Standard Error Calculator

The standard error of the mean tells you how much a sample mean would vary from sample to sample. Get it from raw data or straight from a standard deviation and sample size.

7 values read.

Only have summary numbers? SE = s ÷ √n: ÷ √ = —

Standard error
0.1155
Sample SD (s)
0.3055
Mean
5.1
n
7
±1 SD4.855.25.45.6
▲ mean 5.1 · ◆ median 5.1 · shaded band = mean ± 1 SD (4.7945 to 5.4055)

Step-by-step working

  1. Count the valuesn = 7
  2. Add the values and divide by the count to get the meanx̄ = (5.1 + 4.8 + 5.6 + 5 + 4.7 + 5.3 + 5.2) ÷ 7 = 35.7 ÷ 7 = 5.1
  3. Subtract the mean from each value, then square the result (see the table)(x − x̄)² for each x, e.g. (5.1 − 5.1)² = 7.8886 × 10^-31
  4. Add up the squared deviations (the sum of squares)SS = Σ(x − x̄)² = 0.56
  5. Divide by n − 1 for the sample variance (Bessel’s correction)s² = 0.56 ÷ 6 = 0.093333
  6. Take the square root to get the standard deviations = √0.093333 = 0.305505
  7. Divide the standard deviation by √nSE = 0.305505 ÷ √7 = 0.305505 ÷ 2.645751 = 0.11547
Deviation table
xx − x̄(x − x̄)²
5.1-8.8818 × 10^-167.8886 × 10^-31
4.8−0.30.09
5.60.50.25
5−0.10.01
4.7−0.40.16
5.30.20.04
5.20.10.01
Σx = 35.7≈ 0SS = 0.56

Formula SE = s ÷ √n

Ask the tutor about your result

An AI tutor reads your inputs and results above and explains them in plain English. The numbers come from the calculator; the explanation is AI-written, so check it against your notes.

Standard deviation vs standard error

Standard deviation describes the spread of individual values. Standard error describes the precision of the mean. Quadrupling the sample size halves the standard error, but the standard deviation stays about the same, because individuals don’t become less variable just because you measured more of them.

The standard error is the building block of confidence intervals (estimate ± critical value × SE) and of t-tests (t = difference ÷ SE).

Frequently asked questions

When should I report SE instead of SD?

Report SD to describe how variable the data are; report SE (or better, a confidence interval) to show how precisely you have estimated the mean. Error bars must say which one they show.

What is the standard error of a proportion?

√(p̂(1 − p̂) ÷ n). The margin of error and confidence interval calculators use it for proportions.