Margin of error for proportion:
formula and worked example

What a margin of error is and is not

When you ask a sample of people a yes-or-no question, the share who say yes is an estimate of the share in the whole population. Draw another sample and you will get a slightly different figure. The margin of error describes how much such estimates typically vary because of that sampling luck alone.

A result of 50% with a margin of 4.9 points at 95% confidence means that, if the same poll were repeated many times, about 95% of the intervals built this way would contain the true population share. Roughly, the true figure is likely between about 45% and 55%.

It is a statement about chance variation in who happened to be sampled. It does not cover a badly worded question, a sample that missed some groups, or people who declined to answer.

The formula and the two inputs

Take the square root of p times (1 − p), divided by n. Multiply by 1.96 for 95% confidence and by 100 to express it in percentage points.

• P: the sample proportion as a decimal, for example 0.5 for 50%.
• N: the number of valid responses, 400 in the example.
• 1.96: the multiplier for a 95% confidence level in a normal approximation.

The inner square root is the standard error. It shrinks as n grows, but only with the square root of n, which is where the diminishing returns of large samples come from.

Worked example: 50% support among 400 respondents

A survey of 400 people finds that exactly half agree with a statement.

1. 1Multiply p by its complement: 0.5 × 0.5 = 0.25.
2. 2Divide by n: 0.25 ÷ 400 = 0.000625.
3. 3Square root: √0.000625 = 0.025.
4. 4Multiply by 1.96: 0.049, which is 4.9 percentage points.

The estimate is therefore 50% plus or minus about 4.9 points, which is a range from 45.1% to 54.9%.

How to read the margin in a report

The margin tells you how much two figures must differ before the gap is more than noise. If one party polls 52% and another 48% with a margin of 4.9 points, the lead of 4 is inside the margin and cannot be treated as a firm lead.

Comparing two groups is harsher than the single-figure margin suggests. The uncertainty of the gap between two samples is larger than the margin for either, so differences need to be bigger than the stated margin to be taken seriously.

Also consider who answered. A margin computed from 400 responses is only meaningful if those 400 resemble the population. Weighting, clustering and nonresponse widen the real uncertainty beyond the printed figure.

How the share and the sample size move the margin

The margin is widest when the split is even and narrows as the share moves toward 0 or 100%. Moving P from 0.5 to 0.55 trims it only slightly, since 0.55 × 0.45 = 0.2475 is close to 0.25.

Sample size is the stronger lever. Going from 400 to 1,600 respondents, four times as many, cuts the margin at p = 0.5 from 4.9 to about 2.45 points. Each further halving costs four times the sample, so surveys rarely chase tiny margins.

Mistakes that misstate survey uncertainty

• Entering the percentage rather than the proportion. 50 instead of 0.5 gives a negative number under the root.
• Using the total population in place of the sample size. The margin depends on the number of responses, not the size of the group, unless the sample is a large fraction of it.
• Treating the margin as covering every kind of error. Bias in who was sampled is not included.
• Applying the formula to subgroups with the full sample size, when a subgroup has far fewer responses and a much wider margin.

Checking a poll before quoting it

A report says 56% of 250 customers prefer the new design. What does the margin say?

1. 1Standard error: √(0.56 × 0.44 ÷ 250) = √0.000986 ≈ 0.0314.
2. 2Margin: 1.96 × 0.0314 ≈ 0.0615, about 6.2 points.
3. 3Interval: roughly 50% to 62%.

Since the lower end is about 50%, the survey does not clearly show a majority, even though the headline figure is 56%.