Margin of Error Calculator

This calculator estimates the Margin of Error for a survey statistic using the standard normal approximation: Margin of Error = z × √(Expected Proportion × (1 − Expected Proportion) / Sample Size) (lines 10, 17-18), then widens or narrows the interval with a Finite Population Correction when a positive Population Size at least as large as Sample Size is supplied (lines 21-25). At the calculator's defaults, Expected Proportion sits exactly at 50%, a stationary point of the p×(1−p) term where the first-order (linear) effect vanishes — a small nudge either side of 50% shifts p×(1−p) only to second order, so Margin of Error still moves, just by a tiny, symmetric amount: 5.00% at p=50%, versus 4.98% at both p=45% and p=55%, and 4.90% at p=60%. It's a stationary point, not a flat region — a symmetric ±10% probe around 50% reads zero because the two directions cancel exactly, not because Expected Proportion is inert; away from 50% the effect is larger and no longer symmetric.

Expected Proportion, not Sample Size, is what shifts the Lower and Upper Confidence Interval Bounds the most, since both are built by adding or subtracting the margin from Expected Proportion directly (lines 28-29). The z-score lookup only has exact entries for confidence levels 1 (90%), 2 (95%), and 3 (99%); anything else silently falls back to the 95% value. This calculator does not account for stratified or cluster sampling designs — it assumes simple random sampling, so a survey built on a clustered or stratified design needs the Design Effect from a calculator like Sampling Frame applied on top of this result.