In statistics there are three types of average: the mean, the median and the mode. Measures of spread such as the range and the interquartile range can be used to reach statistical conclusions.

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It can be difficult and time-consuming to calculate mean or median values if there is a large set of data. To make calculations more efficient, tables are often used to condense the data.

The table below shows the amount of goals scored in 10 football matches.

Number of goals | Frequency |
---|---|

0 | 2 |

1 | 2 |

2 | 5 |

3 | 1 |

The **modal value** is the value that occurs most. From a table, this means the modal value is the one with the highest frequency.

Number of goals | Frequency |
---|---|

0 | 2 |

1 | 2 |

2 | 5 |

3 | 1 |

There were five football matches where 2 goals were scored, which is a higher frequency than any other amount of goals.

The modal amount of goals scored is 2.

The **median value** is the middle value when all items are in order. In this table, the amounts of goals are in order, as they start with zero goals and move up to three goals scored.

\(\text{Median} = \frac{n + 1}{2}\). To find the value that is the median for a set of \(n\) items of data, add 1 to \(n\) and then divide by 2. The total frequency is 10, so there are 10 items of data.

Number of goals | Frequency | |
---|---|---|

0 | 2 | |

1 | 2 | |

2 | 5 | |

3 | 1 | |

Total | 10 |

To find the median, work out \(\frac{n + 1}{2} = \frac{10 + 1}{2} = \frac{11}{2} = 5.5\), which means the median will be half way between the 5th and 6th items of data. Go down the frequency column totalling the numbers until you find the category that contains the 5th and 6th items of data.

Number of goals | Frequency | Cumulative frequency | |
---|---|---|---|

0 | 2 | \[2\] | |

1 | 2 | \[2 + 2 = 4\] | |

2 | 5 | \[4 + 5 = 9\] | |

3 | 1 | \[9 + 1 = 10\] | |

Total | 10 |

The third group starts with the 5th item of data, so the 5th and 6th items of data will both be 2. The median number of goals is 2.