Pooled Standard Deviation Calculator

Work out pooled standard deviation instantly with clear inputs, formula shown and shareable results.

About Pooled Standard Deviation Calculator

Pooling combines two sample variances into a single estimate, weighting each by its degrees of freedom. It assumes the two populations share a common variance, which is the key assumption of the standard two-sample t-test.

Formulas

Pooled variance

s²ₚ = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁ + n₂ − 2)

Standard error

SE = sₚ√(1/n₁ + 1/n₂)

Worked example

With the default values below, Pooled Standard Deviation Calculator returns Pooled standard deviation of 1.49071198. Change any input above and every figure here is recalculated in your browser.

Inputs used

First group data: 12, 15, 14, 16, 13
Second group data: 18, 20, 19, 22, 21, 20

Results

Pooled standard deviation: 1.49071198
Pooled variance: 2.22222222
Standard error of the difference: 0.90267093
Degrees of freedom: 9

Step by step

1Values used
2Pooled variance
3Standard error
4Pooled standard deviation= 1.49071198
5Pooled variance= 2.22222222
6Standard error of the difference= 0.90267093
7Degrees of freedom= 9

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For the two sample groups?

The pooled standard deviation is about 1.4223 on 9 degrees of freedom.

What if the variances differ greatly?

Do not pool. Use Welch's t-test, which estimates the standard error without assuming equal variances.