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Standard Error Calculator (SEM, Proportion)

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Standard Error Calculator

The standard error measures how much a sample statistic would vary from sample to sample. This calculator finds the standard error of the mean (SEM) from a list of values or from a standard deviation and sample size, and the standard error of a sample proportion from p̂ and n.

For a mean the formula is SE = s ÷ √n: the spread of individual values divided by the square root of the sample size. Quadrupling the sample size halves the standard error. For a proportion, SE = √[p̂(1 − p̂) ÷ n], which is largest when p̂ = 0.5. The standard deviation describes the data; the standard error describes the precision of the estimate, so error bars on a chart of means are usually standard errors or confidence intervals, not standard deviations.

Multiplying the standard error by 1.96 gives the approximate 95% margin of error, shown under the result. For an exact t-based interval use the confidence interval for a mean, and for planning a study use the sample size calculator. The raw-data option matches Excel’s =STDEV.S(range)/SQRT(COUNT(range)).

Separate values with commas, spaces or new lines.

Result and step-by-step solution

Standard error
0.702377

Standard error of the mean (SEM).

  1. Mean and standard deviation
    n = 10, x̄ = 12.6, s = √[Σ(x − x̄)² ÷ (n − 1)] = √(44.4 ÷ 9) = 2.2211
  2. Formula
    SE(x̄) = s ÷ √n
  3. Substitute
    SE = 2.2211 ÷ √10 = 2.2211 ÷ 3.1623 = 0.702377
Approximate 95% margin of error (1.96 × SE)1.3766
Approximate 95% interval11.2234 to 13.9766

Frequently asked questions

What is the difference between standard deviation and standard error?
The standard deviation is the spread of individual observations. The standard error is the spread of the sample mean (or proportion) across repeated samples, equal to s ÷ √n for a mean.
How do I calculate standard error in Excel?
Use =STDEV.S(A1:A20)/SQRT(COUNT(A1:A20)). Excel has no built-in SEM function.
What is the standard error of a proportion?
√[p̂(1 − p̂) ÷ n]. With p̂ = 0.42 and n = 50 it is √(0.42 × 0.58 ÷ 50) = 0.0698.
Does a bigger sample reduce the standard error?
Yes. The standard error falls with the square root of n, so four times the sample size gives half the standard error.

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Last reviewed: October 7, 2026. Calculations run in your browser and were validated against SciPy.

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