Standard Error of the Mean Calculator

This calculator finds the standard error of the mean (SEM) of a sample. Paste or type the values and you get the sample size, the mean, the sample standard deviation and the standard error, which shows how precisely the sample mean estimates the true mean.

Formula

SEM = s ÷ √n, where s = √( Σ(x − x̄)² ÷ (n − 1) ) is the sample standard deviation and n the number of values.

x: the sample values
x̄: their mean
n: sample size

Worked Example

Sample: 12, 15, 11, 18, 14, 16, 13.

n = 7, sum = 99, mean = 99 ÷ 7 = 14.143
Sum of squared deviations = 34.857
s = √(34.857 ÷ 6) = √5.8095 = 2.410
SEM = 2.410 ÷ √7 = 2.410 ÷ 2.6458 = 0.911

Limits and common mistakes

SEM is not the spread of the data; the standard deviation is. SEM shrinks as the sample grows.
A rough 95% interval for the mean is mean ± 2 × SEM; for small samples use the t value instead of 2.
You need at least two values.

Formula used

list x | Sample values | 12, 15, 11, 18, 14, 16, 13
out se | Standard error of the mean | - | sd(x) / sqrt(count(x)) | 3
out n | Sample size | values | count(x) | 0
out avg | Mean | - | mean(x) | 3
out s | Sample standard deviation | - | sd(x) | 3

Each line is one input (in, sel, list) or one result (out: label, unit, formula, decimals). How the calculators work

Photo collage: glass jars of coloured marbles lined up and a notebook of plain grid paper. — Save this card: a 1000 × 1500 image of Standard Error of the Mean Calculator.

Method: the standard formula for this calculation, rounded as each result states. See how the calculators work or report an error.

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