Pooled standard deviation combines the standard deviations from two or more groups into a single estimate of variability. You calculate it by weighting each group’s variance (standard deviation squared) by its degrees of freedom (sample size minus one), summing those weighted values, dividing by the total degrees of freedom, and taking the square root. The core assumption is that all groups share roughly the same underlying variability, even if their means differ.
The Formula for Two Groups
For two groups, the pooled standard deviation formula is:
s_pooled = √[((n₁ – 1) × s₁² + (n₂ – 1) × s₂²) / (n₁ + n₂ – 2)]
Here’s what each piece means:
- n₁ and n₂ are the sample sizes of group 1 and group 2
- s₁ and s₂ are the standard deviations of group 1 and group 2
- s₁² and s₂² are the variances (just the standard deviations squared)
- (n₁ – 1) and (n₂ – 1) are the degrees of freedom for each group
- (n₁ + n₂ – 2) is the total degrees of freedom
The weighting by (n – 1) rather than just n matters because it accounts for the fact that larger samples give more reliable estimates of variability. A group of 200 contributes more to the pooled estimate than a group of 15, which is exactly what you’d want.
Extending to Three or More Groups
The formula generalizes naturally. For j groups:
s_pooled = √[((n₁ – 1) × s₁² + (n₂ – 1) × s₂² + … + (nⱼ – 1) × sⱼ²) / (n₁ + n₂ + … + nⱼ – j)]
The denominator is the total sample size minus the number of groups. This is the same quantity that appears in a one-way ANOVA as the residual mean square. If you take the square root of that residual mean square, you get the pooled standard deviation.
A Worked Example
Suppose you’re comparing test scores between two classrooms. Classroom A has 25 students with a standard deviation of 8. Classroom B has 30 students with a standard deviation of 10.
Step 1: Square each standard deviation to get the variance.
- s₁² = 8² = 64
- s₂² = 10² = 100
Step 2: Multiply each variance by its degrees of freedom.
- (n₁ – 1) × s₁² = 24 × 64 = 1,536
- (n₂ – 1) × s₂² = 29 × 100 = 2,900
Step 3: Add those products together.
1,536 + 2,900 = 4,436
Step 4: Divide by the total degrees of freedom.
4,436 / (25 + 30 – 2) = 4,436 / 53 = 83.70
Step 5: Take the square root.
√83.70 = 9.15
The pooled standard deviation is 9.15. Notice it falls between 8 and 10 but sits closer to 10, because Classroom B’s larger sample pulled the estimate in its direction.
Calculating It in a Spreadsheet
In Excel or Google Sheets, you don’t need a dedicated function. If you already have each group’s standard deviation and sample size, build the formula directly. For two groups with n₁ in cell A1, s₁ in B1, n₂ in A2, and s₂ in B2:
=SQRT(((A1-1)*B1^2 + (A2-1)*B2^2) / (A1+A2-2))
If you have the raw data instead, an alternative approach is to compute the difference between each value and its group mean, square those differences, sum them across all groups, divide by the total degrees of freedom, and take the square root. You can use DEVSQ() on each group’s data to get the sum of squared deviations, then combine them manually.
When Pooling Is Appropriate
Pooled standard deviation rests on one key assumption: the populations you’re comparing have roughly equal variability. The means can differ, but the spread around those means should be similar. This is sometimes called the assumption of equal variances or homoscedasticity.
A practical rule of thumb: if the two sample standard deviations are close to each other, pooling is reasonable. If one is more than double the other, pooling becomes questionable. Unequal sample sizes make this even more important. When sample sizes differ substantially and the standard deviations aren’t similar, the pooled estimate can be misleading. Specifically, if the smaller standard deviation belongs to the larger group, pooling is not recommended because the calculation will underweight the more variable (smaller) group. If the larger standard deviation belongs to the larger group and both standard deviations are reasonably close, the pooled approach is acceptable and tends to produce conservative results.
Where Pooled Standard Deviation Gets Used
Independent Samples T-Test
The most common application is the pooled (or “equal variance”) two-sample t-test. The pooled standard deviation goes into the denominator of the test statistic, serving as the best single estimate of the shared variability in both groups. When the equal variance assumption doesn’t hold, Welch’s t-test is the alternative. Welch’s version keeps the two standard deviations separate rather than combining them.
Cohen’s d for Effect Size
When researchers report effect sizes, they typically divide the difference between two group means by a standard deviation to put the result on a standardized scale. Cohen’s d uses the pooled standard deviation as that denominator:
Cohen’s d = (M₁ – M₂) / s_pooled
Note that the version of pooled SD used in Cohen’s d is sometimes simplified to √[(s₁² + s₂²) / 2], which gives equal weight to both groups regardless of sample size. This simplified version is common when the two groups are roughly equal in size. For unequal groups, the weighted formula described earlier is more appropriate.
Glass’s Delta: An Alternative Approach
There are situations where pooling doesn’t make sense, particularly when a treatment is expected to change not just the average but also the variability in a group. In those cases, Glass’s delta uses only the control group’s standard deviation as the denominator instead of a pooled value. This avoids contaminating the baseline measure of spread with treatment-induced changes in variability. It comes up frequently in meta-analyses where different studies use different measurement instruments and pooling across instruments would obscure meaningful differences.
ANOVA
In a one-way analysis of variance, the pooled standard deviation is built into the calculation. The residual mean square in an ANOVA table is the pooled variance across all groups. You compute it by finding each value’s deviation from its group mean, squaring those deviations, summing them across every group, and dividing by the total sample size minus the number of groups. The square root of that result is the pooled standard deviation.