344: 12342: 12341: 12336: 12331: 12326: 12321: 12316: 12311: 12305: 12303: 12297: 12296: 12294: 12293: 12288: 12283: 12282: 12281: 12276: 12271: 12266: 12261: 12256: 12251: 12246: 12244:Contraharmonic 12241: 12236: 12225: 12223: 12214: 12204: 12203: 12191: 12190: 12188: 12187: 12182: 12176: 12173: 12172: 12165: 12164: 12157: 12150: 12142: 12136: 12135: 12128: 12108: 12107:External links 12105: 12102: 12101: 12086: 12068: 12061:Miller, Jeff. 12053: 12007: 12000: 11984:Dodge, Yadolah 11975: 11954:(3): 419–420. 11938: 11911: 11904: 11878: 11856: 11796: 11770: 11743:(4): 293–298. 11727: 11700: 11689:. 11 July 2021 11674: 11640: 11615: 11589: 11570: 11545: 11520: 11502: 11480: 11451:(7047): 1654. 11430: 11429: 11427: 11424: 11422: 11421: 11415: 11410: 11408:Standard score 11405: 11403:Standard error 11400: 11395: 11390: 11385: 11380: 11375: 11370: 11365: 11360: 11355: 11350: 11345: 11340: 11334: 11329: 11324: 11318: 11313: 11308: 11303: 11297: 11292: 11287: 11281: 11280: 11279: 11263: 11260: 11241: 11238: 11215: 11204: 11179: 11176: 11155: 11152: 11123: 11118: 11113: 11109: 11102: 11099: 11095: 11092: 11086: 11083: 11077: 11072: 11067: 11063: 11040: 11034: 11031: 11026: 11022: 11015: 11011: 11005: 11000: 10995: 10991: 10963: 10959: 10953: 10949: 10943: 10938: 10933: 10929: 10901: 10895: 10891: 10887: 10882: 10878: 10873: 10868: 10862: 10859: 10856: 10852: 10848: 10843: 10839: 10834: 10828: 10824: 10820: 10815: 10812: 10809: 10805: 10801: 10796: 10791: 10785: 10782: 10779: 10775: 10771: 10766: 10762: 10757: 10748: 10744: 10737: 10734: 10731: 10727: 10721: 10717: 10710: 10705: 10702: 10699: 10695: 10691: 10688: 10686: 10682: 10678: 10674: 10673: 10670: 10667: 10664: 10662: 10658: 10654: 10650: 10649: 10645: 10639: 10636: 10633: 10629: 10625: 10620: 10616: 10611: 10603: 10599: 10593: 10589: 10583: 10578: 10575: 10572: 10568: 10564: 10561: 10559: 10555: 10551: 10547: 10546: 10543: 10540: 10537: 10535: 10531: 10527: 10523: 10522: 10500: 10496: 10491: 10485: 10481: 10467: 10445: 10441: 10437: 10432: 10429: 10426: 10422: 10418: 10415: 10413: 10409: 10405: 10401: 10400: 10397: 10394: 10391: 10389: 10385: 10381: 10377: 10376: 10345: 10342: 10326: 10321: 10316: 10312: 10306: 10302: 10296: 10291: 10288: 10285: 10281: 10277: 10272: 10268: 10252: 10249: 10245: 10238: 10235: 10220: 10216: 10193: 10189: 10176: 10173: 10159: 10154: 10150: 10144: 10139: 10134: 10130: 10104: 10101: 10098: 10092: 10088: 10082: 10077: 10072: 10068: 10051: 10048: 10044: 10041: 10034: 10027: 10024: 10004: 9998: 9994: 9990: 9985: 9981: 9976: 9971: 9965: 9962: 9959: 9955: 9951: 9946: 9942: 9937: 9933: 9928: 9925: 9922: 9918: 9914: 9909: 9904: 9898: 9895: 9892: 9888: 9884: 9879: 9875: 9870: 9863: 9859: 9856: 9853: 9847: 9842: 9839: 9836: 9832: 9828: 9825: 9823: 9819: 9815: 9811: 9810: 9807: 9804: 9801: 9799: 9795: 9791: 9787: 9786: 9754: 9748: 9745: 9742: 9738: 9734: 9729: 9725: 9718: 9713: 9710: 9707: 9703: 9699: 9696: 9694: 9690: 9686: 9682: 9681: 9678: 9675: 9672: 9670: 9666: 9662: 9658: 9657: 9642: 9638: 9604: 9603: 9600: 9585: 9578: 9575: 9572: 9569: 9566: 9563: 9556: 9551: 9547: 9543: 9538: 9534: 9530: 9523: 9520: 9502: 9499: 9478: 9472: 9467: 9463: 9459: 9454: 9450: 9446: 9440: 9437: 9418: 9415: 9409: 9406: 9387: 9381: 9376: 9372: 9365: 9360: 9357: 9354: 9350: 9346: 9341: 9337: 9320: 9319: 9316: 9312: 9309: 9295: 9292: 9286: 9283: 9272: 9269: 9258: 9255: 9240: 9234: 9230: 9225: 9216: 9206:Resulting in: 9189: 9186: 9183: 9180: 9177: 9174: 9169: 9166: 9161: 9158: 9155: 9152: 9149: 9146: 9139: 9135: 9131: 9126: 9123: 9118: 9114: 9110: 9107: 9104: 9099: 9094: 9091: 9088: 9084: 9076: 9072: 9068: 9063: 9060: 9058: 9056: 9052: 9046: 9042: 9036: 9031: 9028: 9025: 9021: 9016: 9012: 9009: 9002: 8998: 8994: 8989: 8985: 8979: 8975: 8969: 8964: 8961: 8958: 8954: 8948: 8945: 8939: 8935: 8932: 8929: 8926: 8924: 8922: 8914: 8911: 8908: 8905: 8904: 8882: 8877: 8873: 8869: 8866: 8863: 8857: 8853: 8849: 8846: 8841: 8837: 8833: 8830: 8827: 8824: 8798: 8793: 8789: 8785: 8782: 8779: 8776: 8773: 8768: 8764: 8760: 8757: 8754: 8751: 8748: 8746: 8744: 8739: 8735: 8731: 8726: 8722: 8718: 8715: 8712: 8709: 8708: 8703: 8698: 8694: 8690: 8687: 8685: 8683: 8680: 8677: 8674: 8671: 8668: 8667: 8637: 8631: 8627: 8622: 8613: 8594:Main article: 8591: 8588: 8564: 8558: 8555: 8549: 8546: 8530: 8526: 8503: 8496: 8491: 8487: 8484: 8479: 8475: 8470: 8463: 8458: 8455: 8452: 8448: 8441: 8438: 8435: 8431: 8424: 8421: 8418: 8415: 8412: 8396: 8395: 8392: 8388: 8385: 8372: 8369: 8366: 8365: 8352:1 / 8350: 8335: 8321: 8314: 8313: 8300:1 / 8298: 8286: 8274: 8262: 8261: 8248:1 / 8246: 8234: 8222: 8210: 8209: 8196:1 / 8194: 8185: 8176: 8164: 8163: 8153:1 / 8151: 8139: 8127: 8116: 8115: 8105:1 / 8103: 8094: 8085: 8073: 8072: 8062:1 / 8060: 8051: 8042: 8030: 8029: 8019:1 / 8017: 8005: 7993: 7986: 7985: 7975:1 / 7973: 7967: 7961: 7949: 7948: 7929:1 / 7927: 7913: 7899: 7890: 7889: 7882:1 / 7880: 7877: 7874: 7862: 7861: 7854:1 / 7852: 7843: 7834: 7827: 7826: 7819:1 / 7817: 7814: 7811: 7799: 7798: 7794:1 / 7792: 7789: 7786: 7774: 7773: 7770: 7761: 7752: 7745: 7744: 7741: 7738: 7735: 7723: 7722: 7715:1 / 7713: 7704: 7695: 7688: 7687: 7684: 7681: 7678: 7666: 7665: 7662: 7659: 7656: 7644: 7643: 7640: 7637: 7634: 7622: 7621: 7614:1 / 7612: 7603: 7594: 7587: 7586: 7583: 7580: 7577: 7565: 7564: 7561: 7558: 7555: 7543: 7542: 7538:1 / 7536: 7530: 7524: 7512: 7511: 7508: 7505: 7502: 7490: 7489: 7486: 7483: 7479: 7478: 7475: 7472: 7433: 7430: 7426: 7423: 7419: 7396: 7392: 7386: 7382: 7372: 7368: 7353: 7349: 7344: 7338: 7334: 7329: 7325: 7322: 7319: 7316: 7312: 7306: 7303: 7298: 7294: 7289: 7281: 7276: 7271: 7268: 7265: 7259: 7255: 7252: 7249: 7246: 7242: 7236: 7233: 7228: 7225: 7222: 7196:error function 7182: 7158: 7152: 7148: 7143: 7139: 7136: 7133: 7098: 7090:expected value 7067: 7062: 7057: 7053: 7050: 7047: 7041: 7034: 7031: 7026: 7022: 7013: 7010: 7005: 7001: 6996: 6992: 6986: 6982: 6978: 6975: 6972: 6969: 6965: 6961: 6926: 6923: 6920: 6919: 6908: 6898: 6887: 6880: 6877: 6874: 6870: 6857: 6856: 6841: 6837: 6833: 6828: 6825: 6815: 6804: 6801: 6790: 6789: 6786: 6775: 6772: 6761: 6760: 6757: 6746: 6743: 6732: 6731: 6728: 6717: 6714: 6703: 6702: 6699: 6688: 6685: 6674: 6673: 6670: 6659: 6656: 6645: 6644: 6641: 6630: 6624: 6611: 6610: 6607: 6593:Main article: 6590: 6587: 6580: 6577: 6573: 6570: 6566: 6563: 6545: 6540: 6533: 6530: 6524: 6519: 6515: 6510: 6503: 6499: 6462: 6461: 6445: 6442: 6439: 6437: 6432: 6429: 6423: 6422: 6419: 6416: 6413: 6411: 6407: 6403: 6397: 6393: 6387: 6384: 6379: 6378: 6375: 6372: 6369: 6367: 6365: 6362: 6359: 6354: 6350: 6344: 6340: 6336: 6335: 6332: 6329: 6326: 6324: 6321: 6317: 6314: 6311: 6306: 6302: 6296: 6292: 6287: 6283: 6280: 6279: 6276: 6273: 6270: 6268: 6266: 6263: 6260: 6255: 6251: 6247: 6244: 6241: 6236: 6232: 6228: 6225: 6222: 6217: 6213: 6209: 6206: 6203: 6202: 6199: 6196: 6193: 6191: 6189: 6186: 6183: 6178: 6174: 6170: 6167: 6164: 6159: 6155: 6151: 6148: 6145: 6140: 6136: 6132: 6129: 6126: 6123: 6120: 6117: 6114: 6111: 6108: 6105: 6104: 6101: 6098: 6095: 6093: 6091: 6088: 6085: 6082: 6079: 6076: 6073: 6070: 6069: 6031: 6027: 6024: 6004: 6001: 5998: 5995: 5992: 5989: 5986: 5983: 5980: 5960: 5940: 5928: 5927: 5914: 5907: 5904: 5898: 5892: 5889: 5883: 5877: 5874: 5867: 5863: 5860: 5851:Derivation of 5850: 5845: 5829: 5822: 5819: 5813: 5807: 5804: 5798: 5792: 5789: 5782: 5778: 5775: 5727: 5723: 5719: 5715: 5711: 5698: 5695: 5691: 5688: 5684: 5681: 5677: 5671: 5668: 5664: 5661: 5657: 5654: 5649: 5646: 5617: 5614: 5608: 5605: 5576: 5573: 5568: 5565: 5511: 5508: 5490: 5483: 5478: 5472: 5468: 5462: 5457: 5454: 5451: 5447: 5441: 5438: 5432: 5427: 5423: 5417: 5412: 5408: 5402: 5397: 5394: 5391: 5387: 5381: 5378: 5372: 5366: 5359: 5352: 5349: 5342: 5338: 5332: 5327: 5323: 5317: 5312: 5309: 5306: 5302: 5297: 5291: 5288: 5281: 5274: 5269: 5262: 5259: 5253: 5248: 5244: 5239: 5232: 5227: 5224: 5221: 5217: 5211: 5208: 5180: 5174: 5168: 5164: 5160: 5157: 5154: 5151: 5148: 5145: 5142: 5139: 5135: 5131: 5128: 5119: 5116: 5113: 5109: 5103: 5100: 5097: 5094: 5091: 5069: 5062: 5058: 5054: 5051: 5048: 5045: 5042: 5039: 5036: 5032: 5027: 5023: 5019: 5015: 5012: 5007: 5001: 4995: 4991: 4987: 4984: 4981: 4978: 4975: 4972: 4969: 4966: 4962: 4958: 4955: 4950: 4947: 4944: 4941: 4938: 4900: 4876: 4872: 4867: 4863: 4839: 4834: 4831: 4828: 4825: 4822: 4819: 4816: 4812: 4809: 4806: 4803: 4800: 4797: 4794: 4791: 4788: 4785: 4782: 4779: 4776: 4771: 4768: 4765: 4762: 4759: 4756: 4753: 4742:between them: 4721: 4718: 4715: 4712: 4709: 4705: 4701: 4697: 4693: 4690: 4688: 4686: 4683: 4680: 4677: 4674: 4671: 4670: 4667: 4664: 4661: 4658: 4655: 4652: 4649: 4647: 4645: 4642: 4639: 4636: 4633: 4630: 4627: 4626: 4623: 4620: 4617: 4615: 4613: 4610: 4607: 4604: 4601: 4600: 4568: 4565: 4554: 4550: 4546: 4542: 4535: 4531: 4520: 4516: 4504: 4498: 4494: 4479: 4474: 4471: 4449: 4441: 4431: 4428: 4421: 4418: 4411: 4396: 4393: 4390: 4387: 4384: 4380: 4371: 4368: 4363: 4357: 4353: 4347: 4344: 4339: 4335: 4331: 4322: 4319: 4314: 4311: 4307: 4301: 4297: 4291: 4287: 4283: 4268: 4244: 4240: 4217: 4214: 4211: 4208: 4205: 4201: 4193: 4190: 4185: 4182: 4178: 4174: 4167: 4163: 4157: 4153: 4147: 4144: 4138: 4135: 4130: 4125: 4121: 4101: 4074: 4071: 4061: 4058: 4043: 4036: 4031: 4024: 4021: 4015: 4010: 4006: 4001: 3994: 3989: 3986: 3983: 3979: 3970: 3966: 3960: 3957: 3952: 3949: 3946: 3943: 3939: 3932: 3926: 3923: 3899: 3884: 3877: 3867: 3860: 3850: 3825: 3818: 3813: 3806: 3803: 3797: 3792: 3788: 3783: 3776: 3771: 3768: 3765: 3761: 3754: 3751: 3748: 3744: 3737: 3731: 3728: 3711: 3707: − 1 3704: 3682: 3675: 3670: 3666: 3663: 3660: 3654: 3650: 3644: 3639: 3636: 3631: 3627: 3614: 3611: 3608: 3604: 3597: 3593: 3590: 3587: 3582: 3578: 3568:, and equals: 3566:Gamma function 3552: 3549: 3541: 3524: 3521: 3505: 3502: 3499: 3495: 3471: 3468: 3431: 3424: 3419: 3412: 3409: 3403: 3398: 3394: 3389: 3382: 3377: 3374: 3371: 3367: 3360: 3357: 3354: 3350: 3343: 3340: 3311: 3308: 3305: 3302: 3297: 3294: 3289: 3284: 3281: 3254: 3251: 3245: 3242: 3236: 3231: 3227: 3222: 3219: 3215: 3209: 3206: 3200: 3195: 3191: 3187: 3156: 3151: 3146: 3139: 3136: 3130: 3125: 3121: 3116: 3109: 3104: 3101: 3098: 3094: 3087: 3084: 3081: 3077: 3072: 3067: 3063: 3005: 2998: 2993: 2986: 2983: 2977: 2972: 2968: 2963: 2956: 2951: 2948: 2945: 2941: 2935: 2932: 2925: 2920: 2916: 2904:central moment 2892: 2889: 2872: 2869: 2866: 2816: 2813: 2790: 2785: 2781: 2776: 2773: 2769: 2764: 2760: 2755: 2750: 2746: 2742: 2720: 2713: 2708: 2701: 2698: 2692: 2687: 2683: 2678: 2671: 2666: 2663: 2660: 2656: 2650: 2647: 2640: 2635: 2631: 2608: 2596: 2593: 2528:Main article: 2517: 2514: 2502: 2493: 2489: 2485: 2482: 2479: 2475: 2470: 2466: 2463: 2456: 2452: 2447: 2442: 2426: 2384: 2381: 2377: 2372: 2369: 2366: 2363: 2359: 2353: 2348: 2344: 2341: 2333: 2328: 2324: 2319: 2316: 2313: 2310: 2304: 2300: 2296: 2293: 2290: 2287: 2281: 2276: 2270: 2267: 2253: 2249: 2234: 2231: 2219: 2214: 2210: 2204: 2200: 2194: 2189: 2186: 2183: 2179: 2175: 2172: 2164: 2157: 2153: 2149: 2146: 2141: 2137: 2133: 2128: 2124: 2118: 2113: 2110: 2107: 2103: 2097: 2094: 2080: 2079: 2076: 2069: 2068: 2065: 2061: 2058: 2052: 2049: 2043: 2040: 2034: 2031: 2016: 2011: 2007: 2001: 1996: 1993: 1990: 1986: 1980: 1977: 1972: 1969: 1961: 1954: 1950: 1946: 1943: 1938: 1934: 1930: 1925: 1920: 1917: 1914: 1910: 1904: 1901: 1894: 1891: 1858: 1855: 1850: 1846: 1842: 1839: 1836: 1831: 1827: 1823: 1818: 1815: 1810: 1807: 1799: 1793: 1787: 1783: 1779: 1776: 1771: 1767: 1763: 1760: 1757: 1754: 1749: 1745: 1741: 1738: 1733: 1729: 1725: 1722: 1717: 1713: 1709: 1706: 1701: 1697: 1693: 1689: 1683: 1680: 1673: 1670: 1654: 1653: 1650: 1646: 1643: 1639: 1636: 1627: 1624: 1607: 1604: 1601: 1598: 1595: 1592: 1589: 1541: 1534: 1530: 1526: 1523: 1520: 1517: 1514: 1511: 1508: 1504: 1499: 1495: 1491: 1487: 1484: 1462: 1457: 1453: 1448: 1445: 1442: 1439: 1434: 1430: 1426: 1423: 1420: 1417: 1412: 1409: 1404: 1401: 1397: 1391: 1385: 1379: 1375: 1371: 1368: 1365: 1362: 1358: 1354: 1351: 1346: 1343: 1315: 1311: 1306: 1303: 1300: 1297: 1294: 1289: 1286: 1281: 1278: 1274: 1270: 1267: 1264: 1261: 1258: 1255: 1252: 1249: 1231: 1220:expected value 1211: 1208: 1172:69 inches 1159: 1156: 1137:{\textstyle n} 1133: 1113: 1110: 1107: 1087: 1084: 1068:{\textstyle s} 1064: 1040: 1037: 1032: 1028: 1024: 1019: 1016: 977: 974: 969: 964: 961: 935: 932: 927: 924: 919: 914: 910: 907: 904: 901: 898: 895: 892: 889: 886: 883: 880: 877: 874: 871: 868: 862: 857: 853: 822: 819: 814: 810: 806: 801: 797: 793: 790: 787: 784: 781: 778: 775: 770: 766: 762: 759: 756: 753: 748: 744: 740: 737: 734: 731: 728: 727: 724: 721: 716: 712: 708: 703: 699: 695: 692: 689: 686: 683: 680: 677: 672: 668: 664: 661: 658: 655: 650: 646: 642: 639: 636: 633: 630: 629: 626: 623: 618: 614: 610: 605: 601: 597: 594: 591: 588: 585: 582: 579: 574: 570: 566: 563: 560: 557: 552: 548: 544: 541: 538: 535: 532: 531: 528: 525: 520: 516: 512: 507: 503: 499: 496: 493: 490: 487: 484: 481: 476: 472: 468: 465: 462: 459: 454: 450: 446: 443: 440: 437: 434: 433: 407: 404: 399: 396: 391: 386: 382: 379: 376: 373: 370: 367: 364: 361: 358: 355: 352: 349: 346: 343: 340: 334: 331: 305: 299: 296: 290: 287: 281: 278: 272: 269: 263: 260: 254: 251: 245: 242: 218: 215: 213: 212:Basic examples 210: 166:standard error 89:expected value 26: 9: 6: 4: 3: 2: 14494: 14483: 14480: 14478: 14475: 14474: 14472: 14465: 14454: 14453: 14444: 14442: 14441: 14432: 14430: 14429: 14424: 14418: 14416: 14415: 14406: 14405: 14402: 14388: 14385: 14383: 14382:Geostatistics 14380: 14378: 14375: 14373: 14370: 14368: 14365: 14364: 14362: 14360: 14356: 14350: 14349:Psychometrics 14347: 14345: 14342: 14340: 14337: 14335: 14332: 14330: 14327: 14325: 14322: 14320: 14317: 14315: 14312: 14310: 14307: 14305: 14302: 14301: 14299: 14297: 14293: 14287: 14284: 14282: 14279: 14277: 14273: 14270: 14268: 14265: 14263: 14260: 14258: 14255: 14254: 14252: 14250: 14246: 14240: 14237: 14235: 14232: 14230: 14226: 14223: 14221: 14218: 14217: 14215: 14213: 14212:Biostatistics 14209: 14205: 14201: 14196: 14192: 14174: 14173:Log-rank test 14171: 14170: 14168: 14164: 14158: 14155: 14154: 14152: 14150: 14146: 14140: 14137: 14135: 14132: 14130: 14127: 14125: 14122: 14121: 14119: 14117: 14113: 14110: 14108: 14104: 14094: 14091: 14089: 14086: 14084: 14081: 14079: 14076: 14074: 14071: 14070: 14068: 14066: 14062: 14056: 14053: 14051: 14048: 14046: 14044:(Box–Jenkins) 14040: 14038: 14035: 14033: 14030: 14026: 14023: 14022: 14021: 14018: 14017: 14015: 14013: 14009: 14003: 14000: 13998: 13997:Durbin–Watson 13995: 13993: 13987: 13985: 13982: 13980: 13979:Dickey–Fuller 13977: 13976: 13974: 13970: 13964: 13961: 13959: 13956: 13954: 13953:Cointegration 13951: 13949: 13946: 13944: 13941: 13939: 13936: 13934: 13931: 13929: 13928:Decomposition 13926: 13925: 13923: 13919: 13916: 13914: 13910: 13900: 13897: 13896: 13895: 13892: 13891: 13890: 13887: 13883: 13880: 13879: 13878: 13875: 13873: 13870: 13868: 13865: 13863: 13860: 13858: 13855: 13853: 13850: 13848: 13845: 13843: 13840: 13839: 13837: 13835: 13831: 13825: 13822: 13820: 13817: 13815: 13812: 13810: 13807: 13805: 13802: 13800: 13799:Cohen's kappa 13797: 13796: 13794: 13792: 13788: 13784: 13780: 13776: 13772: 13768: 13763: 13759: 13745: 13742: 13740: 13737: 13735: 13732: 13730: 13727: 13726: 13724: 13722: 13718: 13712: 13708: 13704: 13698: 13696: 13693: 13692: 13690: 13688: 13684: 13678: 13675: 13673: 13670: 13668: 13665: 13663: 13660: 13658: 13655: 13653: 13652:Nonparametric 13650: 13648: 13645: 13644: 13642: 13638: 13632: 13629: 13627: 13624: 13622: 13619: 13617: 13614: 13613: 13611: 13609: 13605: 13599: 13596: 13594: 13591: 13589: 13586: 13584: 13581: 13579: 13576: 13575: 13573: 13571: 13567: 13561: 13558: 13556: 13553: 13551: 13548: 13546: 13543: 13542: 13540: 13538: 13534: 13530: 13523: 13520: 13518: 13515: 13514: 13510: 13506: 13490: 13487: 13486: 13485: 13482: 13480: 13477: 13475: 13472: 13468: 13465: 13463: 13460: 13459: 13458: 13455: 13454: 13452: 13450: 13446: 13436: 13433: 13429: 13423: 13421: 13415: 13413: 13407: 13406: 13405: 13402: 13401:Nonparametric 13399: 13397: 13391: 13387: 13384: 13383: 13382: 13376: 13372: 13371:Sample median 13369: 13368: 13367: 13364: 13363: 13361: 13359: 13355: 13347: 13344: 13342: 13339: 13337: 13334: 13333: 13332: 13329: 13327: 13324: 13322: 13316: 13314: 13311: 13309: 13306: 13304: 13301: 13299: 13296: 13294: 13292: 13288: 13286: 13283: 13282: 13280: 13278: 13274: 13268: 13266: 13262: 13260: 13258: 13253: 13251: 13246: 13242: 13241: 13238: 13235: 13233: 13229: 13219: 13216: 13214: 13211: 13209: 13206: 13205: 13203: 13201: 13197: 13191: 13188: 13184: 13181: 13180: 13179: 13176: 13172: 13169: 13168: 13167: 13164: 13162: 13159: 13158: 13156: 13154: 13150: 13142: 13139: 13137: 13134: 13133: 13132: 13129: 13127: 13124: 13122: 13119: 13117: 13114: 13112: 13109: 13107: 13104: 13103: 13101: 13099: 13095: 13089: 13086: 13082: 13079: 13075: 13072: 13070: 13067: 13066: 13065: 13062: 13061: 13060: 13057: 13053: 13050: 13048: 13045: 13043: 13040: 13038: 13035: 13034: 13033: 13030: 13029: 13027: 13025: 13021: 13018: 13016: 13012: 13006: 13003: 13001: 12998: 12994: 12991: 12990: 12989: 12986: 12984: 12981: 12977: 12976:loss function 12974: 12973: 12972: 12969: 12965: 12962: 12960: 12957: 12955: 12952: 12951: 12950: 12947: 12945: 12942: 12940: 12937: 12933: 12930: 12928: 12925: 12923: 12917: 12914: 12913: 12912: 12909: 12905: 12902: 12900: 12897: 12895: 12892: 12891: 12890: 12887: 12883: 12880: 12878: 12875: 12874: 12873: 12870: 12866: 12863: 12862: 12861: 12858: 12854: 12851: 12850: 12849: 12846: 12844: 12841: 12839: 12836: 12834: 12831: 12830: 12828: 12826: 12822: 12818: 12814: 12809: 12805: 12791: 12788: 12786: 12783: 12781: 12778: 12776: 12773: 12772: 12770: 12768: 12764: 12758: 12755: 12753: 12750: 12748: 12745: 12744: 12742: 12738: 12732: 12729: 12727: 12724: 12722: 12719: 12717: 12714: 12712: 12709: 12707: 12704: 12702: 12699: 12698: 12696: 12694: 12690: 12684: 12681: 12679: 12678:Questionnaire 12676: 12674: 12671: 12667: 12664: 12662: 12659: 12658: 12657: 12654: 12653: 12651: 12649: 12645: 12639: 12636: 12634: 12631: 12629: 12626: 12624: 12621: 12619: 12616: 12614: 12611: 12609: 12606: 12604: 12601: 12600: 12598: 12596: 12592: 12588: 12584: 12579: 12575: 12561: 12558: 12556: 12553: 12551: 12548: 12546: 12543: 12541: 12538: 12536: 12533: 12531: 12528: 12526: 12523: 12521: 12518: 12516: 12513: 12511: 12508: 12506: 12505:Control chart 12503: 12501: 12498: 12496: 12493: 12491: 12488: 12487: 12485: 12483: 12479: 12473: 12470: 12466: 12463: 12461: 12458: 12457: 12456: 12453: 12451: 12448: 12446: 12443: 12442: 12440: 12438: 12434: 12428: 12425: 12423: 12420: 12418: 12415: 12414: 12412: 12408: 12402: 12399: 12398: 12396: 12394: 12390: 12378: 12375: 12373: 12370: 12368: 12365: 12364: 12363: 12360: 12358: 12355: 12354: 12352: 12350: 12346: 12340: 12337: 12335: 12332: 12330: 12327: 12325: 12322: 12320: 12317: 12315: 12312: 12310: 12307: 12306: 12304: 12302: 12298: 12292: 12289: 12287: 12284: 12280: 12277: 12275: 12272: 12270: 12267: 12265: 12262: 12260: 12257: 12255: 12252: 12250: 12247: 12245: 12242: 12240: 12237: 12235: 12232: 12231: 12230: 12227: 12226: 12224: 12222: 12218: 12215: 12213: 12209: 12205: 12201: 12196: 12192: 12186: 12183: 12181: 12178: 12177: 12174: 12170: 12163: 12158: 12156: 12151: 12149: 12144: 12143: 12140: 12133: 12129: 12125: 12121: 12120: 12115: 12111: 12110: 12097: 12093: 12089: 12083: 12079: 12072: 12064: 12057: 12048: 12043: 12039: 12035: 12031: 12027: 12026: 12021: 12017: 12016:Pearson, Karl 12011: 12003: 11997: 11992: 11991: 11985: 11979: 11971: 11967: 11962: 11957: 11953: 11949: 11948:Technometrics 11942: 11926: 11922: 11915: 11907: 11905:9780130113290 11901: 11897: 11892: 11891: 11882: 11866: 11860: 11853: 11849: 11845: 11841: 11837: 11833: 11829: 11825: 11820: 11815: 11812:(6): 061102, 11811: 11807: 11800: 11784: 11780: 11774: 11766: 11762: 11758: 11754: 11750: 11746: 11742: 11738: 11731: 11723: 11719: 11715: 11711: 11704: 11688: 11684: 11678: 11671: 11667: 11663: 11659: 11655: 11651: 11644: 11629: 11625: 11619: 11604: 11600: 11593: 11582: 11581: 11574: 11559: 11555: 11549: 11540: 11539: 11534: 11531: 11524: 11516: 11512: 11511:Walker, Helen 11506: 11498: 11494: 11490: 11484: 11476: 11472: 11467: 11462: 11458: 11454: 11450: 11446: 11442: 11435: 11431: 11419: 11416: 11414: 11411: 11409: 11406: 11404: 11401: 11399: 11396: 11394: 11391: 11389: 11386: 11384: 11381: 11379: 11376: 11374: 11371: 11369: 11366: 11364: 11361: 11359: 11356: 11354: 11351: 11349: 11346: 11344: 11341: 11338: 11335: 11333: 11330: 11328: 11325: 11322: 11319: 11317: 11314: 11312: 11309: 11307: 11304: 11301: 11298: 11296: 11293: 11291: 11288: 11286: 11283: 11282: 11277: 11271: 11266: 11259: 11257: 11256: 11246: 11237: 11213: 11202: 11189: 11185: 11175: 11173: 11169: 11165: 11161: 11151: 11148: 11134: 11121: 11116: 11111: 11107: 11100: 11097: 11093: 11090: 11084: 11081: 11075: 11070: 11065: 11061: 11051: 11038: 11032: 11029: 11024: 11020: 11013: 11009: 11003: 10998: 10993: 10989: 10979: 10961: 10957: 10951: 10947: 10941: 10936: 10931: 10927: 10917: 10899: 10893: 10889: 10885: 10880: 10876: 10871: 10866: 10860: 10857: 10854: 10850: 10846: 10841: 10837: 10832: 10826: 10822: 10818: 10813: 10810: 10807: 10803: 10799: 10794: 10789: 10783: 10780: 10777: 10773: 10769: 10764: 10760: 10755: 10746: 10742: 10735: 10732: 10729: 10725: 10719: 10715: 10708: 10703: 10700: 10697: 10693: 10689: 10687: 10680: 10676: 10668: 10665: 10663: 10656: 10652: 10643: 10637: 10634: 10631: 10627: 10623: 10618: 10614: 10609: 10601: 10597: 10591: 10587: 10581: 10576: 10573: 10570: 10566: 10562: 10560: 10553: 10549: 10541: 10538: 10536: 10529: 10525: 10498: 10494: 10489: 10483: 10479: 10462: 10443: 10439: 10435: 10430: 10427: 10424: 10420: 10416: 10414: 10407: 10403: 10395: 10392: 10390: 10383: 10379: 10357: 10354: 10338: 10324: 10319: 10314: 10310: 10304: 10300: 10294: 10289: 10286: 10283: 10279: 10275: 10270: 10266: 10257: 10218: 10214: 10191: 10187: 10172: 10157: 10152: 10148: 10142: 10137: 10132: 10128: 10118: 10102: 10099: 10096: 10090: 10086: 10080: 10075: 10070: 10066: 10056: 10037:− 1 = 0 10020: 10002: 9996: 9992: 9988: 9983: 9979: 9974: 9969: 9963: 9960: 9957: 9953: 9949: 9944: 9940: 9935: 9931: 9926: 9923: 9920: 9916: 9912: 9907: 9902: 9896: 9893: 9890: 9886: 9882: 9877: 9873: 9868: 9861: 9857: 9854: 9851: 9845: 9840: 9837: 9834: 9830: 9826: 9824: 9817: 9813: 9805: 9802: 9800: 9793: 9789: 9771: 9752: 9746: 9743: 9740: 9736: 9732: 9727: 9723: 9716: 9711: 9708: 9705: 9701: 9697: 9695: 9688: 9684: 9676: 9673: 9671: 9664: 9660: 9647: 9634: 9620: 9616: 9612: 9596: 9583: 9573: 9570: 9567: 9561: 9554: 9549: 9545: 9541: 9536: 9532: 9528: 9521: 9518: 9510: 9507: 9491: 9476: 9470: 9465: 9461: 9457: 9452: 9448: 9444: 9438: 9435: 9427: 9425: 9398: 9385: 9379: 9374: 9370: 9363: 9358: 9355: 9352: 9348: 9344: 9339: 9335: 9326: 9307:, denoted as 9278: 9268: 9251: 9238: 9232: 9228: 9223: 9214: 9204: 9187: 9181: 9175: 9172: 9167: 9164: 9159: 9153: 9147: 9144: 9137: 9133: 9129: 9124: 9116: 9112: 9105: 9102: 9097: 9092: 9089: 9086: 9082: 9074: 9070: 9066: 9061: 9059: 9050: 9044: 9040: 9034: 9029: 9026: 9023: 9019: 9014: 9010: 9007: 9000: 8996: 8992: 8987: 8983: 8977: 8973: 8967: 8962: 8959: 8956: 8952: 8946: 8943: 8937: 8933: 8930: 8927: 8925: 8909: 8906: 8893: 8875: 8871: 8864: 8861: 8855: 8851: 8847: 8839: 8835: 8831: 8825: 8822: 8813: 8791: 8787: 8780: 8777: 8774: 8766: 8762: 8755: 8752: 8749: 8747: 8737: 8733: 8729: 8724: 8720: 8713: 8710: 8701: 8696: 8692: 8688: 8686: 8678: 8672: 8669: 8657: 8648: 8635: 8629: 8625: 8620: 8611: 8602: 8597: 8587: 8585: 8581: 8576: 8562: 8553: 8547: 8544: 8536: 8523: 8519: 8514: 8501: 8494: 8489: 8485: 8482: 8477: 8473: 8468: 8461: 8456: 8453: 8450: 8446: 8439: 8436: 8433: 8429: 8422: 8416: 8410: 8402: 8382: 8378: 8351: 8336: 8322: 8316: 8315: 8299: 8287: 8275: 8264: 8263: 8247: 8235: 8223: 8212: 8211: 8195: 8186: 8177: 8166: 8165: 8152: 8140: 8128: 8126: 8118: 8117: 8104: 8095: 8086: 8075: 8074: 8061: 8052: 8043: 8032: 8031: 8018: 8006: 7994: 7988: 7987: 7974: 7968: 7962: 7951: 7950: 7928: 7914: 7900: 7892: 7891: 7881: 7878: 7875: 7864: 7863: 7853: 7844: 7835: 7829: 7828: 7818: 7815: 7812: 7801: 7800: 7793: 7790: 7787: 7776: 7775: 7771: 7762: 7753: 7747: 7746: 7742: 7739: 7736: 7725: 7724: 7714: 7705: 7696: 7690: 7689: 7685: 7682: 7679: 7668: 7667: 7663: 7660: 7657: 7646: 7645: 7641: 7638: 7635: 7624: 7623: 7613: 7604: 7595: 7589: 7588: 7584: 7581: 7578: 7567: 7566: 7562: 7559: 7556: 7545: 7544: 7537: 7531: 7525: 7514: 7513: 7509: 7506: 7503: 7492: 7491: 7487: 7484: 7481: 7480: 7473: 7467: 7464: 7459: 7455: 7448: 7443: 7439: 7411: 7409: 7405: 7404: 7364: 7351: 7347: 7342: 7336: 7332: 7327: 7323: 7320: 7317: 7314: 7310: 7304: 7301: 7296: 7292: 7287: 7279: 7274: 7269: 7266: 7263: 7257: 7253: 7250: 7247: 7244: 7240: 7234: 7231: 7226: 7223: 7220: 7207: 7205: 7197: 7180: 7170: 7156: 7150: 7146: 7141: 7137: 7134: 7131: 7118: 7111: 7109: 7091: 7082: 7065: 7060: 7055: 7051: 7048: 7045: 7039: 7032: 7029: 7024: 7020: 7011: 7008: 7003: 6999: 6994: 6990: 6984: 6980: 6976: 6973: 6970: 6967: 6963: 6959: 6951: 6949: 6945: 6937: 6931: 6906: 6899: 6885: 6878: 6875: 6872: 6868: 6859: 6858: 6839: 6835: 6831: 6826: 6823: 6816: 6802: 6799: 6792: 6791: 6787: 6773: 6770: 6763: 6762: 6758: 6744: 6741: 6734: 6733: 6729: 6715: 6712: 6705: 6704: 6700: 6686: 6683: 6676: 6675: 6671: 6657: 6654: 6647: 6646: 6642: 6628: 6622: 6613: 6612: 6608: 6605: 6604: 6601: 6596: 6586: 6543: 6538: 6528: 6522: 6517: 6513: 6508: 6501: 6497: 6483:and the line 6478: 6460: 6443: 6440: 6438: 6427: 6417: 6414: 6412: 6405: 6401: 6395: 6391: 6385: 6382: 6373: 6370: 6368: 6363: 6360: 6357: 6352: 6348: 6342: 6338: 6330: 6327: 6325: 6319: 6315: 6312: 6309: 6304: 6300: 6294: 6290: 6285: 6281: 6274: 6271: 6269: 6261: 6258: 6253: 6249: 6245: 6242: 6239: 6234: 6230: 6226: 6223: 6220: 6215: 6211: 6204: 6197: 6194: 6192: 6184: 6181: 6176: 6172: 6168: 6165: 6162: 6157: 6153: 6149: 6146: 6143: 6138: 6134: 6127: 6121: 6118: 6115: 6112: 6109: 6099: 6096: 6094: 6086: 6083: 6080: 6074: 6071: 6059: 6058:. Therefore: 6044: 6025: 6022: 5999: 5996: 5993: 5990: 5987: 5981: 5978: 5958: 5938: 5930: 5929: 5912: 5902: 5896: 5887: 5881: 5872: 5865: 5861: 5858: 5849: 5848: 5844: 5841: 5827: 5817: 5811: 5802: 5796: 5787: 5780: 5776: 5773: 5765: 5760:to the point 5747: 5740:would lie on 5733: 5707: 5645: 5641: 5637: 5635: 5630: 5628: 5623: 5613: 5604: 5602: 5598: 5594: 5590: 5586: 5581: 5572: 5564: 5562: 5558: 5553: 5551: 5546: 5542: 5537: 5533: 5526: 5521: 5517: 5507: 5504: 5501: 5488: 5481: 5476: 5470: 5466: 5460: 5455: 5452: 5449: 5445: 5439: 5436: 5430: 5425: 5421: 5415: 5410: 5406: 5400: 5395: 5392: 5389: 5385: 5379: 5376: 5370: 5364: 5357: 5347: 5340: 5336: 5330: 5325: 5321: 5315: 5310: 5307: 5304: 5300: 5295: 5289: 5286: 5279: 5272: 5267: 5257: 5251: 5246: 5242: 5237: 5230: 5225: 5222: 5219: 5215: 5209: 5206: 5194: 5191: 5178: 5172: 5166: 5155: 5149: 5143: 5140: 5133: 5129: 5117: 5114: 5111: 5107: 5101: 5095: 5089: 5080: 5067: 5060: 5049: 5043: 5034: 5030: 5025: 5021: 5017: 5013: 5005: 4999: 4993: 4982: 4976: 4970: 4967: 4960: 4956: 4948: 4942: 4936: 4928: 4922: 4917: 4915: 4898: 4874: 4870: 4865: 4861: 4851: 4837: 4829: 4826: 4823: 4817: 4814: 4810: 4807: 4801: 4795: 4792: 4789: 4783: 4777: 4774: 4769: 4763: 4760: 4757: 4751: 4743: 4741: 4736: 4719: 4713: 4707: 4699: 4691: 4689: 4681: 4678: 4672: 4665: 4659: 4653: 4650: 4648: 4640: 4637: 4634: 4628: 4621: 4618: 4616: 4608: 4602: 4578: 4574: 4564: 4528: 4477:For a set of 4470: 4468: 4460: 4459:least squares 4455: 4437: 4407: 4394: 4391: 4388: 4385: 4382: 4378: 4369: 4366: 4361: 4355: 4351: 4345: 4342: 4337: 4333: 4329: 4320: 4317: 4312: 4309: 4305: 4299: 4295: 4289: 4285: 4273: 4242: 4238: 4228: 4215: 4212: 4209: 4206: 4203: 4199: 4191: 4188: 4183: 4180: 4176: 4172: 4165: 4161: 4155: 4151: 4145: 4142: 4136: 4133: 4128: 4123: 4111: 4109: 4097: 4095: 4088: 4084: 4080: 4070: 4068: 4054: 4041: 4034: 4029: 4019: 4013: 4008: 4004: 3999: 3992: 3987: 3984: 3981: 3977: 3968: 3964: 3958: 3955: 3950: 3947: 3944: 3941: 3937: 3930: 3921: 3909: 3873: 3836: 3823: 3816: 3811: 3801: 3795: 3790: 3786: 3781: 3774: 3769: 3766: 3763: 3759: 3752: 3749: 3746: 3742: 3735: 3726: 3700: 3698: 3693: 3680: 3673: 3668: 3664: 3661: 3658: 3652: 3642: 3637: 3634: 3629: 3612: 3609: 3606: 3602: 3595: 3588: 3580: 3576: 3567: 3530: 3520: 3503: 3500: 3497: 3493: 3469: 3466: 3456: 3451: 3447: 3442: 3429: 3422: 3417: 3407: 3401: 3396: 3392: 3387: 3380: 3375: 3372: 3369: 3365: 3358: 3355: 3352: 3348: 3341: 3338: 3331: 3327: 3306: 3300: 3295: 3287: 3279: 3271: 3266: 3252: 3240: 3234: 3229: 3225: 3220: 3217: 3213: 3204: 3198: 3193: 3189: 3176: 3172: 3167: 3154: 3149: 3144: 3134: 3128: 3123: 3119: 3114: 3107: 3102: 3099: 3096: 3092: 3085: 3082: 3079: 3075: 3070: 3065: 3061: 3052: 3048: 3045:to yield the 3044: 3040: 3036: 3032: 3027: 3025: 3021: 3016: 3003: 2996: 2991: 2981: 2975: 2970: 2966: 2961: 2954: 2949: 2946: 2943: 2939: 2933: 2930: 2923: 2918: 2914: 2905: 2901: 2900: 2888: 2886: 2870: 2867: 2864: 2856: 2852: 2848: 2844: 2839: 2837: 2833: 2811: 2783: 2779: 2774: 2771: 2767: 2762: 2758: 2753: 2748: 2744: 2731: 2718: 2711: 2706: 2696: 2690: 2685: 2681: 2676: 2669: 2664: 2661: 2658: 2654: 2648: 2645: 2638: 2633: 2629: 2620: 2616: 2611: 2607: 2602: 2592: 2590: 2586: 2582: 2578: 2577: 2572: 2568: 2564: 2560: 2555: 2553: 2549: 2545: 2541: 2537: 2531: 2525: 2520: 2513: 2500: 2491: 2487: 2483: 2480: 2477: 2473: 2468: 2464: 2461: 2454: 2450: 2445: 2440: 2419: 2415: 2410: 2400: 2395: 2382: 2379: 2367: 2361: 2357: 2346: 2342: 2339: 2331: 2326: 2314: 2308: 2302: 2294: 2291: 2288: 2274: 2268: 2265: 2247: 2240: 2230: 2217: 2212: 2208: 2202: 2198: 2192: 2187: 2184: 2181: 2177: 2173: 2170: 2162: 2155: 2147: 2144: 2139: 2135: 2126: 2122: 2116: 2111: 2108: 2105: 2101: 2095: 2092: 2027: 2014: 2009: 2005: 1999: 1994: 1991: 1988: 1984: 1978: 1975: 1970: 1967: 1959: 1952: 1944: 1941: 1936: 1932: 1923: 1918: 1915: 1912: 1908: 1902: 1899: 1892: 1889: 1881: 1879: 1876:or, by using 1874: 1872: 1856: 1848: 1844: 1840: 1837: 1834: 1829: 1825: 1816: 1813: 1808: 1805: 1797: 1791: 1785: 1777: 1774: 1769: 1765: 1758: 1755: 1752: 1747: 1739: 1736: 1731: 1727: 1720: 1715: 1707: 1704: 1699: 1695: 1687: 1681: 1678: 1671: 1668: 1660: 1623: 1621: 1602: 1599: 1596: 1590: 1587: 1579: 1575: 1571: 1566: 1563: 1557: 1552: 1539: 1532: 1521: 1515: 1506: 1502: 1497: 1493: 1489: 1485: 1460: 1455: 1443: 1437: 1432: 1424: 1421: 1418: 1407: 1399: 1395: 1389: 1383: 1377: 1369: 1366: 1363: 1356: 1352: 1344: 1341: 1313: 1301: 1295: 1292: 1284: 1276: 1272: 1268: 1262: 1256: 1250: 1247: 1229:with density 1225: 1221: 1217: 1207: 1205: 1201: 1197: 1190:of the mean ( 1188:6 inches 1181: 1176:3 inches 1169: 1168:United States 1165: 1155: 1153: 1149: 1148: 1131: 1111: 1108: 1105: 1085: 1082: 1062: 1054: 1038: 1035: 1030: 1026: 1022: 1017: 1014: 1004: 996: 988: 975: 972: 967: 962: 959: 951: 946: 933: 930: 925: 922: 917: 912: 908: 905: 902: 899: 896: 893: 890: 887: 884: 881: 878: 875: 872: 869: 866: 860: 855: 851: 842: 837: 820: 817: 812: 808: 804: 799: 791: 788: 785: 776: 773: 768: 760: 757: 751: 746: 738: 735: 732: 722: 719: 714: 710: 706: 701: 693: 690: 687: 678: 675: 670: 662: 659: 653: 648: 640: 637: 634: 624: 621: 616: 612: 608: 603: 595: 592: 589: 580: 577: 572: 564: 561: 555: 550: 542: 539: 536: 526: 523: 518: 514: 510: 505: 497: 494: 491: 482: 479: 474: 466: 463: 457: 452: 444: 441: 438: 423: 418: 405: 402: 397: 394: 389: 384: 380: 377: 374: 371: 368: 365: 362: 359: 356: 353: 350: 347: 344: 341: 338: 332: 329: 321: 316: 303: 297: 294: 288: 285: 279: 276: 270: 267: 261: 258: 252: 249: 243: 240: 232: 228: 224: 209: 207: 203: 199: 195: 190: 188: 184: 179: 178:population. 176: 171: 167: 162: 160: 156: 152: 151:algebraically 148: 144: 140: 136: 132: 128: 123: 121: 120: 119: 114: 110: 109: 105: 101: 96: 94: 90: 86: 82: 78: 74: 70: 66: 57: 50: 46: 41: 37: 33: 19: 14463: 14450: 14438: 14419: 14412: 14324:Econometrics 14274: / 14257:Chemometrics 14234:Epidemiology 14227: / 14200:Applications 14042:ARIMA model 13989:Q-statistic 13938:Stationarity 13834:Multivariate 13777: / 13773: / 13771:Multivariate 13769: / 13709: / 13705: / 13479:Bayes factor 13378:Signed rank 13290: 13264: 13256: 13244: 12939:Completeness 12775:Cohort study 12673:Opinion poll 12608:Missing data 12595:Study design 12550:Scatter plot 12472:Scatter plot 12465:Spearman's ρ 12427:Grouped data 12333: 12117: 12077: 12071: 12056: 12029: 12023: 12010: 11989: 11978: 11951: 11947: 11941: 11931:30 September 11929:. Retrieved 11924: 11914: 11889: 11881: 11869:. Retrieved 11859: 11809: 11805: 11799: 11787:. Retrieved 11783:the original 11773: 11740: 11736: 11730: 11716:(3): 84–86. 11713: 11709: 11703: 11693:14 September 11691:. Retrieved 11686: 11677: 11656:(4): 30–32, 11653: 11649: 11643: 11631:. Retrieved 11627: 11618: 11606:. Retrieved 11602: 11592: 11579: 11573: 11561:. Retrieved 11557: 11548: 11536: 11523: 11514: 11505: 11496: 11492: 11483: 11448: 11444: 11434: 11253: 11251: 11181: 11171: 11164:Karl Pearson 11159: 11157: 11149: 11135: 11052: 10980: 10918: 10463: 10358: 10355: 10339: 10258: 10178: 10119: 10057: 10021: 9772: 9648: 9635: 9597: 9511: 9508: 9492: 9428: 9423: 9399: 9327: 9280: 9252: 9205: 8894: 8814: 8649: 8603: 8599: 8577: 8537: 8515: 8403: 8374: 7463: 7457: 7446: 7412: 7407: 7401: 7365: 7208: 7171: 7119: 7112: 7083: 6952: 6941: 6598: 6466: 6060: 6045: 5931: 5842: 5766: 5745: 5731: 5705: 5651: 5642: 5638: 5631: 5619: 5610: 5588: 5582: 5578: 5570: 5554: 5541:measurements 5538: 5534: 5531: 5505: 5502: 5195: 5192: 5081: 4929: 4918: 4852: 4744: 4737: 4570: 4492:is given by 4476: 4456: 4438: 4408: 4274: 4229: 4112: 4098: 4090: 4055: 3910: 3874: 3837: 3716:, yielding: 3701: 3694: 3526: 3454: 3449: 3445: 3443: 3329: 3325: 3267: 3170: 3168: 3050: 3046: 3042: 3038: 3030: 3028: 3017: 2902:(the second 2896: 2894: 2854: 2840: 2831: 2732: 2618: 2614: 2609: 2605: 2600: 2598: 2588: 2584: 2580: 2574: 2556: 2551: 2539: 2533: 2519: 2411: 2396: 2236: 2028: 1882: 1875: 1661: 1629: 1567: 1564: 1553: 1215: 1213: 1203: 1161: 1151: 1145: 1098:Dividing by 1052: 1002: 1001:8 (which is 994: 993:7 (which is 989: 949: 947: 838: 419: 317: 220: 205: 201: 197: 192:When only a 191: 180: 163: 124: 116: 115: 113:Latin letter 106: 104:Greek letter 99: 97: 68: 62: 36: 14452:WikiProject 14367:Cartography 14329:Jurimetrics 14281:Reliability 14012:Time domain 13991:(Ljung–Box) 13913:Time-series 13791:Categorical 13775:Time-series 13767:Categorical 13702:(Bernoulli) 13537:Correlation 13517:Correlation 13313:Jarque–Bera 13285:Chi-squared 13047:M-estimator 13000:Asymptotics 12944:Sufficiency 12711:Interaction 12623:Replication 12603:Effect size 12560:Violin plot 12540:Radar chart 12520:Forest plot 12510:Correlogram 12460:Kendall's τ 11388:Sample size 7485:Percentage 7482:Percentage 7469:Confidence 5593:Higgs boson 4527:sample size 4469:for error. 3882:above with 3328:denoted by 2559:sample mean 1075:instead of 999:instead of 223:square root 157:, than the 147:square root 14471:Categories 14319:Demography 14037:ARMA model 13842:Regression 13419:(Friedman) 13380:(Wilcoxon) 13318:Normality 13308:Lilliefors 13255:Student's 13131:Resampling 13005:Robustness 12993:divergence 12983:Efficiency 12921:(monotone) 12916:Likelihood 12833:Population 12666:Stratified 12618:Population 12437:Dependence 12393:Count data 12324:Percentile 12301:Dispersion 12234:Arithmetic 12169:Statistics 12032:: 71–110. 11871:29 October 11867:. Pristine 11819:1602.03837 11633:10 October 11499:: 187–197. 11426:References 11363:Percentile 11172:mean error 10363:from 1 to 9641:= 1, ..., 9303:values of 9275:See also: 7217:Proportion 7128:Proportion 5971:therefore 5543:gives the 4914:covariance 4740:covariance 4424:= 0.000982 4077:See also: 2841:This is a 2601:population 2522:See also: 2516:Estimation 2401:taken for 1880:notation, 950:population 65:statistics 43:A plot of 13700:Logistic 13467:posterior 13393:Rank sum 13141:Jackknife 13136:Bootstrap 12954:Bootstrap 12889:Parameter 12838:Statistic 12633:Statistic 12545:Run chart 12530:Pie chart 12525:Histogram 12515:Fan chart 12490:Bar chart 12372:L-moments 12259:Geometric 12124:EMS Press 12096:818846942 11956:CiteSeerX 11927:. Wolfram 11925:MathWorld 11852:124959784 11765:122328846 11608:21 August 11563:21 August 11538:MathWorld 11398:Six Sigma 11327:Error bar 11214:− 11158:The term 11108:σ 11098:− 11030:− 10928:σ 10886:− 10858:− 10847:− 10811:− 10781:− 10770:− 10733:− 10701:− 10635:− 10624:− 10574:− 10428:− 10280:∑ 10129:σ 10100:− 9989:− 9961:− 9950:− 9924:− 9894:− 9883:− 9855:− 9838:− 9744:− 9733:− 9709:− 9571:− 9542:− 9458:− 9436:σ 9349:∑ 9229:σ 9215:σ 9176:⁡ 9148:⁡ 9106:⁡ 9083:∑ 9020:∑ 9011:⁡ 8953:∑ 8934:⁡ 8910:⁡ 8865:⁡ 8848:≡ 8826:⁡ 8781:⁡ 8756:⁡ 8750:≡ 8714:⁡ 8693:σ 8689:≡ 8673:⁡ 8636:σ 8612:σ 8557:¯ 8483:− 8447:∑ 8437:− 8411:σ 7560:33.3333% 7557:66.6667% 7488:Fraction 7471:interval 7324:⁡ 7275:σ 7270:μ 7267:− 7254:⁡ 7221:≤ 7138:⁡ 7056:σ 7052:μ 7049:− 7025:− 7012:π 7004:σ 6981:σ 6974:μ 6907:ℓ 6886:σ 6879:ℓ 6876:− 6827:− 6803:σ 6774:σ 6745:σ 6716:σ 6687:σ 6658:σ 6629:σ 6532:¯ 6523:− 6498:∑ 6444:ℓ 6431:¯ 6418:ℓ 6392:∑ 6364:ℓ 6358:− 6339:∑ 6316:ℓ 6310:− 6291:∑ 6262:ℓ 6259:− 6243:ℓ 6240:− 6224:ℓ 6221:− 6185:ℓ 6182:− 6166:ℓ 6163:− 6147:ℓ 6144:− 6128:⋅ 6084:− 6075:⋅ 6046:The line 6026:∈ 6023:ℓ 6015:for some 6000:ℓ 5994:ℓ 5988:ℓ 5906:¯ 5891:¯ 5876:¯ 5821:¯ 5806:¯ 5791:¯ 5726:) : 5545:precision 5446:∑ 5426:− 5386:∑ 5351:¯ 5341:− 5301:∑ 5261:¯ 5252:− 5216:∑ 5150:⁡ 5144:− 5130:⁡ 5115:− 5044:⁡ 5035:− 5014:⁡ 4977:⁡ 4971:− 4957:⁡ 4937:σ 4871:σ 4818:⁡ 4796:⁡ 4778:⁡ 4752:σ 4708:σ 4673:σ 4654:σ 4629:σ 4603:σ 4392:α 4389:− 4367:α 4334:σ 4318:α 4313:− 4213:α 4210:− 4189:α 4184:− 4162:σ 4134:α 4023:¯ 4014:− 3978:∑ 3965:γ 3951:− 3945:− 3925:^ 3922:σ 3805:¯ 3796:− 3760:∑ 3750:− 3730:^ 3727:σ 3662:− 3649:Γ 3626:Γ 3610:− 3501:− 3411:¯ 3402:− 3366:∑ 3356:− 3296:≠ 3244:¯ 3235:− 3218:… 3208:¯ 3199:− 3138:¯ 3129:− 3093:∑ 3083:− 2985:¯ 2976:− 2940:∑ 2815:¯ 2772:… 2700:¯ 2691:− 2655:∑ 2567:efficient 2548:estimator 2544:statistic 2488:σ 2481:μ 2462:− 2451:σ 2347:∫ 2340:μ 2295:μ 2292:− 2275:∫ 2266:σ 2178:∑ 2171:μ 2148:μ 2145:− 2102:∑ 2093:σ 1985:∑ 1968:μ 1945:μ 1942:− 1909:∑ 1890:σ 1878:summation 1838:⋯ 1806:μ 1778:μ 1775:− 1756:⋯ 1740:μ 1737:− 1708:μ 1705:− 1669:σ 1591:∈ 1588:α 1570:fat tails 1516:⁡ 1507:− 1486:⁡ 1425:μ 1422:− 1411:∞ 1403:∞ 1400:− 1396:∫ 1370:μ 1367:− 1353:⁡ 1345:≡ 1342:σ 1288:∞ 1280:∞ 1277:− 1273:∫ 1257:⁡ 1251:≡ 1248:μ 1202:, or the 1170:is about 1109:− 1083:σ 1036:≈ 960:σ 852:σ 789:− 758:− 736:− 691:− 660:− 638:− 593:− 562:− 540:− 495:− 464:− 442:− 330:μ 81:deviation 73:variation 14414:Category 14107:Survival 13984:Johansen 13707:Binomial 13662:Isotonic 13249:(normal) 12894:location 12701:Blocking 12656:Sampling 12535:Q–Q plot 12500:Box plot 12482:Graphics 12377:Skewness 12367:Kurtosis 12339:Variance 12269:Heronian 12264:Harmonic 12018:(1894). 11986:(2003). 11844:26918975 11513:(1931). 11368:Raw data 11311:Cumulant 11262:See also 11094:′ 11085:′ 8518:calculus 7876:99.999% 7376:, where 6479:between 5746:distance 4573:location 4507:> 100 4096:or CI. 3887:− 1.5 + 3049:denoted 3037:, using 3031:variance 2563:unbiased 1556:variance 948:and the 841:variance 170:infinite 139:data set 14440:Commons 14387:Kriging 14272:Process 14229:studies 14088:Wavelet 13921:General 13088:Plug-in 12882:L space 12661:Cluster 12362:Moments 12180:Outline 12126:, 2001 12034:Bibcode 11824:Bibcode 11757:2685690 11670:2682923 11475:8664723 11466:2351401 11154:History 9424:current 9315:, ..., 8391:, ..., 7964:99.9999 7879:0.001% 7813:99.99% 7194:is the 7088:is the 6600:table. 5616:Finance 5607:Weather 5589:5 sigma 4921:moments 4434:= 5.024 4257:is the 3905:⁠ 3889:⁠ 3855:⁠ 3841:⁠ 3557:⁠ 3538:⁠ 3270:commute 2897:biased 2895:If the 2579:(using 2064:, ..., 1649:, ..., 1218:be the 1166:in the 227:average 225:of the 183:science 145:is the 93:outlier 14309:Census 13899:Normal 13847:Manova 13667:Robust 13417:2-way 13409:1-way 13247:-test 12918: 12495:Biplot 12286:Median 12279:Lehmer 12221:Center 12094: 12084: 11998: 11958: 11902: 11850: 11842: 11789:30 May 11763: 11755: 11668: 11473: 11463: 11136:where 10032:since 10022:Note: 9773:where 9617:, and 9493:Where 8895:hence 8650:where 8520:or by 8516:Using 8324:99.999 8277:99.999 8225:99.999 8179:99.999 8130:99.999 8088:99.999 8045:99.999 7996:99.999 7970:0.0001 7902:99.999 7837:99.993 7816:0.01% 7788:99.9% 7755:99.730 7717:21.977 7698:95.449 7606:31.731 7597:68.268 7510:3 / 4 7172:where 7102:, and 7084:where 5951:is on 4853:where 4482:> 4 4230:where 4085:, and 4056:where 2733:where 1214:Let 1196:normal 1053:sample 422:square 301: 292: 283: 274: 265: 256: 247: 194:sample 155:robust 131:sample 67:, the 13933:Trend 13462:prior 13404:anova 13293:-test 13267:-test 13259:-test 13166:Power 13111:Pivot 12904:shape 12899:scale 12349:Shape 12329:Range 12274:Heinz 12249:Cubic 12185:Index 11848:S2CID 11814:arXiv 11761:S2CID 11753:JSTOR 11666:JSTOR 11584:(PDF) 11170:used 11168:Gauss 8338:0.000 8332:7440% 8289:0.000 8266:6.806 8237:0.000 8214:6.466 8188:0.000 8168:6.109 8142:0.000 8097:0.000 8077:5.730 8054:0.000 8034:5.326 8008:0.000 7953:4.891 7935:.5358 7924:6249% 7916:0.000 7910:3751% 7866:4.417 7846:0.006 7803:3.890 7791:0.1% 7778:3.290 7764:0.269 7727:2.575 7707:4.550 7670:1.959 7648:1.644 7626:1.281 7616:3.151 7569:0.994 7547:0.977 7516:0.674 7494:0.318 7406:, or 4577:scale 4497:= 0.6 4452:= 100 4432:0.975 4422:0.025 4409:With 3880:− 1.5 3709:with 2244:with 141:, or 14166:Test 13366:Sign 13218:Wald 12291:Mode 12229:Mean 12092:OCLC 12082:ISBN 11996:ISBN 11933:2014 11900:ISBN 11873:2011 11840:PMID 11791:2015 11695:2021 11635:2022 11610:2020 11565:2020 11471:PMID 11053:or 10981:and 9636:For 9290:and 9259:mean 9219:mean 8917:mean 8616:mean 8190:0001 8181:9999 8147:1973 8135:8027 8013:3303 8001:6697 7796:1000 7766:9796 7757:0204 7737:99% 7709:0264 7700:9736 7680:95% 7661:10% 7658:90% 7639:20% 7636:80% 7618:4872 7608:0508 7599:9492 7582:32% 7579:68% 7507:75% 7504:25% 7422:= (− 6942:The 6788:97% 6759:96% 6730:94% 6701:89% 6672:75% 6643:50% 6471:and 5714:= {( 5622:risk 5597:CERN 5518:and 4890:and 4587:and 4444:= 10 4426:and 4343:< 4330:< 4267:1 − 4173:< 4143:< 3902:− 1) 3527:For 3484:and 2868:> 2424:and 1039:2.1. 997:− 1) 839:The 320:mean 85:mean 77:mean 13346:BIC 13341:AIC 12042:doi 12030:185 11966:doi 11896:438 11832:doi 11810:116 11745:doi 11718:doi 11658:doi 11461:PMC 11453:doi 11449:312 11445:BMJ 11258:). 11199:SDI 10039:or 10030:= 0 9506:). 9173:var 9145:var 9103:var 9008:var 8931:var 8907:var 8862:var 8823:var 8778:var 8753:var 8711:var 8670:var 8658:): 8362:445 8359:215 8356:682 8354:390 8346:256 8343:000 8340:000 8329:999 8326:999 8310:000 8307:000 8304:000 8302:100 8294:001 8291:000 8282:999 8279:999 8268:502 8258:000 8255:000 8252:000 8239:000 8227:999 8216:951 8206:000 8203:000 8200:000 8170:410 8160:346 8157:797 8155:506 8144:000 8132:999 8112:000 8109:000 8107:100 8099:001 8090:999 8079:729 8069:000 8066:000 8036:724 8026:278 8023:744 8010:057 7998:942 7982:000 7979:000 7955:638 7945:000 7942:000 7933:159 7931:147 7921:534 7918:679 7907:465 7904:320 7894:4.5 7886:000 7884:100 7868:173 7858:787 7848:334 7839:666 7823:000 7805:592 7780:527 7740:1% 7729:829 7719:895 7683:5% 7672:964 7650:854 7628:552 7571:458 7549:925 7518:490 7496:639 7395:± 3 7385:± 2 7321:erf 7251:erf 7181:erf 7135:erf 6950:of 6054:to 5752:to 5748:of 5703:in 5680:= ( 4899:cov 4862:var 4815:cov 4793:var 4775:var 4414:= 1 4104:= 2 3948:1.5 3870:= 9 3863:= 3 3753:1.5 2258:is 1558:of 1330:of 991:by 821:16. 200:or 181:In 63:In 14473:: 12122:, 12116:, 12090:. 12040:. 12028:. 12022:. 11964:. 11950:. 11923:. 11898:. 11846:, 11838:, 11830:, 11822:, 11808:, 11759:. 11751:. 11741:55 11739:. 11712:. 11685:. 11664:, 11654:25 11652:, 11626:. 11601:. 11556:. 11535:. 11495:. 11469:. 11459:. 11447:. 11443:. 11174:. 10513:: 10466:1/ 10367:: 10353:. 10248:, 10241:, 10055:. 10047:= 9646:: 9613:, 9413:, 9404:, 9325:: 8586:. 8349:% 8297:% 8285:% 8250:10 8245:% 8242:01 8233:% 8230:99 8193:% 8184:% 8150:% 8138:% 8102:% 8093:% 8064:10 8059:% 8056:01 8050:% 8047:99 8016:% 8004:% 7972:% 7966:% 7856:15 7851:% 7842:% 7821:10 7769:% 7760:% 7712:% 7703:% 7611:% 7602:% 7535:% 7533:50 7529:% 7527:50 7429:, 7420:CI 7410:. 7371:± 7206:: 7110:. 6576:, 6569:, 6487:) 6043:. 5730:∈ 5722:, 5718:, 5694:, 5687:, 5667:, 5660:, 5603:. 5552:. 4591:: 4545:≈ 4523:/4 4519:≈ 4416:, 4282:Pr 4120:Pr 4110:: 4081:, 3908:. 3898:8( 3330:s: 3053:: 2871:75 2565:, 2409:. 2047:, 1642:, 1562:. 1240:: 1023:32 976:2. 934:4. 923:32 909:16 406:5. 395:40 304:9. 137:, 133:, 129:, 100:SD 13291:G 13265:F 13257:t 13245:Z 12964:V 12959:U 12161:e 12154:t 12147:v 12134:" 12130:" 12098:. 12065:. 12050:. 12044:: 12036:: 12004:. 11972:. 11968:: 11952:4 11935:. 11908:. 11875:. 11834:: 11826:: 11816:: 11793:. 11767:. 11747:: 11724:. 11720:: 11714:2 11697:. 11660:: 11637:. 11612:. 11567:. 11541:. 11497:1 11477:. 11455:: 11203:= 11144:′ 11142:n 11138:n 11122:, 11117:2 11112:n 11101:1 11091:n 11082:n 11076:= 11071:2 11066:n 11062:s 11039:, 11033:1 11025:n 11021:W 11014:n 11010:Q 11004:= 10999:2 10994:n 10990:s 10962:n 10958:W 10952:n 10948:Q 10942:= 10937:2 10932:n 10900:) 10894:k 10890:A 10881:k 10877:x 10872:( 10867:) 10861:1 10855:k 10851:A 10842:k 10838:x 10833:( 10827:k 10823:w 10819:+ 10814:1 10808:k 10804:Q 10800:= 10795:2 10790:) 10784:1 10778:k 10774:A 10765:k 10761:x 10756:( 10747:k 10743:W 10736:1 10730:k 10726:W 10720:k 10716:w 10709:+ 10704:1 10698:k 10694:Q 10690:= 10681:k 10677:Q 10669:0 10666:= 10657:0 10653:Q 10644:) 10638:1 10632:k 10628:A 10619:k 10615:x 10610:( 10602:k 10598:W 10592:k 10588:w 10582:+ 10577:1 10571:k 10567:A 10563:= 10554:k 10550:A 10542:0 10539:= 10530:0 10526:A 10499:k 10495:W 10490:/ 10484:k 10480:w 10468:k 10444:k 10440:w 10436:+ 10431:1 10425:k 10421:W 10417:= 10408:k 10404:W 10396:0 10393:= 10384:0 10380:W 10365:n 10361:k 10351:N 10346:0 10343:s 10325:. 10320:j 10315:k 10311:x 10305:k 10301:w 10295:N 10290:1 10287:= 10284:k 10276:= 10271:j 10267:s 10253:2 10250:s 10246:1 10243:s 10239:0 10236:s 10219:k 10215:w 10192:k 10188:x 10158:n 10153:n 10149:Q 10143:= 10138:2 10133:n 10103:1 10097:n 10091:n 10087:Q 10081:= 10076:2 10071:n 10067:s 10052:1 10049:A 10045:1 10042:x 10035:k 10028:1 10025:Q 10003:) 9997:k 9993:A 9984:k 9980:x 9975:( 9970:) 9964:1 9958:k 9954:A 9945:k 9941:x 9936:( 9932:+ 9927:1 9921:k 9917:Q 9913:= 9908:2 9903:) 9897:1 9891:k 9887:A 9878:k 9874:x 9869:( 9862:k 9858:1 9852:k 9846:+ 9841:1 9835:k 9831:Q 9827:= 9818:k 9814:Q 9806:0 9803:= 9794:0 9790:Q 9775:A 9753:k 9747:1 9741:k 9737:A 9728:k 9724:x 9717:+ 9712:1 9706:k 9702:A 9698:= 9689:k 9685:A 9677:0 9674:= 9665:0 9661:A 9643:n 9639:k 9631:n 9627:n 9623:n 9605:j 9601:s 9584:. 9577:) 9574:1 9568:N 9565:( 9562:N 9555:2 9550:1 9546:s 9537:2 9533:s 9529:N 9522:= 9519:s 9503:0 9500:s 9495:N 9477:N 9471:2 9466:1 9462:s 9453:2 9449:s 9445:N 9439:= 9419:2 9416:s 9410:1 9407:s 9402:N 9386:. 9380:j 9375:k 9371:x 9364:N 9359:1 9356:= 9353:k 9345:= 9340:j 9336:s 9321:N 9317:x 9313:1 9310:x 9305:x 9301:N 9296:2 9293:s 9287:1 9284:s 9264:σ 9256:σ 9239:. 9233:N 9224:= 9188:. 9185:) 9182:X 9179:( 9168:N 9165:1 9160:= 9157:) 9154:X 9151:( 9138:2 9134:N 9130:N 9125:= 9122:) 9117:i 9113:X 9109:( 9098:N 9093:1 9090:= 9087:i 9075:2 9071:N 9067:1 9062:= 9051:) 9045:i 9041:X 9035:N 9030:1 9027:= 9024:i 9015:( 9001:2 8997:N 8993:1 8988:= 8984:) 8978:i 8974:X 8968:N 8963:1 8960:= 8957:i 8947:N 8944:1 8938:( 8928:= 8921:) 8913:( 8881:) 8876:1 8872:X 8868:( 8856:2 8852:c 8845:) 8840:1 8836:X 8832:c 8829:( 8797:) 8792:2 8788:X 8784:( 8775:+ 8772:) 8767:1 8763:X 8759:( 8743:) 8738:2 8734:X 8730:+ 8725:1 8721:X 8717:( 8702:2 8697:X 8682:) 8679:X 8676:( 8652:N 8630:N 8626:1 8621:= 8563:. 8554:x 8548:= 8545:r 8533:) 8531:r 8529:( 8527:σ 8502:. 8495:2 8490:) 8486:r 8478:i 8474:x 8469:( 8462:N 8457:1 8454:= 8451:i 8440:1 8434:N 8430:1 8423:= 8420:) 8417:r 8414:( 8397:n 8393:x 8389:1 8386:x 8319:σ 8317:7 8272:σ 8220:σ 8198:1 8174:σ 8124:σ 8121:6 8083:σ 8040:σ 8021:1 7991:σ 7989:5 7977:1 7959:σ 7940:1 7897:σ 7872:σ 7832:σ 7830:4 7809:σ 7784:σ 7750:σ 7748:3 7733:σ 7693:σ 7691:2 7676:σ 7654:σ 7632:σ 7592:σ 7590:1 7575:σ 7553:σ 7540:2 7522:σ 7500:σ 7458:z 7449:) 7447:z 7436:) 7434:σ 7431:z 7427:σ 7424:z 7415:z 7397:σ 7393:μ 7387:σ 7383:μ 7378:μ 7373:σ 7369:μ 7352:. 7348:] 7343:) 7337:2 7333:z 7328:( 7318:+ 7315:1 7311:[ 7305:2 7302:1 7297:= 7293:] 7288:) 7280:2 7264:x 7258:( 7248:+ 7245:1 7241:[ 7235:2 7232:1 7227:= 7224:x 7200:x 7157:) 7151:2 7147:z 7142:( 7132:= 7115:z 7104:n 7099:n 7094:σ 7086:μ 7066:2 7061:) 7046:x 7040:( 7033:2 7030:1 7021:e 7009:2 7000:1 6995:= 6991:) 6985:2 6977:, 6971:, 6968:x 6964:( 6960:f 6938:. 6873:1 6869:1 6840:2 6836:k 6832:1 6824:1 6800:k 6771:6 6742:5 6713:4 6684:3 6655:2 6623:2 6583:) 6581:3 6578:x 6574:2 6571:x 6567:1 6564:x 6562:( 6544:2 6539:) 6529:x 6518:i 6514:x 6509:( 6502:i 6485:L 6481:P 6473:M 6469:P 6441:= 6428:x 6415:= 6406:i 6402:x 6396:i 6386:3 6383:1 6374:0 6371:= 6361:3 6353:i 6349:x 6343:i 6331:0 6328:= 6320:) 6313:3 6305:i 6301:x 6295:i 6286:( 6282:r 6275:0 6272:= 6265:) 6254:3 6250:x 6246:+ 6235:2 6231:x 6227:+ 6216:1 6212:x 6208:( 6205:r 6198:0 6195:= 6188:) 6177:3 6173:x 6169:, 6158:2 6154:x 6150:, 6139:1 6135:x 6131:( 6125:) 6122:r 6119:, 6116:r 6113:, 6110:r 6107:( 6100:0 6097:= 6090:) 6087:M 6081:P 6078:( 6072:L 6056:P 6052:M 6048:L 6030:R 6003:) 5997:, 5991:, 5985:( 5982:= 5979:M 5959:L 5939:M 5913:) 5903:x 5897:, 5888:x 5882:, 5873:x 5866:( 5862:= 5859:M 5828:) 5818:x 5812:, 5803:x 5797:, 5788:x 5781:( 5777:= 5774:M 5762:P 5758:L 5754:L 5750:P 5742:L 5738:P 5734:} 5732:R 5728:r 5724:r 5720:r 5716:r 5712:L 5706:R 5701:) 5699:3 5696:x 5692:2 5689:x 5685:1 5682:x 5678:P 5672:3 5669:x 5665:2 5662:x 5658:1 5655:x 5489:, 5482:2 5477:) 5471:i 5467:x 5461:N 5456:1 5453:= 5450:i 5440:N 5437:1 5431:( 5422:) 5416:2 5411:i 5407:x 5401:N 5396:1 5393:= 5390:i 5380:N 5377:1 5371:( 5365:= 5358:2 5348:x 5337:) 5331:2 5326:i 5322:x 5316:N 5311:1 5308:= 5305:i 5296:( 5290:N 5287:1 5280:= 5273:2 5268:) 5258:x 5247:i 5243:x 5238:( 5231:N 5226:1 5223:= 5220:i 5210:N 5207:1 5179:. 5173:] 5167:2 5163:) 5159:] 5156:X 5153:[ 5147:E 5141:X 5138:( 5134:[ 5127:E 5118:1 5112:N 5108:N 5102:= 5099:) 5096:X 5093:( 5090:s 5068:. 5061:2 5057:) 5053:] 5050:X 5047:[ 5041:E 5038:( 5031:] 5026:2 5022:X 5018:[ 5011:E 5006:= 5000:] 4994:2 4990:) 4986:] 4983:X 4980:[ 4974:E 4968:X 4965:( 4961:[ 4954:E 4949:= 4946:) 4943:X 4940:( 4925:E 4875:2 4866:= 4838:. 4833:) 4830:Y 4827:, 4824:X 4821:( 4811:2 4808:+ 4805:) 4802:Y 4799:( 4790:+ 4787:) 4784:X 4781:( 4770:= 4767:) 4764:Y 4761:+ 4758:X 4755:( 4720:. 4717:) 4714:X 4711:( 4704:| 4700:c 4696:| 4692:= 4685:) 4682:X 4679:c 4676:( 4666:, 4663:) 4660:X 4657:( 4651:= 4644:) 4641:c 4638:+ 4635:X 4632:( 4622:0 4619:= 4612:) 4609:c 4606:( 4589:Y 4585:X 4581:c 4561:N 4557:) 4555:N 4553:( 4551:K 4549:/ 4547:R 4543:s 4538:) 4536:N 4534:( 4532:K 4521:R 4517:s 4512:R 4505:N 4499:R 4495:s 4490:s 4486:R 4480:N 4463:k 4450:N 4442:N 4429:q 4419:q 4412:k 4395:. 4386:1 4383:= 4379:) 4370:2 4362:q 4356:2 4352:s 4346:k 4338:2 4321:2 4310:1 4306:q 4300:2 4296:s 4290:k 4286:( 4269:α 4263:k 4259:p 4243:p 4239:q 4216:, 4207:1 4204:= 4200:) 4192:2 4181:1 4177:q 4166:2 4156:2 4152:s 4146:k 4137:2 4129:q 4124:( 4102:N 4062:2 4059:γ 4042:, 4035:2 4030:) 4020:x 4009:i 4005:x 4000:( 3993:N 3988:1 3985:= 3982:i 3969:2 3959:4 3956:1 3942:N 3938:1 3931:= 3900:N 3895:/ 3892:1 3885:N 3878:N 3868:N 3861:N 3851:N 3847:/ 3844:1 3824:, 3817:2 3812:) 3802:x 3791:i 3787:x 3782:( 3775:N 3770:1 3767:= 3764:i 3747:N 3743:1 3736:= 3712:N 3705:N 3681:. 3674:) 3669:2 3665:1 3659:N 3653:( 3643:) 3638:2 3635:N 3630:( 3613:1 3607:N 3603:2 3596:= 3592:) 3589:N 3586:( 3581:4 3577:c 3562:N 3553:4 3550:c 3546:/ 3542:s 3533:s 3504:1 3498:N 3494:1 3470:N 3467:1 3455:N 3453:( 3450:s 3446:s 3430:. 3423:2 3418:) 3408:x 3397:i 3393:x 3388:( 3381:N 3376:1 3373:= 3370:i 3359:1 3353:N 3349:1 3342:= 3339:s 3310:] 3307:X 3304:[ 3301:E 3293:] 3288:X 3283:[ 3280:E 3253:. 3250:) 3241:x 3230:n 3226:x 3221:, 3214:, 3205:x 3194:1 3190:x 3186:( 3171:N 3155:. 3150:2 3145:) 3135:x 3124:i 3120:x 3115:( 3108:N 3103:1 3100:= 3097:i 3086:1 3080:N 3076:1 3071:= 3066:2 3062:s 3051:s 3043:N 3039:N 3004:. 2997:2 2992:) 2982:x 2971:i 2967:x 2962:( 2955:N 2950:1 2947:= 2944:i 2934:N 2931:1 2924:= 2919:N 2915:s 2865:N 2855:N 2832:N 2812:x 2789:} 2784:N 2780:x 2775:, 2768:, 2763:2 2759:x 2754:, 2749:1 2745:x 2741:{ 2719:, 2712:2 2707:) 2697:x 2686:i 2682:x 2677:( 2670:N 2665:1 2662:= 2659:i 2649:N 2646:1 2639:= 2634:N 2630:s 2610:N 2606:s 2589:N 2585:N 2581:N 2552:s 2540:σ 2501:. 2492:2 2484:+ 2478:2 2474:e 2469:) 2465:1 2455:2 2446:e 2441:( 2427:σ 2422:μ 2407:X 2403:x 2383:, 2380:x 2376:d 2371:) 2368:x 2365:( 2362:p 2358:x 2352:X 2343:= 2332:, 2327:x 2323:d 2318:) 2315:x 2312:( 2309:p 2303:2 2299:) 2289:x 2286:( 2280:X 2269:= 2256:) 2254:x 2252:( 2250:p 2242:X 2218:. 2213:i 2209:x 2203:i 2199:p 2193:N 2188:1 2185:= 2182:i 2174:= 2163:, 2156:2 2152:) 2140:i 2136:x 2132:( 2127:i 2123:p 2117:N 2112:1 2109:= 2106:i 2096:= 2081:N 2077:p 2070:N 2066:x 2062:2 2059:p 2053:2 2050:x 2044:1 2041:p 2035:1 2032:x 2015:. 2010:i 2006:x 2000:N 1995:1 1992:= 1989:i 1979:N 1976:1 1971:= 1960:, 1953:2 1949:) 1937:i 1933:x 1929:( 1924:N 1919:1 1916:= 1913:i 1903:N 1900:1 1893:= 1857:, 1854:) 1849:N 1845:x 1841:+ 1835:+ 1830:1 1826:x 1822:( 1817:N 1814:1 1809:= 1798:, 1792:] 1786:2 1782:) 1770:N 1766:x 1762:( 1759:+ 1753:+ 1748:2 1744:) 1732:2 1728:x 1724:( 1721:+ 1716:2 1712:) 1700:1 1696:x 1692:( 1688:[ 1682:N 1679:1 1672:= 1655:N 1651:x 1647:2 1644:x 1640:1 1637:x 1632:X 1606:] 1603:2 1600:, 1597:1 1594:( 1560:X 1540:. 1533:2 1529:) 1525:] 1522:X 1519:[ 1513:E 1510:( 1503:] 1498:2 1494:X 1490:[ 1483:E 1461:, 1456:x 1452:d 1447:) 1444:x 1441:( 1438:f 1433:2 1429:) 1419:x 1416:( 1408:+ 1390:= 1384:] 1378:2 1374:) 1364:X 1361:( 1357:[ 1350:E 1332:X 1328:σ 1314:x 1310:d 1305:) 1302:x 1299:( 1296:f 1293:x 1285:+ 1269:= 1266:] 1263:X 1260:[ 1254:E 1238:) 1236:x 1234:( 1232:f 1227:X 1216:μ 1152:n 1132:n 1112:1 1106:n 1086:. 1063:s 1031:7 1027:/ 1018:= 1015:s 1005:) 1003:n 995:n 973:= 968:4 963:= 931:= 926:8 918:= 913:8 906:+ 903:4 900:+ 897:0 894:+ 891:0 888:+ 885:1 882:+ 879:1 876:+ 873:1 870:+ 867:9 861:= 856:2 818:= 813:2 809:4 805:= 800:2 796:) 792:5 786:9 783:( 777:1 774:= 769:2 765:) 761:1 755:( 752:= 747:2 743:) 739:5 733:4 730:( 723:4 720:= 715:2 711:2 707:= 702:2 698:) 694:5 688:7 685:( 679:1 676:= 671:2 667:) 663:1 657:( 654:= 649:2 645:) 641:5 635:4 632:( 625:0 622:= 617:2 613:0 609:= 604:2 600:) 596:5 590:5 587:( 581:1 578:= 573:2 569:) 565:1 559:( 556:= 551:2 547:) 543:5 537:4 534:( 527:0 524:= 519:2 515:0 511:= 506:2 502:) 498:5 492:5 489:( 483:9 480:= 475:2 471:) 467:3 461:( 458:= 453:2 449:) 445:5 439:2 436:( 403:= 398:8 390:= 385:8 381:9 378:+ 375:7 372:+ 369:5 366:+ 363:5 360:+ 357:4 354:+ 351:4 348:+ 345:4 342:+ 339:2 333:= 298:, 295:7 289:, 286:5 280:, 277:5 271:, 268:4 262:, 259:4 253:, 250:4 244:, 241:2 118:s 108:σ 51:. 34:. 20:)

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