The formulas
Mean: x̄ ± t × s ÷ √n, with t the two-sided critical value of the Student t distribution on n − 1 degrees of freedom. With a population of N items the standard error is multiplied by √(1 − n ÷ N). Proportion: Wald is p ± z × √(p(1 − p) ÷ n). Wilson is (p + z² ÷ 2n ± z × √(p(1 − p) ÷ n + z² ÷ 4n²)) ÷ (1 + z² ÷ n). The Wilson score interval comes from Wilson, 1927, Journal of the American Statistical Association, and the t distribution from Gosset ("Student"), 1908.
When to use it
Use it to put a range around a sample mean, such as the average audited value per item or the mean of a recalculated ratio, and around an observed exception rate. In sampling terms, AU-C 530 requires you to consider sampling risk when you project results, and an interval is one way to show it.
Limits
The mean interval assumes the sample is random and the sample mean is close to normally distributed, which is a stretch for small samples from skewed populations such as receivable balances. Intervals describe sampling error only. They do not cover bias, a wrong population, or items that were never in the frame. Illustrative figures only. Check the result against your firm's methodology before relying on it.
Common questions
What does a 95% confidence interval mean?
If you drew many samples the same way and built an interval from each, about 95% of those intervals would contain the true population value. It does not say there is a 95% probability that this one interval holds the true value, and it says nothing about bias in how the sample was chosen.
When do I use t and when do I use z?
Use t for a mean when the standard deviation comes from the sample, which is nearly always. The t critical value is larger than z for small samples (2.228 against 1.960 at 95% with 10 degrees of freedom) and approaches it as the sample grows. This calculator uses t for means and z for proportions.
Which proportion interval should I use?
The Wald interval is the textbook formula and works when n is large and the rate is away from 0% and 100%. It fails exactly where auditors work: low deviation rates and modest samples. If you find zero deviations, Wald reports a zero-width interval. The Wilson interval does not have that flaw and is the safer default.
When should I apply the finite population correction?
When the sample is a meaningful share of the population, a common rule is more than 5%. The correction multiplies the standard error by √(1 − n ÷ N) and narrows the interval. For a sample of 40 from 100 items it reduces the standard error by about 23%.
Is this the same as the upper deviation limit in a test of controls?
No. A test of controls uses a one-sided upper limit, usually from binomial or Poisson tables in attribute sampling. This page gives two-sided intervals. Use the attribute sampling calculator for the upper deviation limit.
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