# Averages from frequency tables

## In a nutshell

Mode, median and mean can all be calculated from frequency tables. If the data being considered is discrete then all of these averages can be calculated, but for any situation where the data is categoric, only the mode can be found.

## Calculating averages

### Mode

#### Procedure

1. | Find the greatest frequency value in the table. |

2. | The mode is the data value corresponding to this greatest frequency. |

### Median

#### Procedure

1. | Work out the total number of samples taken by summing the frequencies. This value is $n$. |

2. | Ensure the rows of the table are written in ascending order for the data value they represent. |

3. | The median is the data value that corresponds to the $\dfrac{n+1}{2} th$ item. |

### Mean

#### Procedure

1. | Calculate $n$ and fill in a third column of the table corresponding to frequency multiplied by the data value. |

2. | Calculate the sum of the third column. This value is $\text{sum } x$. |

3. | The mean is calculated as $\dfrac{\text{sum } x}{n}$. |

##### Example 1

*Calculate the mode, median and mean from the following frequency table:*

#### Size | #### Frequency |

$6$ | $4$ |

$6.5$ | $7$ |

$7$ | $6$ |

$8$ | $3$ |

*Mode:** The most frequent value is *$\underline{6.5}$

*Median**: There are *$4+7+6+3=20$* pieces of data. Hence the median occurs at the *$\dfrac{20+1}{2}=10.5th$* data item. This is the item between the $10th$ piece of data (size $6.5$) and the $11th$ piece of data (size $6.5$). Hence the median is *$\underline{6.5}$

*Mean**: Create a third column as follows,*

**SIZE** | **FREQUENCY** | **SIZE x FREQUENCY** |

$6$ | $4$ | $24$ |

$6.5$ | $7$ | *$45.5$* |

$7$ | $6$ | *$42$* |

$8$ | $3$ | *$24$* |

*This third column has a total of $24+45.5+42+24=135.5$, and so the mean is:*

*$\dfrac{135.5}{20}=\underline{6.775}$*