Averages: Mean, Median, Mode and Range

This lesson covers the three measures of average — mean, median, and mode — along with the range as a measure of spread. These are among the most commonly tested topics in the AQA GCSE Mathematics Statistics paper and appear in a wide variety of question types, from simple calculations to more challenging problem-solving scenarios.

---

The Three Averages

An average is a single value that represents or summarises a set of data. There are three types of average you need to know.

Average	How to Calculate	When to Use
Mean	Add up all the values and divide by how many values there are	When all values are important and there are no extreme outliers
Median	Arrange values in order and find the middle value	When there are extreme values (outliers) that would distort the mean
Mode	Find the value that appears most often	When data is categorical (non-numerical) or you want the most popular value

---

Calculating the Mean

Mean = sum of all values / number of values

Worked Example

Find the mean of: 3, 7, 5, 9, 6, 4, 8

Step 1: Add the values: 3 + 7 + 5 + 9 + 6 + 4 + 8 = 42

Step 2: Count the values: 7

Step 3: Divide: 42 / 7 = 6

The mean is 6.

Using the Mean to Find a Missing Value

If you know the mean and the number of values, you can work backwards to find a missing value.

Worked Example

The mean of five numbers is 12. Four of the numbers are 8, 15, 10, and 13. Find the fifth number.

Step 1: Total of all five numbers = mean x count = 12 x 5 = 60

Step 2: Total of known numbers = 8 + 15 + 10 + 13 = 46

Step 3: Missing number = 60 - 46 = 14

Exam Tip: "Working backwards from the mean" is a very common AQA exam question. Remember: total = mean x number of values. This formula lets you find the sum first, then subtract the known values.

---

Calculating the Median

The median is the middle value when all values are arranged in order from smallest to largest.

• If there is an odd number of values, the median is the single middle value.
• If there is an even number of values, the median is the mean of the two middle values.

To find the position of the median: (n + 1) / 2, where n is the number of values.

Worked Example — Odd Number of Values

Find the median of: 12, 5, 8, 3, 15, 9, 7

Step 1: Arrange in order: 3, 5, 7, 8, 9, 12, 15

Step 2: There are 7 values. Position = (7 + 1) / 2 = 4th value.

Step 3: The 4th value is 8.

The median is 8.

Worked Example — Even Number of Values

Find the median of: 4, 9, 2, 7, 11, 6

Step 1: Arrange in order: 2, 4, 6, 7, 9, 11

Step 2: There are 6 values. Position = (6 + 1) / 2 = 3.5th value, so the median is between the 3rd and 4th values.

Step 3: 3rd value = 6, 4th value = 7. Median = (6 + 7) / 2 = 6.5

The median is 6.5.

Exam Tip: The most common mistake with the median is forgetting to put the values in order first. Always sort the data before finding the median. The formula (n + 1) / 2 gives you the position, not the median itself.

---

Finding the Mode

The mode is the value that appears most frequently in a data set.

• A data set can have no mode (if all values appear the same number of times).
• A data set can have more than one mode (if two or more values share the highest frequency — this is called bimodal or multimodal).

Worked Example

Find the mode of: 3, 5, 7, 3, 8, 5, 3, 9, 5, 3

Count the frequencies:

• 3 appears 4 times
• 5 appears 3 times
• 7 appears 1 time
• 8 appears 1 time
• 9 appears 1 time

The mode is 3 (it appears most often).

---

Calculating the Range

The range is a measure of spread (not an average). It tells you how spread out the data is.

Range = largest value - smallest value

Worked Example

Find the range of: 3, 7, 15, 2, 9

Range = 15 - 2 = 13

Exam Tip: The range is affected by outliers (extreme values). A single very high or very low value can make the range misleadingly large. If asked about limitations, always mention this.

---

Choosing the Best Average

Different averages are appropriate in different situations.

Average	Advantages	Disadvantages
Mean	Uses all the data; most commonly used	Affected by extreme values (outliers)
Median	Not affected by outliers; good for skewed data	Does not use all the data
Mode	Can be used with non-numerical data; represents the most common value	May not exist; may not be representative; ignores most of the data

When to Use Each Average

graph TD
    A[Which average should I use?] --> B{Is the data categorical?}
    B -- Yes --> C[Use the Mode]
    B -- No --> D{Are there extreme values or outliers?}
    D -- Yes --> E[Use the Median]
    D -- No --> F[Use the Mean]

Worked Example

The weekly wages of employees at a small company are:

250, 280, 290, 300, 310, 320, 1500

• Mean = (250 + 280 + 290 + 300 + 310 + 320 + 1500) / 7 = 3250 / 7 = 464.29 (to 2 d.p.)
• Median = 300 (the 4th value out of 7)
• Mode = no mode (all values appear once)

The mean (464.29) is misleadingly high because of the outlier (1500, likely the manager's wage). The median (300) is the best average here because it is not affected by the extreme value and better represents the typical wage.

---

Averages from a Frequency Table

When data is presented in a frequency table, the method for calculating averages changes slightly.

Worked Example

Score	Frequency
1	3
2	7
3	5
4	4
5	1
Total	20

Mean:

Add a "Score x Frequency" column: