Standard Error Calculator
Find the standard error of the mean from a sample standard deviation and size — how much the sample mean is expected to vary.
How it works
SE = sd ÷ √n
The standard error of the mean is the standard deviation of the sample mean itself — a measure of how much your average would jump around if you drew the sample again. It is the data’s standard deviation divided by the square root of the sample size, so it always ends up smaller than the raw spread. The standard error is the engine behind confidence intervals and hypothesis tests about a mean.
Worked example
With a standard deviation of 15 and a sample of 100, the standard error is 15 ÷ √100 = 15 ÷ 10 = 1.5. Growing the sample to 400 would cut it to 15 ÷ 20 = 0.75 — four times the data for half the error.
Frequently asked questions
What is the difference between standard deviation and standard error?
Standard deviation describes the spread of individual data points. Standard error describes the spread of a summary statistic — usually the mean — across repeated samples, and it shrinks as the sample grows, while standard deviation does not.
Why divide by the square root of n and not n?
Averaging reduces variance in proportion to n, and standard error is a standard deviation, so it moves with the square root of variance. That √n law is why doubling precision costs four times the sample.
Can the standard error be larger than the standard deviation?
Only when n = 1, where they are equal. For any sample of two or more, dividing by √n makes the standard error strictly smaller than the standard deviation.
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