# Grouped Data Mean and Standard Deviation Calculator

We want to determine the mean and standard deviation of ungrouped data in practically all circumstances. However, how can you do that with grouped data?

## Grouped Data Mean and Standard Deviation

Let’s look at an example of how to compute mean and SD.

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Range | Frequency |

1-20 | 4 |

21-40 | 8 |

41-60 | 6 |

61-80 | 2 |

81-100 | 9 |

### Grouped Data Mean

The formula for grouped mean is

Mean: Σmini / N

where:

mi: Midpoint of the ith group

ni: Frequency of the ith group

N: Total sample size

The midpoint is determined by averaging the range’s bottom and upper values.

Range | Frequency | Midpoint | MidPoint*Frequency |

1-20 | 4 | 10.5 | 42 |

21-40 | 8 | 30.5 | 244 |

41-60 | 6 | 50.5 | 303 |

61-80 | 2 | 70.5 | 141 |

81-100 | 9 | 90.5 | 814.5 |

Mean=(42+244+303+141+814.5)/29=53.26

Grouped data set mean is 53.26.

### Grouped Data Standard Deviation

Let’s look at the formula for computing the standard deviation of grouped data.

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Range | Frequency | Midpoint | Midpoint*Freq | Mean | Midpoint-Mean | (Midpoint-Mean)*(Midpoint-Mean) | Frequency[(Midpoint-Mean)(Midpoint-Mean)] |

1-20 | 4 | 10.5 | 42 | 53.26 | 6.5 | 42.25 | 169 |

21-40 | 8 | 30.5 | 244 | 53.26 | 22.5 | 506.25 | 4050 |

41-60 | 6 | 50.5 | 303 | 53.26 | 44.5 | 1980.25 | 11881.5 |

61-80 | 2 | 70.5 | 141 | 53.26 | 68.5 | 4692.25 | 9384.5 |

81-100 | 9 | 90.5 | 814.5 | 53.26 | 81.5 | 6642.25 | 59780.25 |

Standard Deviation: Sqrt(Σni(mi-μ)2 / (N-1))

where:

ni: Frequency of the ith group

mi: Midpoint of the ith group

μ: Average value

N: Total sample size

Let’s execute the formula.

Standard Deviation=sqrt(169+4050+11881.5+9384.5+59780.25)/28=55.18

Grouped data standard is 55.18.

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