Margin of Error Calculator

Frequently Asked Questions

Should I enter p as a decimal or a percentage?

Enter p as a proportion between 0 and 1 (e.g. 0.5 for 50%, 0.3 for 30%). The margin of error output is reported as a percentage for readability.

When does the finite population correction matter?

The FPC is meaningful when your sample is more than about 5% of the total population. For national polls where n = 1,000 out of millions, the correction is negligible. For a company survey of 200 employees in a workforce of 1,000, it reduces the MoE noticeably.

Why do polls always seem to use about 1,000 respondents?

A random sample of around 1,000 gives a MoE of about ±3% at 95% confidence with p = 0.5 - precise enough for most political and social research, and cost-efficient since going to 2,000 only drops MoE to ±2.2%.

Does a smaller MoE guarantee a more accurate poll?

No. A smaller MoE only means tighter bounds on random sampling error. A poll with a tiny MoE can still be wrong by many points if it suffers from nonresponse bias, bad sampling frames, or poor question design.

Provided by AllCalculators.io
Free online calculators for everyday. No registration required.

Estimates for informational purposes only.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

JavaScript is required to use the interactive calculator above. The questions and answers below remain readable without JavaScript.

How This Calculator Works

The margin of error (MoE) quantifies the uncertainty in a survey proportion p̂ = x/n. The formula is:

SE = √[p(1−p) ÷ n] and MoE = z* × SE

where z* is the critical value for the chosen confidence level: 1.645 (90%), 1.96 (95%), or 2.576 (99%). If you know the total population size N and it is not much larger than your sample, the finite population correction (FPC) reduces the SE: SE_FPC = SE × √[(N−n) / (N−1)]. The resulting interval is p ± MoE, clamped to [0%, 100%].

Why the Default Proportion Is 0.5

The variance p(1−p) is maximized when p = 0.5, giving the widest, most conservative margin of error. Using p = 0.5 when the true proportion is unknown ensures the calculated n or MoE is large enough regardless of the actual outcome.

When you have a reliable prior estimate (for example, from a pilot study or previous wave), using that value instead of 0.5 can substantially reduce the required sample or narrow the stated margin. For example, if you expect about 10% support for an issue, p = 0.1 reduces p(1−p) to 0.09 versus 0.25 for p = 0.5, cutting the variance by 64%.

Worked Example

A poll of n = 1,000 voters uses p = 0.5 at 95% confidence. SE = √(0.5 × 0.5 ÷ 1000) = √0.00025 ≈ 0.01581. MoE = 1.96 × 0.01581 ≈ 0.031, i.e. ±3.1%. The interval runs from 46.9% to 53.1%.

If the population is only 5,000 (e.g. a school of 5,000 students), the FPC applies: SE_FPC = 0.01581 × √[(5000−1000) / (5000−1)] = 0.01581 × 0.8944 ≈ 0.01414. MoE ≈ ±2.8% - meaningfully narrower.

What the Margin of Error Does Not Measure

The MoE only accounts for random sampling error - the natural variation from drawing a random subset of the population. It does not capture:

Nonresponse bias: people who decline to participate may differ systematically from those who do.

Coverage bias: the sampling frame may exclude important subgroups (e.g., online surveys miss non-internet users).

Question wording effects: framing, order, and wording of questions can shift answers by more than the MoE.

Social desirability bias: respondents may not answer honestly on sensitive topics.

A MoE of ±3% from a biased convenience sample is not meaningful; only probability-based sampling gives the MoE a valid probabilistic interpretation.

Diminishing Returns of Larger Samples

Because MoE shrinks with 1/√n, the returns to larger samples diminish quickly. Doubling from 1,000 to 2,000 only reduces MoE from ±3.1% to ±2.2%. Quadrupling from 1,000 to 4,000 brings it down to ±1.5%. This is why most national polls use 1,000-1,200 respondents - beyond that point, further sampling cost is rarely justified by the incremental precision gained.

Margin of Error Calculator

Frequently Asked Questions

Related Statistics Calculators