Standard Error Calculator
Standard error calculator. SE of the mean (s/√n) with a t-based 95% interval, and SE of a proportion (√(p(1−p)/n)) with the Wilson interval.
Standard Error
How to use the standard error calculator
Pick the right mode
SE of the mean is for a measured, continuous outcome (a test score, a weight, a dollar amount) where you have the sample mean, the sample standard deviation and the sample size. SE of a proportion is for yes/no outcomes, where you have the share of successes p (e.g. 0.42) and the sample size.
Enter sample statistics
For the mean you need x̄, s and n, where s is the sample standard deviation with n − 1 in its divisor (the STDEV.S figure in a spreadsheet). For a proportion, type p as a decimal (0.42) or a percentage (42%). Grouped numbers such as 1,234.5 are read correctly.
Read the SE and the 95% interval
The standard error is the standard deviation of the sampling distribution of your estimate. For a mean the 95% interval is x̄ ± t·SE, with t taken from Student's t distribution on n − 1 degrees of freedom; at n = 25 that t is 2.064. For a proportion the page gives the textbook interval p ± 1.96·SE and the Wilson interval, which stays between 0 and 1.
Sample size beats sample variance
To halve the SE you need four times the sample, because SE scales with 1/√n. Doubling the sample cuts the SE by only about 29%.
Know which interval you are quoting
The "1.96" interval assumes the population SD is known. When you estimate it from the data, as you nearly always do, the t interval is the correct one, and the page shows both so you can see the gap: about 5% at n = 25, and under 1% once n passes 200.
Standard error — the precision of your estimate
Standard error answers one question: if you drew another random sample of the same size, how much would your estimate move? It is not the standard deviation of the data — that describes the spread between individuals. The standard error describes the spread of the estimate across hypothetical repeats of the whole study. A sample of n = 25 with s = 12 has SE = 12/√25 = 2.4. With a mean of 100, the 95% interval is 100 ± 2.064 × 2.4, or about 95.05 to 104.95.
Mean vs proportion — two different formulas
For continuous data, SE = s/√n. For a proportion, SE = √(p(1−p)/n), the standard deviation of a binomial share. That SE is largest at p = 0.5 and shrinks as p approaches 0 or 1, which is why survey planners often size a sample for p = 0.5: it is the worst case. At exactly 0 or 1 the formula gives an SE of zero, which is not certainty but a sign the formula has run out of road — the Wilson interval still gives a sensible range there.
"To halve the standard error, quadruple the sample size." Doubling a survey does not halve its margin of error; quadrupling it does.
Why the interval for a mean uses t, not 1.96
1.96 is the z value that captures the central 95% of a normal distribution, and it is right when the population standard deviation is known. In practice you estimate it with s, and NIST/SEMATECH's e-Handbook (§1.3.5.2) gives the interval as x̄ ± t(1−α/2, N−1)·s/√N, with t from Student's t distribution on N − 1 degrees of freedom. t is larger than 1.96 for small samples (2.262 at 9 degrees of freedom, 2.064 at 24) and approaches it as N grows (1.984 at 100). The interpretation is about the procedure: if you repeated the sampling many times, about 95% of the intervals built this way would contain the true mean. It does not mean there is a 95% probability that this particular interval contains it.
The interval for a proportion
The interval most often printed for a proportion is p ± 1.96·SE. It can produce a lower limit below 0 or an upper limit above 1, and NIST's e-Handbook (§7.2.4.1) calls an interval that yields an impossible value an inferior approach. It recommends the Wilson score interval instead (Wilson 1927, later recommended by Agresti and Coull, and by Brown, Cai and DasGupta), which this calculator shows alongside. The two agree closely for middling p and large n; they part company for small samples and for p near 0 or 1.
The poll arithmetic
A poll's "margin of error" is usually 1.96 × √(0.25/n), the proportion formula at its worst case p = 0.5. For 1,000 respondents that is ±3.1 percentage points, and it covers sampling error only: who answered, how the question was worded and who refused are not in it.
SE against SD, why halving it costs 4× the sample, and where p = 0.5 hurts most
SE = s/√n (mean) or √(p(1−p)/n) (proportion).
SE measures the precision of an estimate; SD measures the spread of the data. Different things.
A 95% interval for a mean is x̄ ± t·SE with n − 1 degrees of freedom; 1.96 is the limit t approaches as n grows.
To halve SE, you need 4× the sample size. Diminishing returns.
SE of a proportion is largest at p = 0.5 (most uncertainty).
t for a 95% interval: 12.706 with 1 degree of freedom, 2.262 with 9, 2.064 with 24, 1.984 with 100.
Polling margin for 1,000 respondents at p = 0.5: ±3.1 percentage points at 95% confidence.
SE assumes random sampling. A convenience sample has no probability model for the SE to describe.
For stratified or cluster sampling, the formulas need a design effect.
The Wilson interval for a proportion never leaves 0 to 1; the textbook p ± 1.96·SE can.
Frequently asked questions
SD describes how spread out individual data points are. SE describes how precise your estimate of the mean is. SE = SD/√n. A larger sample does not change the SD, but it does shrink the SE — the estimate gets more precise even though individuals vary as much as before.
The mean of n independent observations with variance σ² has variance σ²/n, so its standard deviation is σ/√n. The √n (not n) is why doubling the sample does not halve the uncertainty — it cuts it by only about 29%.
Whenever the standard deviation comes from your own sample, which is almost always. The often-quoted "n < 30" cut-off is a rule of thumb about when the difference is large enough to notice, not a switch: at n = 10 the 95% t value is 2.262 (9 degrees of freedom), at n = 60 it is 2.001, and it reaches 1.96 only in the limit. This calculator uses t for every n.
The central limit theorem says sample means become approximately normal as n grows, even when the data are not. How large n must be depends on how skewed the data are, so there is no single safe number. For small or strongly skewed samples, a bootstrap interval is the usual check.
Yes, but the formula is different: it depends on the density of the data at the median, so it is usually estimated by bootstrap. For large samples from a normal distribution, SE(median) ≈ 1.253 × SE(mean) (the factor is √(π/2)); for other distributions the factor differs.
SE(x̄₁ − x̄₂) = √(SE₁² + SE₂²) for independent samples. This is the denominator of the two-sample (Welch) t-test. Use the T-Test calculator for the full procedure.
Yes. The SE describes the spread of an estimate across random samples, so it only means something when the observations were drawn at random from the target population. Volunteers or web traffic have no probability mechanism for it to describe.
No. All inputs stay in your browser.
Margin of error = critical value × SE: t × SE for a mean, 1.96 × SE for a proportion in the textbook interval. The interval is the estimate ± the margin. The Wilson interval for a proportion is not symmetric around p, so it has no single margin.
NIST/SEMATECH's e-Handbook of Statistical Methods is free online: §1.3.5.2 for the interval for a mean, §1.3.6.7.2 for the t table, and §7.2.4.1 for intervals for a proportion. For a textbook treatment, Moore, McCabe and Craig's Introduction to the Practice of Statistics is a standard undergraduate text.
Method & sources
How it computes
Mean: SE = s/√n with s the sample SD (n − 1 divisor); 95% interval x̄ ± t(0.975, n − 1)·SE, the 1.96 shortcut shown beside it. Proportion: SE = √(p(1 − p)/n); the textbook interval p ± 1.96·SE reported unclipped and flagged when it leaves [0, 1], and the Wilson score interval (StandardErrorKernel).
What this tool implements
- NIST/SEMATECH e-Handbook §1.3.5.2: confidence limits for the mean are x̄ ± t(1−α/2, N−1)·s/√N. Its worked example (N = 195, x̄ = 9.261460, s = 0.022789, t = 1.9723 → 9.258242 to 9.264679) is a test vector.
- NIST §1.3.6.7.2 t table, 0.975 column: 12.706 (1 df), 2.262 (9), 2.064 (24), 2.001 (59), 1.984 (100), 1.960 (∞) — reproduced to three decimals from the regularised incomplete beta.
- NIST §7.2.4.1: the Wilson interval (Wilson 1927; recommended by Agresti & Coull and by Brown, Cai & DasGupta) cannot leave [0, 1]; an interval giving an impossible value is "an inferior approach". Its example (p̂ = 0.13, n = 200, z = 1.645 → 0.09577) is a test vector.
- Halving the SE needs four times the sample; doubling the sample cuts it by about 29%. A poll of 1,000 at p = 0.5 has a 95% margin of ±3.1 points.
- Corrected 2026-10-19: the mean interval used 1.96 at every n (now t on n − 1 df); the proportion interval was clipped silently (now unclipped and flagged, with Wilson beside it); unsourced ASEAN statistics-agency and polling claims removed; SE(median) ≈ 1.253·SE(mean) restricted to normal data; the "2019 ASA statement" reference removed.
Sources
- NIST/SEMATECH. e-Handbook of Statistical Methods, §1.3.5.2 Confidence Limits for the Mean. https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm
- NIST/SEMATECH. e-Handbook of Statistical Methods, §1.3.6.7.2 Critical Values of the Student's t Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3672.htm
- NIST/SEMATECH. e-Handbook of Statistical Methods, §7.2.4.1 Confidence intervals (proportions). https://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm
What can make this go out of date
- None at runtime; nothing the reader types is sent anywhere. t quantiles come from correlation-kernel.js (betai), loaded with the page.
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