# What is the pooled standard deviation?

The pooled standard deviation is a method for estimating a single standard deviation to represent all independent samples or groups in your study when they are assumed to come from populations with a common standard deviation. The pooled standard deviation is the average spread of all data points about their group mean (not the overall mean). It is a weighted average of each group's standard deviation. The weighting gives larger groups a proportionally greater effect on the overall estimate. Pooled standard deviations are used in 2-sample t-tests, ANOVAs, control charts, and capability analysis.

## Example of a pooled standard deviation

Suppose your study has the following four groups:
Group Mean Standard Deviation N
1 9.7 2.5 50
2 12.1 2.9 50
3 14.5 3.2 50
4 17.3 6.8 200

The first three groups are equal in size (n=50) with standard deviations around 3. The fourth group is much larger (n=200) and has a higher standard deviation (6.8). Because the pooled standard deviation uses a weighted average, its value (5.486) is closer to the standard deviation of the largest group. If you used a simple average, then all groups would have had an equal effect.

## Manually calculating the pooled standard deviation

Suppose C1 contains the response, and C3 contains the mean for each factor level. For example:

C1 C2 C3
Response Factor Mean
18.95 1 14.5033
12.62 1 14.5033
11.94 1 14.5033
14.42 2 10.5567
10.06 2 10.5567
7.19 2 10.5567

Use Calc > Calculator with the following expression:

SQRT((SUM((C1 - C3)^2)) / (total number of observations - number of groups))

For the previous example, the expression for pooled standard deviation would be:

SQRT((SUM(('Response' - 'Mean')^2)) / (6 - 2))

The value that Minitab stores is 3.75489.