Written by Dackohn; last reviewed November 2026.
To calculate the standard error of the mean, divide the standard deviation by the square root of the sample size: SE = s / √n. For s = 20 and n = 100, SE = 2. It measures how precisely the mean is estimated, and it shrinks as the sample grows.
Standard error of the mean
Data: 12, 14, 15, 15, 16, 17, 18, 19, 21 (n = 9).
- Mean: 147 / 9 = 16.33.
- Standard deviation: s = 2.74. See how to calculate standard deviation.
- Standard error: 2.74 / √9 = 2.74 / 3 = 0.91.
The 95% confidence interval uses the t critical value for df = 8: 16.33 ± 2.306 × 0.91, which is 14.23 to 18.44.
Standard error of a proportion
SE = √(p(1 − p) / n). For 52% of 1,000: √(0.52 × 0.48 / 1000) = 0.0158, or 1.58 percentage points. Multiplying by 1.96 gives the familiar margin of error of ±3.1.
Standard error of a difference between means
SE = √(s₁² / n₁ + s₂² / n₂). For SDs of 3.2 and 4.1 with 20 in each group: √(0.512 + 0.8405) = 1.163. This is the denominator of the two-sample t-test.
What it means
Imagine repeating your study many times and calculating the mean each time. Those means would vary, and the standard error estimates how much. Try it with the Central Limit Theorem simulator.
How sample size affects it
| n (with s = 20) | Standard error |
|---|---|
| 25 | 4 |
| 100 | 2 |
| 400 | 1 |
Each halving takes four times the data.
In Excel or Google Sheets
=STDEV.S(A1:A9)/SQRT(COUNT(A1:A9)). There is no built-in function for it.
Common mistakes
- Confusing SE with SD. The SD describes the data, and the SE describes the estimate of the mean. See SD vs SE.
- Dividing by n instead of √n.
- Showing SE error bars to make data look tidier. SE bars are shorter than SD bars, so always label which one you used.
Try it yourself
Standard Error Calculator →From raw data, an SD and n, or a proportion. Confidence Interval Calculator →Turn a standard error into an interval.Frequently asked questions
What is the formula for standard error?
SE = s divided by the square root of n for a mean, and the square root of p(1-p)/n for a proportion.
What is the difference between standard error and standard deviation?
Standard deviation measures how spread out the values are. Standard error measures how precisely the mean is estimated.
What is a good standard error?
There is no universal value. Judge it relative to the size of the effect that matters, usually through a confidence interval.
Related reading: Standard deviation vs standard error · How to calculate a confidence interval · Margin of error explained
Sources & further reading: NIST/SEMATECH e-Handbook of Statistical Methods · OpenStax Introductory Statistics. How we compute & validate: methodology. Written & reviewed by Dackohn; last reviewed August 2026.