How to calculate pooled standard deviation

Minitab offers within-subgroup variation estimates such as Rbar, Sbar and pooled standard deviation. The original article describes pooled standard deviation as the default; available settings depend on the analysis and configuration.

What is pooled standard deviation?

The pooled standard deviation is the square root of a degrees-of-freedom-weighted average of the variances of two or more independent groups.

In statistics, it is commonly used in a two-sample t-test that assumes equal population variances when testing whether two population means are equal.

Formula for pooled standard deviation

For two groups, the formula is:

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

Where:

• n1 and n2: sample sizes of groups 1 and 2, respectively.
• s1 and s2: sample standard deviations of groups 1 and 2, respectively.

Notes on interpretation

• Pooling assumes that the groups can reasonably be treated as having a common population variance.
• Each group is weighted by its degrees of freedom, n-1, so a larger group contributes more to the pooled estimate.
• The uncorrected pooled standard deviation lies between the sample standard deviations being pooled. It is not their simple arithmetic average.

Use in process capability analysis

Pooled within-subgroup estimates can be useful when data are collected in rational subgroups with different sample sizes, for example across several days. The grouping must still represent within-process variation, and the assumptions of the selected capability analysis should be checked.

Minitab methods reference: Normal Capability Analysis — methods and formulas