Find the Mode and Median from a Frequency Table

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.