When we are finding averages, we should easily be able to find the **median** and** mode** from a list but it gets a bit more complicated when we are dealing with frequency tables.

**Definitions.**

**Median: **The value that is in the middle position of a list of ordered data.

**Mode****:** The value with the highest frequency.

The good news when we are dealing with the median is that in a frequency table, the data is already ordered.

**Example 1: Find the median and modal shoe size for this frequency table.**

Shoe Size | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

Frequency | 5 | 7 | 9 | 14 | 21 | 28 | 15 | 6 | 5 | 3 |

**Mode:** The mode is defined as the value with the highest frequency. In this table, the highest frequency is 28.

This means the modal shoe size is size 8.

**Median:** We need to find the number in the middle position, to do this, we first need to find the position of this number.

Step 1: Find the total frequency.

All we need to do is add up all the frequencies. This gives us 113.

Step 2: Find the position of the median.

To find the position of the median, we add 1 to the total ad half it

(113 + 1) ÷ 2 = 57th

This means the median the 57th number in the list

Step 3: Find the value of the median.

To do this, the easiest way is to find the cumulative frequencies then see where the 57th number lies

Shoe Size | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

Frequency (Cum | 5 | 12 | 21 | 35 | 56 | 84 | 99 | 105 | 110 | 113 |

We can now see that shoe size 7 starts at the 56th number and ends at the 98th number.

The 57th number must, therefore, be in the shoe size 7 column.

**Example 2: Find the median and modal groups for this grouped frequency table.**

Pocket Money (p) | 0 ≤ p <10 | 10 ≤ p <20 | 20 ≤ p <30 | 30 ≤ p <40 | 40 ≤ p <50 | 40 ≤ p <60 |

Frequency | 4 | 6 | 11 | 10 | 8 | 6 |

**Mode:** The mode is defined as the value with the highest frequency. In this table, the highest frequency is 11.

This means the modal pocket money is 20 ≤ p <30

**Median:** We need to find the number in the middle position, to do this, we first need to find the position of this number.

Step 1: Find the total frequency.

All we need to do is add up all the frequencies. This gives us 45.

Step 2: Find the position of the median.

To find the position of the median, we add 1 to the total ad half it

(45 + 1) ÷ 2 = 23

This means the median the 23 rd number in the list.

Step 3: Find the value of the median.

To do this, the easiest way is to find the cumulative frequencies then see where the 23rd number lies

Pocket Money (p) | 0 ≤ p <10 | 10 ≤ p <20 | 20 ≤ p <30 | 30 ≤ p <40 | 40 ≤ p <50 | 40 ≤ p <60 |

Frequency | 4 | 6 | 11 | 10 | 8 | 6 |

Frequency (cum) | 4 | 10 | 21 | 31 | 39 | 45 |

We can now see that 20 ≤ p <30 starts at the 20th number and ends at the 30th number.

The 23rd number must, therefore, be in the column 20 ≤ p <30 - This is the median group