What you'll learn
This revision guide covers the three key measures of central tendency: mean, median and mode. You'll learn how to calculate each average from different data presentations, understand when to use each measure, and interpret their meaning in real-world contexts. These skills form the foundation of statistical analysis at GCSE level and appear frequently in AQA examinations.
Key terms and definitions
Mean — the sum of all values divided by the number of values; affected by extreme values (outliers).
Median — the middle value when data is arranged in order; if there are two middle values, the median is their mean.
Mode — the value or category that occurs most frequently in a dataset; the only average that can be used for categorical data.
Modal class — the class interval with the highest frequency in grouped data; you cannot find an exact mode from grouped frequency tables.
Outlier — an extreme value that lies far from the other values in a dataset; can significantly affect the mean but not the median.
Range — the difference between the highest and lowest values; a measure of spread (not an average, but often examined alongside averages).
Frequency — the number of times a particular value or category occurs in a dataset.
Class interval — a group of values in grouped data, written as inequalities (e.g., 10 ≤ x < 20).
Core concepts
Calculating the mean
The mean is the most commonly used average in statistics. The formula is:
Mean = Sum of all values ÷ Number of values
Or using notation: x̄ = Σx ÷ n
Where:
- x̄ represents the mean (pronounced "x bar")
- Σx means "sum of all x values"
- n is the number of values
For ungrouped data:
- Add all values together
- Count how many values there are
- Divide the sum by the count
For data in a frequency table:
- Multiply each value by its frequency (x × f)
- Add all these products to get Σfx
- Add all frequencies to get Σf (total number of values)
- Calculate: Mean = Σfx ÷ Σf
For grouped data (estimation):
- Find the midpoint of each class interval
- Multiply each midpoint by its frequency
- Divide Σ(midpoint × frequency) by Σf
The mean is sensitive to extreme values. A single very large or very small value can pull the mean away from the typical value, making it less representative.
Calculating the median
The median represents the central position in ordered data and is unaffected by outliers.
For ungrouped data:
- Arrange all values in ascending order
- If n is odd: median is the middle value at position (n+1)÷2
- If n is even: median is the mean of the two middle values at positions n÷2 and (n÷2)+1
For data in a frequency table:
- Calculate the total frequency (Σf)
- Find the position: (Σf + 1) ÷ 2
- Use a cumulative frequency column to locate which value contains this position
For grouped data: You cannot calculate an exact median from grouped data, but you can identify the median class — the class interval containing the middle position. Some exam questions may ask you to estimate the median using linear interpolation, though this is more common at A-level.
Calculating the mode
The mode is the simplest measure of central tendency and the only one suitable for categorical (non-numerical) data.
For ungrouped data:
- Identify the value that appears most frequently
- Data can be unimodal (one mode), bimodal (two modes), or multimodal (more than two modes)
- If all values occur with equal frequency, there is no mode
For data in a frequency table:
- Find the value with the highest frequency
For grouped data:
- You cannot find an exact mode
- Instead, identify the modal class — the class interval with the highest frequency
- State the modal class as an inequality, not as a single value
For categorical data:
- The mode is the only appropriate average
- For example: if examining favourite sports, the modal category might be "football"
Choosing the appropriate average
Understanding which average to use is essential for AQA examinations:
Use the mean when:
- You want to use all data values in the calculation
- The data is numerical with no extreme outliers
- You need the most mathematically useful average for further calculations
- Data is reasonably symmetrical
Use the median when:
- There are outliers or extreme values that would distort the mean
- You want to find the "typical" middle value
- The data is skewed (not symmetrical)
- Example contexts: house prices, salaries
Use the mode when:
- The data is categorical (non-numerical)
- You want to identify the most popular or common item
- Dealing with discrete data where the mode makes practical sense
- Example contexts: shoe sizes, favourite colours, most common number of children per family
Comparing datasets using averages
Exam questions frequently require you to compare two or more datasets:
When comparing:
- Calculate the same type of average for each dataset
- Use measures of spread (usually range) alongside averages
- Write comparative statements, not just numerical answers
- Make conclusions in context
For example: "Group A has a higher mean score (15.3) than Group B (12.7), indicating Group A performed better overall. However, Group A also has a larger range (18), suggesting more variation in performance."
Working with different data presentations
AQA exams present data in various formats:
Lists of data:
- Raw numbers requiring you to organize and calculate
Frequency tables:
- Two columns: values and frequencies
- Use Σfx for mean calculations
- Build cumulative frequency column for median
Grouped frequency tables:
- Data organized into class intervals
- Can only estimate mean using midpoints
- Can only identify modal class, not exact mode
Stem-and-leaf diagrams:
- Data already in order, making median easier to find
- Remember to use the key correctly (e.g., 2|3 = 23 if key states "2|3 means 23")
Bar charts and pictograms:
- Extract frequency data first, then calculate averages
Worked examples
Example 1: Mean, median and mode from a list
Question: The number of goals scored in 9 football matches were: 2, 1, 3, 2, 0, 5, 2, 1, 4
Calculate: (a) the mean [2 marks] (b) the median [2 marks] (c) the mode [1 mark]
Solution:
(a) Mean = sum of values ÷ number of values Sum = 2 + 1 + 3 + 2 + 0 + 5 + 2 + 1 + 4 = 20 ✓ Mean = 20 ÷ 9 = 2.22 goals (or 2.2̇) ✓
(b) First, arrange in order: 0, 1, 1, 2, 2, 2, 3, 4, 5 ✓ Position of median = (9 + 1) ÷ 2 = 5th value Median = 2 goals ✓
(c) Mode = 2 goals (appears 3 times, more than any other value) ✓
Example 2: Mean from a frequency table
Question: The table shows the number of text messages sent by 50 students on one day.
| Number of texts (x) | Frequency (f) |
|---|---|
| 0 | 3 |
| 1 | 8 |
| 2 | 15 |
| 3 | 12 |
| 4 | 7 |
| 5 | 5 |
Calculate the mean number of text messages. [3 marks]
Solution:
Add column for x × f:
| x | f | x × f |
|---|---|---|
| 0 | 3 | 0 |
| 1 | 8 | 8 |
| 2 | 15 | 30 |
| 3 | 12 | 36 |
| 4 | 7 | 28 |
| 5 | 5 | 25 |
| Totals | Σf = 50 | Σfx = 127 |
✓ for correct table setup and calculations
Mean = Σfx ÷ Σf ✓ Mean = 127 ÷ 50 = 2.54 text messages ✓
Example 3: Modal class and estimated mean from grouped data
Question: The grouped frequency table shows the times, in minutes, taken by 40 students to complete a test.
| Time (t minutes) | Frequency |
|---|---|
| 10 ≤ t < 15 | 4 |
| 15 ≤ t < 20 | 11 |
| 20 ≤ t < 25 | 18 |
| 25 ≤ t < 30 | 5 |
| 30 ≤ t < 35 | 2 |
(a) Write down the modal class. [1 mark] (b) Calculate an estimate of the mean time. [4 marks]
Solution:
(a) Modal class is 20 ≤ t < 25 (highest frequency = 18) ✓
(b)
| Time (t) | Midpoint | Frequency (f) | Midpoint × f |
|---|---|---|---|
| 10 ≤ t < 15 | 12.5 | 4 | 50 |
| 15 ≤ t < 20 | 17.5 | 11 | 192.5 |
| 20 ≤ t < 25 | 22.5 | 18 | 405 |
| 25 ≤ t < 30 | 27.5 | 5 | 137.5 |
| 30 ≤ t < 35 | 32.5 | 2 | 65 |
| Totals | 40 | 850 |
✓ for correct midpoints ✓ for correct products
Estimated mean = 850 ÷ 40 ✓ = 21.25 minutes ✓
(Note: This is an estimate because we don't know the exact values within each class)
Common mistakes and how to avoid them
Not arranging data in order before finding the median. Always write out the ordered list, especially for small datasets. For larger datasets, use a frequency table with cumulative frequency.
Confusing mode with modal class. From grouped data, you can only state the modal class (an interval), never a single modal value. Write your answer as an inequality.
Forgetting to multiply by frequency when calculating mean from a frequency table. Create a column for x × f and show this working clearly. Examiners award marks for method.
Incorrect midpoint calculations for grouped data. The midpoint is (lower boundary + upper boundary) ÷ 2. For the class 10 ≤ x < 20, the midpoint is (10 + 20) ÷ 2 = 15, not 10 or 20.
Giving mean or median for categorical data. If the data consists of categories (colours, names, types), only the mode is appropriate. Don't attempt to calculate a mean of non-numerical categories.
Rounding too early in calculations. Keep full calculator accuracy throughout calculations and round only the final answer to an appropriate degree of accuracy (usually 2 or 3 significant figures unless told otherwise).
Exam technique for "Processing Data: Averages — Mean, Median and Mode"
Show all working clearly. For mean calculations, write out Σx or Σfx, show the division, and give the final answer. Method marks are available even if your final answer is incorrect.
Include units in your final answer. If the question gives data in minutes, goals, or pounds, your answer must include the unit. Marks are sometimes deducted for missing units.
"Calculate" means show working; "write down" means the answer is obvious from the data. Respond appropriately to command words. For "write down the mode," simply state the answer. For "calculate the mean," you must show the formula and steps.
When comparing datasets, make contextual statements. Don't just write "15.3 and 12.7" — write "The mean for Year 10 (15.3 marks) is higher than Year 9 (12.7 marks), suggesting Year 10 performed better overall."
Quick revision summary
The mean uses all values (sum ÷ count) but is affected by outliers. The median is the middle value when ordered and is resistant to extremes. The mode is the most frequent value and works for categorical data. For grouped data, use midpoints to estimate the mean and identify the modal class instead of an exact mode. Choose the appropriate average based on data type and presence of outliers. Always show working for calculations, include units, and make contextual comparisons when analyzing datasets.