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We now know how to find the averages of data, but what if this data is grouped? This makes things harder for us as there will not be exact numbers that we can find an average of – there will instead be a group of values.

### Mean of grouped frequency

The actual calculation of a mean when data is grouped is exactly the same except we need to add in one more step at the start. We must begin by finding the ‘midpoint’ of each group and then use this as the value. To find a midpoint we must add up the highest and lowest value in the group and then divide by 2. So the midpoint of the group ‘1 to 10’ must be equal to .

It is this midpoint value that we will use to give us the mean of the data set by multiplying it by the frequency in that group.

#### Steps for getting the mean of data when in a grouped frequency table:

- Find each midpoint of the groups and put this as another column.
- Multiply each midpoint by its frequency and write as another column.
- Get the sum of all these values (add them all up).
- Divide this value from step 3 by the total frequency.

#### Example

Find the mean of the following data.

Score | Midpoint | Frequency | MP✕Freq |
---|---|---|---|

0 to 9 | 4.5 | 3 | 13.5 |

10 to 19 | 14.5 | 11 | 159.5 |

20 to 29 | 24.5 | 16 | 392 |

30 to 39 | 34.5 | 17 | 586.5 |

40 to 49 | 44.5 | 9 | 400.5 |

50 to 59 | 54.5 | 2 | 109 |

The midpoints for each group have been calculated by adding the lowest and highest in the group and then dividing by 2. Next, we must multiply each midpoint by its corresponding frequency value. This gives us the values that are shown in the right-hand column. Then adding all of these in the far right column together and dividing by the total frequency gives us as the mean.

### Mode of grouped frequency

The most frequent value when data is set into groups will not be specific. It will instead be the **most frequent group**. This is found in the same way as we would find the mode usually, but instead the answer is simply a certain group instead of a specific number, e.g. we may get a mode of ‘10 to 20’ meaning that values fall into this group most frequently.

#### Example

What is the mode of the following data set?

Score | Frequency |
---|---|

0 to 9 | 8 |

10 to 19 | 24 |

20 to 29 | 19 |

30 to 39 | 27 |

40 to 49 | 11 |

50 to 59 | 8 |

Clearly the data here is split into groups. However, **the way in which we find the mode does not change at all**, we will simply end up with a most frequent group instead of most frequent value. From the table this modal group is clearly **30 to 39**.

### Median of a frequency distribution table

When finding the median of a frequency table we need to realise that the data is really already in the correct order. Therefore, all we need to do is to find the middle value (by finding as usual) and then see which value this will take by looking at the frequencies in each segment.

#### Example

Find the median from the data listed here in a frequency table.

Score | Frequency |
---|---|

1 | 11 |

2 | 3 |

3 | 1 |

4 | 19 |

5 | 7 |

6 | 6 |

To find the median of this data we need to first find the amount of individual values we have, this is done by summing the frequencies to get 47. Then finding the value for the median we get , so we need to find the 24th value when the data is listed in order. This means that we can simply move down the ‘frequency’ column of the table and find where the 24th will be since the table is in the correct order. A frequency of 11 for the score 1 means that we have a block of 11 different 1s. Then we have 3 different 2s, then 1 score of 3 and so on. Clearly , and . This then tells us that the 24th value must be included in the block of 19 values of 4. Therefore, **the median of this data must be 4.**

It is extremely rare that you will need to find the median of data that is grouped. If this does happen, you must first find the midpoints of each group and then use this as the score. Then working through with the exact same method as we have just looked at will give us the correct median for a table of grouped frequencies.

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