Averages from frequency tables are a common GCSE Maths topic. Let’s learn how to calculate the mean, median and mode from a frequency table, at around Grade 4 level.
What is a frequency table?
A frequency table shows how many times each value occurs.
For example, a group of students are asked how many pets they have.
| Number of pets | Frequency |
|---|---|
| 0 | 2 |
| 1 | 4 |
| 2 | 3 |
| 3 | 1 |
| Total | 10 |
This means:
- 2 students have 0 pets.
- 4 students have 1 pet.
- 3 students have 2 pets.
- 1 student has 3 pets.
There are 10 students altogether.
Finding the mode
The mode is the value that occurs most often.
Look at the frequency column and find the biggest number.
The highest frequency is 4, which corresponds to 1 pet.
Answer: Mode = 1
Remember: the mode is the value, not the frequency.
Finding the mean
The mean is calculated by:
\[ \text{Mean}=\frac{\text{Total of all values}}{\text{Number of values}} \]
With a frequency table, we need to multiply each value by its frequency first.
Step 1: Add a value × frequency column.
| Number of pets | Frequency | Value × frequency |
|---|---|---|
| 0 | 2 | $0\times2=0$ |
| 1 | 4 | $1\times4=4$ |
| 2 | 3 | $2\times3=6$ |
| 3 | 1 | $3\times1=3$ |
| Total | 10 | 13 |
Step 2: Divide the total of the third column by the total frequency.
\[ \text{Mean}=\frac{13}{10}=\boxed{1.3} \]
Answer: The mean number of pets is 1.3.
Exam tip: Always multiply the value by its frequency. Do not just add up the values in the first column!
Finding the median
The median is the middle value when all the data is arranged in order.
Our table contains 10 students, so the median is the average of the 5th and 6th values.
Let’s list the positions:
| Number of pets | Positions |
|---|---|
| 0 | 1st–2nd |
| 1 | 3rd–6th |
| 2 | 7th–9th |
| 3 | 10th |
The 5th and 6th values are both 1.
\[ \text{Median}=\frac{1+1}{2}=\boxed{1} \]
Answer: The median is 1.
Remember:
- If there is an odd number of values, find the single middle value.
- If there is an even number of values, find the middle two values and calculate their mean.
5. Summary: the three averages
| Average | Method |
|---|---|
| Mean | Total of value × frequency ÷ total frequency |
| Median | Find the middle value or middle two values |
| Mode | Find the value with the highest frequency |
The most important skill for GCSE Grade 4 is knowing which method to use for each average. Practise the mean first, as it is usually the one that requires the most working.