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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 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.
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).
where ¯ and ¯ are the sample mean and its standard error, with denoting the corrected sample standard deviation, and sample size.
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 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.
To address such potential overfitting, AICc was developed: AICc is AIC with a correction for small sample sizes. The formula for AICc depends upon the statistical model. Assuming that the model is univariate, is linear in its parameters, and has normally-distributed residuals (conditional upon regressors), then the formula for AICc is as follows.
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.
In sampling theory, the sampling fraction is the ratio of sample size to population size or, in the context of stratified sampling, the ratio of the sample size to the size of the stratum. The formula for the sampling fraction is =, where n is the sample size and N is the population size. A sampling fraction value close to 1 will occur if the ...