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 have 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 t-tests, ANOVAs, control charts, and capability analysis.

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.

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

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.