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In statistics, Bessel's correction is the use of n − 1 instead of n in the formula for the sample variance and sample standard deviation, where n is the number of observations in a sample. This method corrects the bias in the estimation of the population variance.
This depends on the sample size n, and is given as follows: c 4 ( n ) = 2 n − 1 Γ ( n 2 ) Γ ( n − 1 2 ) = 1 − 1 4 n − 7 32 n 2 − 19 128 n 3 + O ( n − 4 ) {\displaystyle c_{4}(n)={\sqrt {\frac {2}{n-1}}}{\frac {\Gamma \left({\frac {n}{2}}\right)}{\Gamma \left({\frac {n-1}{2}}\right)}}=1-{\frac {1}{4n}}-{\frac {7}{32n^{2}}}-{\frac ...
To determine the sample size n required for a confidence interval of width W, with W/2 as the margin of error on each side of the sample mean, the equation Z σ n = W / 2 {\displaystyle {\frac {Z\sigma }{\sqrt {n}}}=W/2} can be solved.
Where is the sample size, = / is the fraction of the sample from the population, () is the (squared) finite population correction (FPC), is the unbiassed sample variance, and (¯) is some estimator of the variance of the mean under the sampling design. The issue with the above formula is that it is extremely rare to be able to directly estimate ...
Welch's t-test defines the statistic t by the following formula: t = Δ X ¯ s Δ X ¯ = X ¯ 1 − X ¯ 2 s X ¯ 1 2 + s X ¯ 2 2 {\displaystyle t={\frac {\Delta {\overline {X}}}{s_{\Delta {\bar {X}}}}}={\frac {{\overline {X}}_{1}-{\overline {X}}_{2}}{\sqrt {{s_{{\bar {X}}_{1}}^{2}}+{s_{{\bar {X}}_{2}}^{2}}}}}\,}
For example, in the R statistical computing environment, this value can be obtained as fisher.test(rbind(c(1,9),c(11,3)), alternative="less")$p.value, or in Python, using scipy.stats.fisher_exact(table=[[1,9],[11,3]], alternative="less") (where one receives both the prior odds ratio and the p -value).
This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall Street stock quotes. Moreover, this formula works for positive and negative ρ alike. See also unbiased estimation of standard deviation for more discussion.
The Bonferroni correction compensates for that increase by testing each individual hypothesis at a significance level of , where is the desired overall alpha level and is the number of hypotheses. [4] For example, if a trial is testing hypotheses with a desired overall , then the Bonferroni correction would test each individual hypothesis at .
That is, as sample size increases: n ( X ¯ − μ ) → d N ( 0 , σ 2 ) {\displaystyle {\sqrt {n}}({\bar {X}}-\mu )\xrightarrow {d} N(0,\sigma ^{2})} as per the Central limit theorem , s 2 → p σ 2 {\displaystyle s^{2}\xrightarrow {p} \sigma ^{2}} as per the law of large numbers ,
If the sample mean and uncorrected sample variance are defined as X ¯ = 1 n ∑ i = 1 n X i S 2 = 1 n ∑ i = 1 n ( X i − X ¯ ) 2 {\displaystyle {\overline {X}}\,={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\qquad S^{2}={\frac {1}{n}}\sum _{i=1}^{n}{\big (}X_{i}-{\overline {X}}\,{\big )}^{2}\qquad }