What are averages and range?
An average is a single value used to represent a typical value in a set of data. At GCSE you need to know three types of average — the mean, median and mode — plus the range, which measures spread rather than typical size.
The key definitions:
- Mean — add up all the values and divide by how many there are
- Median — the middle value when the data is written in order
- Mode — the value that occurs most often (there can be more than one, or none)
- Range — the largest value minus the smallest value
Mean, median, mode and range from a list
Worked Example 1
Find the mean, median, mode and range of: 4, 7, 3, 9, 7, 5, 2
1
Mean: add the values (4+7+3+9+7+5+2 = 37) and divide by 7: 37 ÷ 7 = 5.29 (2 d.p.)
2
Median: order the data: 2, 3, 4, 5, 7, 7, 9 — the middle value is 5
3
Mode: 7 appears twice, more than any other value, so the mode is 7
4
Range: 9 − 2 = 7
AnswerMean 5.29, Median 5, Mode 7, Range 7
Averages from a frequency table
Worked Example 2
The table shows the number of pets owned by 20 students. Find the mean number of pets.
| Number of pets | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Frequency (students) | 4 | 7 | 6 | 3 |
1
Multiply each value by its frequency: (0×4) + (1×7) + (2×6) + (3×3) = 0 + 7 + 12 + 9 = 28
2
Divide by the total frequency (4+7+6+3 = 20): 28 ÷ 20 = 1.4
Answer1.4 pets
Estimated mean from grouped data
When data is grouped into class intervals, you cannot find an exact mean because you don't know the individual values. Instead, use the midpoint of each group to calculate an estimated mean.
Worked Example 3
The table shows the time (in minutes) taken by 30 students to complete a puzzle. Estimate the mean time.
| Time (minutes) | 0–10 | 10–20 | 20–30 | 30–40 |
|---|---|---|---|---|
| Frequency (students) | 6 | 12 | 9 | 3 |
1
Find the midpoint of each group: 5, 15, 25, 35
2
Multiply each midpoint by its frequency: (5×6) + (15×12) + (25×9) + (35×3) = 30 + 180 + 225 + 105 = 540
3
Divide by the total frequency (6+12+9+3 = 30): 540 ÷ 30 = 18
AnswerEstimated mean = 18 minutes
Bar height = frequency for each time interval. The midpoint of each bar (dashed line) is the value used to estimate the mean — 5, 15, 25 and 35 minutes.
Working backwards from the mean
Worked Example 4 — Missing value
Five numbers have a mean of 8. Four of the numbers are 5, 9, 6 and 11. Find the fifth number.
1
Find the total of all five numbers: mean × count = 8 × 5 = 40
2
Add the four known numbers: 5 + 9 + 6 + 11 = 31
3
Subtract from the total: 40 − 31 = 9
Answer9
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1Find the mean, median, mode and range of: 6, 2, 9, 4, 6, 8, 5Foundation
Q2Find the median and range of: 12, 7, 15, 9, 11, 7, 20Foundation
Q3A shop records daily sales for a week: £320, £290, £410, £320, £350, £480, £310. Find the mean daily sales.Foundation
Q4A list of six numbers has a mean of 9. Find the total of the six numbers.Foundation
Q5Number of goals scored by a team in 10 matches: 0 goals (2 matches), 1 goal (4 matches), 2 goals (3 matches), 3 goals (1 match). Find the mean number of goals per match.Foundation
Higher (Grade 5–7)
Q6
The table shows shoe sizes of 25 students. Find the mean shoe size.
Higher| Shoe size | 4 | 5 | 6 | 7 |
|---|---|---|---|---|
| Frequency | 3 | 7 | 9 | 6 |
Q7
The table shows the age (in years) of 40 members of a gym. Estimate the mean age.
Higher| Age (years) | 20–30 | 30–40 | 40–50 | 50–60 |
|---|---|---|---|---|
| Frequency | 10 | 16 | 9 | 5 |
Q8Four numbers have a mean of 11. Three of the numbers are 8, 14 and 10. Find the fourth number.Higher
Q9The mean of 6 numbers is 15. A seventh number, 29, is added. Find the new mean.Higher
Q10
The table shows time spent revising (hours) by 50 students. Estimate the mean revision time.
Higher| Time (hours) | 0–2 | 2–4 | 4–6 | 6–8 |
|---|---|---|---|---|
| Frequency | 14 | 20 | 11 | 5 |
Higher — Hard (Grade 8–9)
Q11The mean of 5 numbers is 12. The mean of 3 of these numbers is 10. Find the mean of the remaining 2 numbers.Grade 8–9
Q12A class of 30 students has a mean test score of 64%. If the 10 highest scores have a mean of 80%, find the mean score of the remaining 20 students.Grade 8–9
Q13Two classes sit the same test. Class A (18 students) has a mean of 62%. Class B (22 students) has a mean of 71%. Find the combined mean for both classes, to 1 d.p.Grade 8–9
Q14
The table shows delivery times (minutes) for 25 parcels. The total frequency is 25, and the sum of (frequency × midpoint) across all classes is 620. Find x, then estimate the mean.
Grade 8–9| Time (minutes) | 0–10 | 10–20 | 20–30 | 30–40 |
|---|---|---|---|---|
| Frequency | 5 | 8 | x | 4 |
Q15Explain why the estimated mean calculated from grouped data is not usually exactly equal to the true mean of the raw data, using the idea of midpoints in your answer.Grade 8–9
Answers
Foundation (Q1–Q5)
Q1Mean 5.71, Median 6, Mode 6, Range 7(sum 40 ÷ 7 = 5.71; ordered 2,4,5,6,6,8,9)
Q2Median 11, Range 13(ordered 7,7,9,11,12,15,20; 20 − 7)
Q3£354.29(total £2480 ÷ 7)
Q454(9 × 6)
Q51.3 goals(13 ÷ 10)
Higher (Q6–Q10)
Q65.72(143 ÷ 25)
Q737.25 years(1490 ÷ 40, using midpoints 25, 35, 45, 55)
Q812(44 − 32)
Q917(119 ÷ 7)
Q103.28 hours(164 ÷ 50, using midpoints 1, 3, 5, 7)
Higher — Hard (Q11–Q15)
Q1115(60 − 30 = 30, ÷ 2)
Q1256%(1920 − 800 = 1120, ÷ 20)
Q1367.0%((18×62 + 22×71) ÷ 40 = 2678 ÷ 40)
Q14x = 8; estimated mean = 24.8(25 − 5 − 8 − 4 = 8; 620 ÷ 25)
Q15The estimated mean assumes every value in a class is equal to the midpoint, but the real values are spread throughout the class — so the estimate is an approximation, not the exact mean.
Common mistakes
Common Mistake 1
Forgetting to order the data before finding the median
The median is the middle value of the ordered list, not the list as given. For 8, 3, 9, 4, 6 the median is not 9 (the middle position in the original order) — order the data first: 3, 4, 6, 8, 9, then take the middle value, 6.
Common Mistake 2
Dividing by the number of groups instead of the total frequency
When finding a mean from a frequency table, always divide by the total frequency (the total number of data items), not by the number of rows or categories in the table.
Common Mistake 3
Using the class width instead of the midpoint for grouped data
For an estimated mean, each class must be represented by its midpoint (the average of the upper and lower boundary), not the class width. For the group 20–30, the midpoint is 25, not 10.
Common Mistake 4
Confusing the mode with the modal class
For grouped data you cannot state an exact mode — only the modal class, the group with the highest frequency. Never quote a single number as "the mode" when the data has been grouped into class intervals.
Exam tips
💡 Exam Tip 1
Always state your units
If the data is in minutes, £, cm or goals, your final answer needs the same units. Examiners can withhold the final accuracy mark for a correct number with no unit — write "18 minutes", not just "18".
💡 Exam Tip 2
Show your working for method marks
Even if you can do the arithmetic in your head, write the calculation out — e.g. "sum ÷ count" or "Σfx ÷ Σf". Method marks are awarded even if your final answer has a small arithmetic slip.
💡 Exam Tip 3
Use "estimate" for grouped data
When a question asks for the mean of grouped data, always write "estimated mean" in your answer — examiners look for this word to confirm you understand that midpoints give an approximation, not an exact value.
💡 Exam Tip 4
Only round your final answer
Keep exact values (or several decimal places) through your working, and round only at the very end. Rounding too early — for example rounding a midpoint or a running total — can shift your final answer outside the accepted range.
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