When several groups are measured on the same scale and are believed to have the same underlying spread, their standard deviations can be combined into one, more precise estimate: the pooled standard deviation. Each group’s variance counts in proportion to its degrees of freedom, n − 1, so a group of 50 influences the result much more than a group of 5.
The pooled standard deviation is the denominator of Cohen’s d, the standard error in the classic equal-variance two-sample t test, and the root mean square error of a one-way ANOVA. In laboratory quality control it combines repeatability estimates from several runs or materials into a single figure.
Pooling only makes sense when the groups really have similar spreads. A common rule of thumb is that the largest standard deviation should be no more than about twice the smallest. The calculator shows that ratio so you can see when Welch’s unequal-variance methods would be the safer choice.
How to use the Pooled Standard Deviation Calculator
- Enter
- Number of groups
- Group 1 data
- Group 2 data
- Group 3 data
- Group 1 standard deviation (s₁)
- Group 1 size (n₁)
- Group 2 standard deviation (s₂)
- Group 2 size (n₂)
- Group 3 standard deviation (s₃)
- Group 3 size (n₃)
Results update as you type, and everything is calculated in your browser.
How it works
Enter each group’s raw data, or switch to summary mode and type each standard deviation and sample size. The calculator squares each SD to get the variance, multiplies it by n − 1, adds these sums of squares, divides by the total degrees of freedom N − k and takes the square root.
Worked example
Group 1: 12, 15, 14, 10, 13, 16 (n = 6, s² = 4.667). Group 2: 9, 11, 8, 12, 10 (n = 5, s² = 2.5). The pooled variance is (5 × 4.667 + 4 × 2.5) ÷ (6 + 5 − 2) = (23.333 + 10) ÷ 9 = 3.704, so s_p = √3.704 = 1.9245. The SD ratio is 2.160 ÷ 1.581 = 1.37, well under 2, so pooling is reasonable.
Pooled SD versus other averages
| Method | Formula | Result for the example |
|---|---|---|
| Pooled (df-weighted) | √(Σ(nᵢ − 1)sᵢ² ÷ (N − k)) | 1.9245 |
| Root-mean-square of SDs | √((s₁² + s₂²) ÷ 2) | 1.8930 |
| Plain mean of SDs | (s₁ + s₂) ÷ 2 | 1.8707 |
Inputs explained
| Field | What to enter |
|---|---|
| Enter | Choose one: Raw data for each group, Standard deviation and size of each group |
| Number of groups | Choose one: 2 groups, 3 groups |
| Group 1 data | A list of numbers separated by commas, spaces or new lines |
| Group 2 data | A list of numbers separated by commas, spaces or new lines |
| Group 3 data | A list of numbers separated by commas, spaces or new lines |
| Group 1 standard deviation (s₁) | A number |
| Group 1 size (n₁) | A number |
| Group 2 standard deviation (s₂) | A number |
| Group 2 size (n₂) | A number |
| Group 3 standard deviation (s₃) | A number |
| Group 3 size (n₃) | A number |
What each result means
- Degrees of freedom (N − k) = [Group 1 size] + [Group 2 size] + [Group 3 size] – [Number of groups]
- Pooled standard deviation s_p = sqrt([Pooled variance s_p²])
Formula
s_p = √[((n₁ − 1)s₁² + (n₂ − 1)s₂² + … + (n_k − 1)s_k²) ÷ (n₁ + n₂ + … + n_k − k)]
Worked example
Inputs — Enter: Raw data for each group; Number of groups: 2 groups; Group 1 data: 12, 15, 14, 10, 13, 16; Group 2 data: 9, 11, 8, 12, 10.
Results — Group 1 standard deviation = 2.1602; Group 1 size = 6; Group 2 standard deviation = 1.5811; Group 2 size = 5; Degrees of freedom (N − k) = 9; Pooled variance s_p² = 3.703704; Pooled standard deviation s_p = 1.924501; Unweighted root-mean-square SD = 1.892969; Largest SD ÷ smallest SD = 1.366; Equal-spread check = Ratio ≤ 2 – pooling is reasonable.
Key terms
- Pooled variance
- The weighted average of group variances, weighted by degrees of freedom.
- Degrees of freedom
- The number of independent deviations: n − 1 per group, N − k in total.
- Homogeneity of variance
- The assumption that all groups share the same population variance.
Frequently asked questions
When should I use the pooled standard deviation?
When groups measure the same thing and you can assume the same population spread – for Cohen's d, an equal-variance t test or one-way ANOVA.
What if both groups have the same standard deviation?
Then the pooled SD equals that common value, whatever the sample sizes. Two groups with s = 2.83 give s_p = 2.83.
How do I pool more than two groups?
Add (nᵢ − 1)sᵢ² over all groups and divide by the total sample size minus the number of groups. This calculator handles up to three; the formula extends to any number.
Is the pooled SD the average of the standard deviations?
No. It averages the variances, weighted by degrees of freedom, and then takes the square root. For s = 1 and s = 3 with equal n it is √5 = 2.24, not 2.
How is pooled SD different from the SD of all data combined?
Combining all values into one list also adds the differences between group means. Pooling measures only the spread within groups.
What if the variances are very different?
Do not pool; use Welch's t test or report each group separately. A largest-to-smallest SD ratio above 2 is a common warning sign.
Why use n − 1 weights instead of n?
Each sample variance has n − 1 degrees of freedom, so weighting by n − 1 gives an unbiased estimate of the common population variance.