Comparing Distributions
Comparing distributions means using summary statistics to compare two or more data sets, rather than describing one on its own, with a full comparison needing a statement about average, using the mean, median or mode to say which data set is typically higher or lower, and a statement about spread, using the range to say which is more consistent or varied. Both statements should be written in context, referring to what the data measures, and compare the two sets directly rather than simply stating two numbers side by side.
Before you start
Make sure you're comfortable with these topics first:
Method
- Calculate (or read off) the same average, usually the mean or median, for both data sets, using a consistent measure for a fair comparison.
- Calculate (or read off) the range for both data sets.
- Write an AVERAGE comparison sentence: state which data set has the higher or lower average, in context, not just one number next to another.
- Write a SPREAD comparison sentence: state which data set has the smaller (more consistent) or larger (more varied) range, in context.
- Use comparative language throughout, such as 'on average' and 'more consistently', rather than just repeating the numbers from each set.
- Where relevant, link the two statements to the real-world context, for example explaining why a smaller range might make one option more reliable or predictable than another.
Worked example
Two tutors, Priya and Daniel, each mark 5 mock papers. Priya's marks, out of 60: 40, 44, 46, 42, 48. Daniel's marks, out of 60: 25, 58, 33, 51, 43. Compare the two sets of marks.
- Find Priya's mean: (40 + 44 + 46 + 42 + 48) / 5 = 220 / 5 = 44.
- Find Daniel's mean: (25 + 58 + 33 + 51 + 43) / 5 = 210 / 5 = 42.
- Find Priya's range: 48 - 40 = 8. Find Daniel's range: 58 - 25 = 33.
- Compare the averages: Priya's mean (44) is higher than Daniel's mean (42), so on average Priya's marks were slightly higher.
- Compare the ranges: Priya's range (8) is much smaller than Daniel's range (33), so Priya's marks were far more consistent than Daniel's, which were much more varied.
Practice questions
Try each question, then tap to reveal the answer.
Q1Team A's points in 4 matches: 12, 18, 15, 15. Team B's points in 4 matches: 10, 22, 8, 20. Find the mean points scored for each team.Show answer
Answer: Both teams have a mean of 15 points (Team A: 60/4; Team B: 60/4)
Q2Using the same two teams (Team A: 12, 18, 15, 15; Team B: 10, 22, 8, 20), find the range for each team and state which team was more consistent.Show answer
Answer: Team A's range is 6 (18-12), Team B's range is 14 (22-8), so Team A was more consistent
Q3Class 7X's median test score is 62, with a range of 14. Class 7Y's median test score is 58, with a range of 30. Write one sentence comparing the classes' medians, and one sentence comparing their ranges.Show answer
Answer: On average, Class 7X scored higher than Class 7Y, since its median (62) is higher than Class 7Y's median (58). Class 7X's scores were also more consistent, since its range (14) is much smaller than Class 7Y's range (30)
Q4Bus route A's journey times, in minutes, have a mean of 25 and a range of 4. Bus route B's journey times have a mean of 22 and a range of 18. Which route has the shorter average journey time, and which route has the more reliable (consistent) journey time?Show answer
Answer: Route B has the shorter average journey time (mean 22 minutes, against 25 for Route A). Route A is the more reliable route, since its range (4 minutes) is much smaller than Route B's range (18 minutes)
Q5The number of sweets in each of 5 bags from Shop A: 20, 22, 20, 24, 20. The number of sweets in each of 5 bags from Shop B: 21, 21, 23, 21, 19. Find the modal number of sweets for each shop.Show answer
Answer: Shop A's mode is 20 sweets (appears 3 times); Shop B's mode is 21 sweets (appears 3 times)
Q6Explain why comparing the ranges of two data sets alone does not tell you which set has the higher average.Show answer
Answer: The range only measures spread, not the size or location of the data - two data sets can have very different averages but the same range, or the same average but very different ranges, so the mean or median has to be compared separately to say which set is typically higher
Q7The heights, in cm, of 5 tomato plants grown in Greenhouse 1: 30, 35, 28, 33, 34. The heights of 5 tomato plants grown in Greenhouse 2: 32, 31, 33, 30, 34. Compare the mean heights of the two greenhouses.Show answer
Answer: Both greenhouses have the same mean height, 32 cm (160/5 for each), so on average there is no difference between them; comparing their ranges would be needed to say anything about consistency
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Two sprinters' 100 m race times, in seconds, over 5 races: Amara: 12.1, 12.4, 12.0, 12.3, 12.2. Ben: 11.8, 13.1, 11.5, 12.9, 12.2. (a) Calculate the mean race time for each sprinter. (b) Calculate the range of race times for each sprinter. (c) Compare the two sprinters' performances, referring to both their average time and their consistency.
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A gardener compares the weekly growth, in mm, of tomato plants using two different fertilisers over 6 weeks. Fertiliser X: mean growth 18 mm, range 5 mm. Fertiliser Y: mean growth 15 mm, range 21 mm. The gardener wants a fertiliser that gives strong AND reliable growth. (a) State which fertiliser gives the greater average weekly growth. (b) State which fertiliser gives more consistent growth. (c) Which fertiliser would you recommend? Justify your answer using both measures.
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Five rounds of a golf tournament give Player 1 a mean score of 72. Four of Player 1's five round scores are: 70, 75, 68, 74. (a) Find Player 1's fifth round score. (b) Player 2's five round scores have a mean of 74. State, with a reason, which player had the better average score, given that a LOWER score is better in golf.
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Free printable worksheet
Want more practice on paper? Download the comparing distributions worksheet pack - 8 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 37 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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