Standard Error of the Mean Calculator
Calculate the standard error of the mean from raw data or from SD and sample size, with the t critical value, margin of error and confidence interval.
The standard error of the mean (SEM) measures how precisely a sample mean estimates the true population mean. It shrinks as the sample grows, which is why larger studies give tighter confidence intervals. Paste raw data, or enter a standard deviation and sample size, to get the standard error, the t critical value, the margin of error and a confidence interval.
How to use this tool
- Choose whether you have raw data or summary statistics.
- Paste the data, or enter the sample standard deviation, the sample size and, optionally, the mean.
- Pick a confidence level.
- Read the standard error, margin of error and confidence interval.
Formula
SE = s / √n Margin of error = t* × SE CI = x̄ ± t* × SE
s is the sample standard deviation (dividing by n − 1), n is the sample size and t* is the two-sided critical value of Student’s t distribution with n − 1 degrees of freedom. For large samples t* approaches the normal value of 1.96 at 95% confidence.
Example
A sample of 25 with a standard deviation of 10 has SE = 10 / √25 = 2. With 24 degrees of freedom, t* at 95% is about 2.064, so the margin of error is 2.064 × 2 ≈ 4.13. If the sample mean is 50, the 95% confidence interval runs from about 45.87 to 54.13.
Frequently asked questions
What is the difference between standard deviation and standard error?
The standard deviation describes how spread out the individual values are. The standard error describes how much the sample mean would vary from one sample to the next. Since SE = SD / √n, it is always smaller once n is above 1.
How do I halve the standard error?
Quadruple the sample size. Because the SE falls with the square root of n, going from 25 to 100 observations halves it.
Should I use t or z?
Use t when the standard deviation comes from the sample itself, which is almost always the case. The t value is larger for small samples and approaches z (1.96 at 95%) as n grows.
Can I report mean ± SE?
Many papers do, but mean ± 1 SE is only about a 68% interval. Say which one you report, or give the 95% confidence interval instead.
Last updated: 2026-10-30