Variances and Confidence Intervals
Find the smallest sample size needed for a two-sided confidence interval for a normally distributed population standard deviation to have a specified maximum width. The calculation uses the exact chi-square confidence interval for a standard deviation with an unknown population mean and assumes the future sample standard deviation equals the planning value.
For normally distributed observations with an unknown population mean, the sample variance satisfies (N − 1)S2/σ2 ~ χ2N−1. Therefore, the exact two-sided 100(1 − α)% confidence interval for the population standard deviation is obtained by taking the square root of the corresponding chi-square variance limits.
The calculator searches integer sample sizes starting at N = 2 and returns the first N whose exact confidence-interval width is at most the specified target. Because the lower and upper standard-deviation limits are asymmetric, the requested width is the distance between the two limits rather than twice a single margin of error.
The procedure assumes independent observations from a normal population and uses the supplied standard deviation as the planning value for the future sample. As PASS notes, if the standard deviation observed in the eventual sample differs from the planning value, the achieved confidence-interval width may be narrower or wider than specified.
PASS User's Guide III, Example 2, based on Hahn and Meeker (1991, p. 56), uses a 95% confidence level, a standard deviation of 1.31, and a target two-sided interval width of 2.9795. PASS reports a required sample size of 5, with an actual width of 2.979 and confidence limits of 0.785 and 3.764.
Hahn, G. J. & Meeker, W. Q. (1991). Statistical Intervals: A Guide for Practitioners. John Wiley & Sons, New York.
Hintze, J. L. (2008). PASS User's Guide III: Standard Deviations, Variances, Normality Tests, Survival Analysis, Correlations, Regression, Design of Experiments, and Tools, Helps, and Aids. NCSS. Chapter 640, “Confidence Intervals for One Standard Deviation using Standard Deviation.”