Sample Size Calculator

Sample size for estimating a proportion comes from inverting the confidence interval formula. The half-width of the interval is z times the square root of p times (1-p) divided by n, and solving for n gives n0 = z squared times p(1-p) divided by e squared. The variance term p(1-p) reaches its maximum of 0.25 at p = 0.50, which is why 50 percent is the conservative default when nothing is known in advance. At 95 percent confidence and a margin of 5 percentage points the requirement is 384.16, rounded to 385, regardless of how large the population is; tightening the margin to 3 points raises it to 1,068 and to 1 point raises it to 9,604, since n scales with the inverse square of the margin. The finite population correction, n0 divided by 1 plus (n0-1)/N, matters only when the sample is a substantial share of the population, conventionally more than about 5 percent. For a herd, ward or school of 500 the corrected requirement drops from 385 to 217, while for any population above roughly 20,000 the correction changes almost nothing. Two limitations are routinely overlooked. The formula assumes simple random sampling, so cluster or multistage designs must multiply n by a design effect that is commonly between 1.5 and 3.0. And the margin of error applies to the whole-sample estimate only; any subgroup you intend to report needs its own calculation. Inflating for non-response by dividing by the expected response rate protects the sample size but does nothing about non-response bias.

Until the late nineteenth century official statistics meant complete enumeration. Anders Nicolai Kiaer, director of the Norwegian statistical bureau, proposed a representative method based on partial investigation at the International Statistical Institute in 1895 and met heavy resistance because he could offer no measure of error. Arthur Bowley supplied part of the missing theory in the early 1900s, applying standard error reasoning to random selection in his survey work in Reading, published in 1915 as Livelihood and Poverty. The question was settled by Jerzy Neyman's 1934 paper for the Journal of the Royal Statistical Society on the two different aspects of the representative method, which introduced confidence intervals for survey estimates and the theory of stratification with optimal allocation, and made probability sampling rather than purposive selection the professional standard. Practice followed quickly: the failure of the 1936 Literary Digest straw poll against the much smaller Gallup sample dramatised the point, and the United States Census Bureau, through the work of Morris Hansen and William Hurwitz, built probability sampling into the 1940 census and the Current Population Survey. William G. Cochran's textbook Sampling Techniques, published in 1953 and revised through a third edition in 1977, codified the proportion formula and the finite population correction in the form still taught in biostatistics and used in ecological and clinical survey design.