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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.
To determine an appropriate sample size n for estimating proportions, the equation below can be solved, where W represents the desired width of the confidence interval. The resulting sample size formula, is often applied with a conservative estimate of p (e.g., 0.5): = /
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).
To reduce the error in approximation, Frank Yates, an English statistician, suggested a correction for continuity that adjusts the formula for Pearson's chi-squared test by subtracting 0.5 from the difference between each observed value and its expected value in a 2 × 2 contingency table.
where ¯ and ¯ are the sample mean and its standard error, with denoting the corrected sample standard deviation, and sample size.
The use of n − 1 instead of n in the formula for the sample variance is known as Bessel's correction, which corrects the bias in the estimation of the population variance, and some, but not all of the bias in the estimation of the population standard deviation.
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.
This also influences the sample size (overall, per stratum, per cluster, etc.). When planning the sample size, work may be done to correct the design effect so as to separate the interviewer effect (measurement error) from the effects of the sampling design on the sampling variance.
It is used for comparing two or more independent samples of equal or different sample sizes. It extends the Mann–Whitney U test , which is used for comparing only two groups. The parametric equivalent of the Kruskal–Wallis test is the one-way analysis of variance (ANOVA).
The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.