Standard error of a proportion
Describe sampling variation of a sample proportion.
Standard error of a proportion is one of 2 sampling formulas in the statistics section of this library, and it is used at ap · university level.
Why standard error of a proportion works
A proportion is just the mean of a column of zeros and ones. One such value has a variance of p times 1 minus p, and averaging n independent copies divides that by n, so the square root gives the typical distance between the sample proportion and the truth.
What each symbol means
$p$ is population success proportion and $n$ sample size.
Standard error of a proportion: when it holds
Independence and a suitable random-sampling model are required; use $\hat p$ when estimating for a confidence interval.
When it stops applying
It assumes independent trials that all share the same success probability. It also becomes badly misleading near 0 or 1: with 20 trials and no successes at all it reports a standard error of exactly zero, pretending to certainty where the data give almost none.
Standard error of a proportion: a worked example
For $p=.5,n=100$, $SE=.05$.
The mistake to avoid
What people do: Taking the square root of the top before dividing by n, instead of dividing inside the root.
Why it goes wrong: With p of 0.5 and 100 trials the correct standard error is 0.05, while the misplaced root gives 0.005. That makes the estimate look ten times more precise than it really is.
Do this instead: Compute p times 1 minus p, divide that by n while still inside the radical, and only then take the square root.
Standard error of a proportion: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $p$ is population success proportion and $n$ sample size.
- Check the conditions before substituting. Independence and a suitable random-sampling model are required; use $\hat p$ when estimating for a confidence interval.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most statistics slips.
Where this formula fits
- Subject
- Statistics formulas — 21 entries in this library
- Topic
- Sampling
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where standard error of a proportion comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Standard Deviation — the lesson behind this formula: measure typical distance from the mean.
- Statistics Calculator — check your substitution and the value it produces.
- Study statistics — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 21 statistics formulas — the full grouped reference, or the complete formula library.
Questions about standard error of a proportion
Which proportion do I put in, the true one or the sample one?
Use a planned value of p when designing a study and deciding sample size, and use the observed sample proportion when building an interval after the data are collected.
Why does a proportion of 0.5 give the biggest standard error?
Because p times 1 minus p peaks at 0.25 there and falls away toward both ends. A near-unanimous population is easy to measure; a split one needs the most data.
How large a sample do I need for a 3 percent margin?
Using the worst case of 0.5 and 95 percent confidence, the requirement works out to 1067.1, so you round up to 1068 people. That is why so many national polls report about a thousand respondents.
What should I do if my sample had zero successes?
Do not use this formula, since it collapses to zero width. Use an adjusted method such as adding two successes and two failures, or a Wilson interval, which stay sensible at the boundary.