When we are finding averages, it is easy 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.

When we are dealing with the median in a frequency table, the good news is that 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 that the modal shoe size is size 8.

**Median:** We need to find the number in the middle position.

**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 and halve it:

(113 + 1) ÷ 2 = 57th

This means that the median is 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 |

Cumulative frequency | 5 | 12 | 21 | 35 | 56 | 84 | 99 | 105 | 110 | 113 |

We can now see that shoe size 8 starts at the 57th number and ends at the 84th number.

The 57th number must, therefore, be in the shoe size 8 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 | 50 ≤ p < 60 |

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

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.

**To find the median**, we need to find the number in the middle position.

**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 and halve it:

(45 + 1) ÷ 2 = 23

This means that the median is the** 23rd **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 | 50 ≤ p < 60 |

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

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

We can now see that 30 ≤ p < 40 starts at the 22nd number and ends at the 31st number.

The 23rd number must, therefore, be in the column 30 ≤ p < 40.

**This is the median group.**

Let's move on to some questions.