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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.
