# Grouped Data Mean and Standard Deviation Calculator

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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.

How to Calculate Root Mean Square Error (RMSE) in R »

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.

How to Calculate Mean Absolute Percentage Error (MAPE) in R »

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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