Quick answer
The standard error of the mean is SE = s ÷ √n, and for a proportion SE = √(p(1 − p) ÷ n). With s = 10 and n = 25, SE = 2.
Choose mean or proportion, then enter the standard deviation (or p) and the sample size. The calculator returns the standard error.
Related tools: Margin of Error Calculator: Margin of error for a mean or a proportion. Sample Size Calculator: Sample size for estimating a mean or a proportion. Confidence Interval Calculator: Confidence interval for a mean (t or z).
How to use the Standard Error Calculator
- Choose statistic: Mean and Proportion.
- Enter standard deviation (mean) or proportion p (0–1) (above 0).
- Enter sample size (a whole number, 1 or more).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Statistic | Choose one: Mean; Proportion | Starts at Mean |
| Standard deviation (mean) or proportion p (0–1) | A number | > 0 |
| Sample size | A whole number | ≥ 1 |
Formula and method
Formula
SE(x̄) = s ÷ √n; SE(p̂) = √(p(1 − p) ÷ n)
IEEE-754 double precision; shown to at most 6 decimal places.
Large-sample formulas.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Statistic = Mean; Standard deviation (mean) or proportion p (0–1) = 10; Sample size = 25.
| Result | Value |
|---|---|
| Standard error | 2 |
Example 2
Inputs: Statistic = Proportion; Standard deviation (mean) or proportion p (0–1) = 0.5; Sample size = 400.
| Result | Value |
|---|---|
| Standard error | 0.025 |
The first line is the headline result (standard error). IEEE-754 double precision; shown to at most 6 decimal places.
What SE tells you
SE measures how much sample means vary from sample to sample. With n = 100 it would be 1; to halve it you need four times the data. Confidence intervals are ± about 2 SE.
SD versus SE
SD describes the spread of individual values; SE describes the precision of the mean. Report the SD for data variability and the SE (or a confidence interval) for the accuracy of an estimate. A proportion must be below 1.
How the result changes with standard deviation or proportion p
Holding every other input at the first worked example, this table changes only standard deviation or proportion p. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Standard deviation or proportion p | Standard error |
|---|---|
| 5 | 1 |
| 10 | 2 |
| 20 | 4 |
| 40 | 8 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Standard deviation (mean) or proportion p (0–1): entering "1.5" shows "A proportion must be between 0 and 1." beside that field and no result.
- Sample size: entering "0" shows "Sample size must be at least 1." beside that field and no result.
Choosing between the Standard Error Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Margin of Error Calculator](/statistics/margin-of-error-calculator/) | for a, confidence level and σ or proportion p | Margin of error |
| [Sample Size Calculator](/statistics/sample-size-calculator/) | estimate a, confidence level and margin of error | Required sample size |
| [Confidence Interval Calculator](/statistics/confidence-interval-calculator/) | sample mean, standard deviation and sample size | Confidence interval |
| [Point Estimate Calculator](/statistics/point-estimate/) | successes and trials | Sample proportion |
| [Odds Ratio Calculator](/statistics/odds-ratio-calculator/) | exposed with outcome, exposed without outcome and unexposed with outcome | Odds ratio |
Background: hypothesis testing and inference
A hypothesis test compares a test statistic with what the null hypothesis predicts. The decision rule is to reject the null hypothesis when the p-value is at or below the significance level α, which you should choose before seeing the data (0.05 is conventional, not special). For a two-tailed z test the p-value is 2 × (1 − Φ(|z|)); for a one-sample t test the statistic is (x̄ − μ₀) ÷ (s ÷ √n) with n − 1 degrees of freedom.
Failing to reject the null hypothesis does not prove it true, and a statistically significant result can still be too small to matter in practice.
Assumptions and limitations
- Proportion must be below 1
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for standard deviation (mean) or proportion p (0–1) and sample size to confirm this.
Frequently asked questions
What is SE for p = 0.5 and n = 100?
For p = 0.5 and n = 100, SE = √(0.25 ÷ 100) = 0.05, or 5 percentage points.
Why does SE fall with n?
Averaging more observations smooths out random variation, so SE = s ÷ √n shrinks as n grows: quadrupling n halves it.
Is SE the same as SD?
No. SD describes the spread of individual values; SE = SD ÷ √n describes how precisely the sample mean is known.
What does the Standard Error Calculator calculate?
Standard error of a mean or a proportion. It returns Standard error.
What formula does the Standard Error Calculator use?
SE(x̄) = s ÷ √n; SE(p̂) = √(p(1 − p) ÷ n). Large-sample formulas.
What do I need to enter in the standard Error Calculator?
You enter Statistic, Standard deviation (mean) or proportion p (0–1) and Sample size.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Proportion must be below 1.
What does the “Statistic” option change?
It sets how the standard Error is worked out. The choices are Mean and Proportion.
What is an example of a standard Error result?
With statistic Mean, standard deviation (mean) or proportion p (0–1) 10 and sample size 25, the standard error is 2.
Related calculators
- Margin of Error Calculator — Margin of error for a mean or a proportion.
- Sample Size Calculator — Sample size for estimating a mean or a proportion.
- Confidence Interval Calculator — Confidence interval for a mean (t or z).
- Point Estimate Calculator — Point estimates of a proportion (MLE, Laplace and Wilson).
- Odds Ratio Calculator — Odds ratio from a 2×2 table.
- Relative Risk Calculator — Relative risk from a 2×2 table.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods (itl.nist.gov/div898/handbook)
- Standard textbook definitions (e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.