The pooled standard deviation is a quantity that is used to measure the combined standard deviations for two or more samples.

**Applications of Pooled Standard Deviation**:

- The pooled standard deviation is calculated whenever we perform two sample T test to check for equality of means.
- Also, the pooled sample variance is used as an unbiased estimator for the combined population variance for two or more populations.

**How to calculate Pooled Standard Deviation?**

Suppose you have ‘k’ number of samples with each having standard deviation S_{i}. The formula for pooled standard deviation is given as,

The denominator in the above formula represents the combined degrees of freedom of the test statistic.

**Example**: Suppose we are given the following data about four samples. Calculate the pooled standard deviation.

Sample size(n_{i}) | Sample variance(S_{i}^{2}) |

4 | 12 |

3 | 14 |

6 | 16 |

8 | 11 |

**Solution**: We calculate the quantities required as below,

Sample size(n_{i}) | Sample variance(S_{i}^{2}) | (n_{i }-1) | (n_{i }-1)* S_{i}^{2} |

4 | 12 | 3 | 36 |

3 | 14 | 2 | 28 |

6 | 16 | 5 | 80 |

8 | 11 | 7 | 77 |

∑(n_{i }-1)= 17 | ∑(n_{i }-1)* S_{i}^{2} = 221 |

Substituting all this in the above formula we get,

Pooled Standard Deviation = √(221/17) = √13 = 3.61