**Contents:**

## What is Grouped Data?

**Grouped data** is data that has been bundled together in categories. Histograms and frequency tables can be used to show this type of data:

The data is grouped together by *classes *or *bins*.

## Grouped vs. Ungrouped Data

Ungrouped data is the data you first gather from an experiment or study. The data is raw — that is, it’s not sorted into categories, classified, or otherwise grouped. An **ungrouped **set of data is basically a list of numbers.

## Calculating the Sample Mean for Grouped Data

When you have a frequency table or other group of data, the original set of data is lost — replaced with statistics for the group. You can’t find the exact sample mean (as you don’t have the original data) but you *can* find an estimate. The formula for estimating the sample mean for data that has been grouped is:

Where:

- x̄ is the sample mean,
- x is the class (or category) midpoint,
- f is the class frequency.

**Example question:** Find the sample mean for the following frequency table.

Score | Frequency ( f ) |
---|---|

Between 5 and 10 | 1 |

10 ≤ t < 15 | 4 |

15 ≤ t < 20 | 6 |

20 ≤ t < 25 | 4 |

25 ≤ t < 30 | 2 |

30 ≤ t < 35 | 3 |

TOTALS | 20 |

Step 1: Find the midpoint for each class interval. the midpoint is just the middle of each interval. For example, the middle of 10 and 15 is 12.5:

Score | Frequency ( f ) | Midpoint ( x ) |
---|---|---|

Between 5 and 10 | 1 | 7.5 |

10 ≤ t < 15 | 4 | 12.5 |

15 ≤ t < 20 | 6 | 17.5 |

20 ≤ t < 25 | 4 | 22.5 |

25 ≤ t < 30 | 2 | 27.5 |

30 ≤ t < 35 | 3 | 32.5 |

TOTALS | 20 |

Step 2: Multiply the midpoint (x) by the frequency (f):

Score | Frequency ( f ) | Midpoint ( x ) | Midpoint x * frequency f |
---|---|---|---|

Between 5 and 10 | 1 | 7.5 | 7.5 |

10 ≤ t < 15 | 4 | 12.5 | 50 |

15 ≤ t < 20 | 6 | 17.5 | 105 |

20 ≤ t < 25 | 4 | 22.5 | 90 |

25 ≤ t < 30 | 2 | 27.5 | 55 |

30 ≤ t < 35 | 3 | 32.5 | 97.5 |

TOTALS | 20 | 405 |

Add up all of the totals for this step. In other words, add up all the values in the last column (you should get 405).

Step 3: Divide the last column (f*x) by the second column (f):

The mean (x̄) = 405 / 20 = 20.25.

## References

Agresti A. (1990) Categorical Data Analysis. John Wiley and Sons, New York.

Klein, G. (2013). The Cartoon Introduction to Statistics. Hill & Wamg.