How this instrument works
When you draw a sample of size n from a population and measure the proportion of items with some characteristic — support for a policy, defective units, or heads in a series of coin flips — that sample proportion, written p̂ ("p-hat"), is itself a random quantity. Draw a different sample of the same size, and you'd very likely get a slightly different p̂, even if the true underlying population proportion p never changed. This instrument describes that variability directly.
This is the Central Limit Theorem applied specifically to proportions rather than to raw numeric means. The expected value, E[p̂] = p, says that on average across many repeated samples, the sample proportion is unbiased — it neither systematically overstates nor understates the true population proportion. The standard error, SE(p̂) = √(p(1−p)/n), measures the typical spread of p̂ around that true value for samples of size n.
Notice that p(1−p) is largest when p = 0.5 and shrinks as p moves toward 0 or 1 — a population evenly split at 50/50 produces the most sample-to-sample variability in p̂, while a population that's nearly all one way or the other (say p = 0.05 or p = 0.95) produces comparatively little. This is why pollsters often report their widest margins of error for close, roughly-even races, and why n needs to grow faster to pin down a near-even split precisely.
- Enter the true (or assumed) population proportion into Population proportion (p), a value between 0 and 1.
- Enter your sample size into Sample size (n), the number of individuals or items measured.
- Read Expected value of the sample proportion, E[p̂] — under repeated sampling, this always equals p exactly.
- Read Standard error of the sample proportion, SE(p̂) — the typical amount a sample proportion of this size would differ from p.
- For a rough 95% margin of error, multiply SE(p̂) by about 1.96 — this instrument reports SE itself, the building block for that wider interval.
Worked example — population proportion 0.3, sample size 100
Enter p = 0.3 into Population proportion and n = 100 into Sample size — a true population proportion of 30%, sampled 100 at a time. Expected value of the sample proportion, E[p̂], reads 0.3 — exactly the population proportion, confirming the sample proportion is an unbiased estimator regardless of n.
Standard error of the sample proportion, SE(p̂), reads 0.0458257569496, computed as the square root of (0.3 × 0.7) / 100, or √0.0021. That means a sample of 100 drawn from a population with a true 30% proportion would typically produce a sample proportion within roughly 4.6 percentage points of 30%, purely from ordinary sampling variability — before any consideration of bias, wording effects, or other real-world survey error.
Questions
How is this different from the Central Limit Theorem calculator on this site?
Same underlying theorem, different statistic. This site's Central Limit Theorem calculator covers the sampling distribution of a sample mean, x̄, computed from numeric data, with standard error σ/√n. This instrument covers the sampling distribution of a sample proportion, p̂, computed from categorical (yes/no, success/failure) data, with standard error √(p(1−p)/n) — the proportion version replaces the population standard deviation σ with the proportion-specific spread term √(p(1−p)).
Why is the standard error largest when p = 0.5?
Because p(1−p) is maximized at p = 0.5 (where it equals 0.25) and shrinks toward zero as p approaches either 0 or 1. A population split evenly between two outcomes produces the most sample-to-sample bounce in the sample proportion, while a population that's nearly unanimous one way produces comparatively little — most samples from a 95%-yes population will land close to 95% yes just by chance.
How do I turn SE(p̂) into a confidence interval or margin of error?
A common approximate 95% confidence interval is p̂ ± 1.96 × SE(p̂), based on the normal approximation to the sampling distribution of p̂ (valid when both n×p and n×(1−p) are reasonably large, often taken as at least 10). A published poll's 'margin of error' is typically exactly that: SE multiplied by a critical value like 1.96, so it will be somewhat wider than the raw SE(p̂) this instrument reports.
Do I need to know the true population proportion p to use this?
In practice, you often use your best estimate or a conservative assumption. If planning a survey before collecting data, using p = 0.5 gives the most conservative (largest) standard error, which is a common precaution since it guarantees you're not underestimating the sampling variability. If you already have an observed sample proportion from prior data, plugging that in as p gives a data-informed estimate of the standard error instead.
How much does doubling my sample size shrink the standard error?
Not by half — SE shrinks by a factor of √2 ≈ 1.41, since it's divided by √n, not n. To actually cut SE(p̂) in half, you need to quadruple the sample size. This site's example moving from n = 100 to n = 400 at p = 0.5 shows it directly: SE falls from about 0.05 down to exactly 0.025, a halving that required 4× the sample, not 2×.