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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.
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 ...
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.
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).
With these data, the sample size (161 patients) is not small, however results from the McNemar test and other versions are different. The exact binomial test gives p = 0.053 and McNemar's test with continuity correction gives = 3.68 and p = 0.055.
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 }
While Mauchly's test is one of the most commonly used to evaluate sphericity, the test fails to detect departures from sphericity in small samples and over-detects departures from sphericity in large samples. Consequently, the sample size has an influence on the interpretation of the results.
If there are k samples with sizes and sample variances then Bartlett's test statistic is χ 2 = ( N − k ) ln ( S p 2 ) − ∑ i = 1 k ( n i − 1 ) ln ( S i 2 ) 1 + 1 3 ( k − 1 ) ( ∑ i = 1 k ( 1 n i − 1 ) − 1 N − k ) {\displaystyle \chi ^{2}={\frac {(N-k)\ln(S_{p}^{2})-\sum _{i=1}^{k}(n_{i}-1)\ln(S_{i}^{2})}{1+{\frac {1}{3 ...