Comparing Data Sets
Compare two data sets properly, using both a measure of centre and a measure of spread, read comparisons off graphs, and write a comparative statement in the form markers expect.
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Think First
Two basketball teams each averaged 82 points per game. Does that mean they performed the same? What else would you need to know to compare their performances fairly?
Comparing Data Sets
The mean only tells part of the story. Two groups can share the same average but be completely different, one rock-steady, one all over the place. Real analysis always uses both centre and spread.
What You'll Master
- Calculate mean, median, mode, and range for two data sets
- Compare data sets using both centre (mean/median) and spread (range)
- Interpret back-to-back stem-and-leaf plots and side-by-side dot plots
- Write structured comparative statements using statistical language
- Explain what it means for one group to be more consistent than another
Words You Need
⚠ Spot the Trap
Two groups can have the exact same mean but be totally different. Team A (mean 80.8, range 20) is far more consistent than Team B (mean 80.4, range 50). Never compare data sets using only one statistic. Always report both centre and spread.
What to Compare
When comparing two data sets, statisticians always report two types of information:
- Centre (mean or median): which group performed better on average?
- Spread (range): which group was more consistent and predictable?
A group with a higher mean performed better overall. A group with a smaller range was more consistent. Both pieces of information matter, a team that scores 90 occasionally but 50 other times is hard to rely on.
If two groups have similar means, the spread becomes the deciding factor. Consistency is highly valued in sport, business, and science.
Comparing Using Statistics, Worked Example
Team A scores: 70, 78, 82, 84, 90
Team B scores: 55, 65, 82, 95, 105
For Team A: $$\text{Mean} = \frac{70+78+82+84+90}{5} = \frac{404}{5} = 80.8 \qquad \text{Range} = 90-70 = 20$$
For Team B: $$\text{Mean} = \frac{55+65+82+95+105}{5} = \frac{402}{5} = 80.4 \qquad \text{Range} = 105-55 = 50$$
Both teams have almost identical means (80.8 vs 80.4). But Team A's range of 20 is much smaller than Team B's range of 50. Team A is more consistent.
Comparing Using Graphs
Graphs let you see centre and spread at a glance. Two main options for comparing groups:
- Back-to-back stem-and-leaf plot: shared stem in the middle, Group A leaves go left, Group B leaves go right. Read Group A right-to-left when listing values.
- Side-by-side dot plots: both groups plotted on the same number line, different colours. Easy to compare clustering and outliers.
When reading comparison graphs, ask: Which group clusters higher? Which has dots further apart? Are there any outliers pulling the mean away from the median?
Writing Comparative Statements
Strong comparison answers use this three-part structure:
- Compare centre: "On average, Group A scored higher (median 72 vs 65)."
- Compare spread: "However, Group B had a smaller range (20 vs 35), making Group B more consistent."
- Draw a conclusion: "Group A performed better overall, but Group B was more reliable."
Always name the statistic, give both numerical values, and explain what it means in context. Examiners deduct marks for comparative statements that lack values or lack a conclusion.
Common Pitfalls
- Forgetting to order data before finding the median, always sort first
- Comparing means without also comparing ranges, that is incomplete analysis
- Writing "Group A is better" without giving the actual statistical values
- Confusing a small range (consistent) with a small mean (low scores), they are different ideas
- Using mode to compare numerical data, mean and median are almost always more informative
Copy This Into Your Book
Comparing Data Sets, the essentials:
$$\text{Mean} = \frac{\text{sum of values}}{\text{number of values}} \qquad \text{Range} = \text{max} - \text{min}$$
Median = middle value after ordering. For an even count, average the two middle values.
Template: "On average, [Group X] scored higher ([stat] vs [stat]). However, [Group Y] had a smaller range ([val] vs [val]), so [Group Y] was more consistent."
Team A scores: 70, 78, 82, 84, 90. Team B scores: 55, 65, 82, 95, 105. Which team is more consistent?
Group A: 18, 25, 12, 15, 20. Group B: 22, 8, 19, 27, 14. Which statement correctly compares the medians?
Class A: mean 72, range 30. Class B: mean 65, range 12. Which is the best comparative statement?
Which graph is best suited to comparing the shape, centre, and spread of two data sets at the same time?
Two classes both have a mean exam score of 68. Which conclusion is definitely correct?
Q6. Two groups completed a running test (time in seconds).
Group X: 45, 52, 48, 61, 57, 53. Group Y: 38, 71, 49, 60, 42, 66.
(a) Calculate the mean and range for each group.
(b) Write two comparison statements, one about centre (mean), one about spread (range).
Q7. A back-to-back stem-and-leaf plot shows test marks for Year 8 and Year 9 (leaves represent units).
Year 8 (read right to left) | Stem | Year 9
9 8 5 | 5 | 2 4 7
7 4 2 1 | 6 | 0 3 5 8
6 3 | 7 | 1 4 6
5 | 8 | 2
Compare the two groups using: (a) median, (b) range, (c) one comment about the shape or distribution.
Q8. A school compares Year 8 and Year 9 science scores (out of 50).
Year 8: mean = 36, median = 37, range = 28.
Year 9: mean = 39, median = 40, range = 14.
Write a complete comparison using all three statistics. Which year group performed better overall? Explain your reasoning clearly.
Show Answers
Q6
Group X: sum = 316, mean = 316 ÷ 6 ≈ 52.7; range = 61 − 45 = 16.
Group Y: sum = 326, mean = 326 ÷ 6 ≈ 54.3; range = 71 − 38 = 33.
Centre: On average, Group Y had a slightly higher mean time (54.3 s vs 52.7 s), meaning Group X was faster on average.
Spread: Group X had a much smaller range (16 s vs 33 s), so Group X was significantly more consistent.
Q7
Year 8 values: 55, 58, 59, 61, 62, 64, 67, 73, 76, 85 → order has 10 values; median = (64+67)÷2 = 65.5; range = 85−55 = 30.
Year 9 values: 52, 54, 57, 60, 63, 65, 68, 71, 74, 76, 82 → 11 values; median = 65; range = 82−52 = 30.
Both groups have a similar median (~65–65.5) and identical range (30). Year 9 has more values spread into the upper 70s and 80s, while Year 8 concentrates more in the 60s.
Q8
On average, Year 9 performed higher than Year 8 (mean 39 vs 36; median 40 vs 37). Year 9 also had a much smaller range (14 vs 28), meaning Year 9 was both higher-scoring AND more consistent. Year 9 performed better overall on both measures of centre and spread.
Stretch Challenge
Class A: 45, 52, 58, 63, 67, 70, 72, 80. Class B: 30, 40, 55, 65, 72, 78, 85, 95.
(a) Calculate the mean for each class (show working).
(b) Find the median for each class (remember: even number of values, average the two middle ones).
(c) Calculate the range for each class.
(d) Which class performed better overall? Justify using your statistics.
(e) Which class was more consistent? Justify.
(f) Are there any potential outliers? (A value is a suspected outlier if it is more than about 1.5× the range away from the nearest value.) Justify.