How this calculator works
A central confidence interval extends a critical multiple of the standard error on each side of an estimate. Known population SD uses a normal critical value; an estimated SD for a normal mean uses a t value with n - 1 degrees of freedom. The proportion option is the simple Wald approximation, which can fail near zero or one. Enter population size 0 to omit finite correction. For known population SD or proportions, the correction uses divisor N; a sample-SD t margin uses sqrt(1 - n/N). Reverse proportion modes solve sample size, confidence, both compatible proportions, population size or a success count. A census has zero sampling margin, though measurement and non-response error can remain.
Formula
E = critical × SE; mean SE = SD/√n; proportion SE = √(p(1 - p)/n). Finite factor = √((N - n)/(N - 1)) for population SD or proportions; √(1 - n/N) for a sample-SD mean margin. Inverting p(1 - p) gives two proportion solutions.
Worked example
At 95% confidence, known SD 10 and n = 100 give margin 1.959964. A proportion 0.5 with n = 100 gives a Wald margin about 0.0979982, or 9.79982 percentage points.
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Frequently asked questions
Does this cover every source of survey error?
No. Sampling margins do not include selection bias, measurement error or non-response bias.
Which SD convention does finite correction use?
Known population SD and proportion modes use sqrt((N - n)/(N - 1)). The sample-SD t mode uses sqrt(1 - n/N), matching a variance estimate with divisor n - 1.
Why can a proportion margin be zero at an endpoint?
The Wald plug-in standard error vanishes at p = 0 or 1. That does not establish certainty; prefer Wilson or exact intervals for such data.