Skip to content
mathlab
0
0
0 XP
Lvl 1
KJ
Lesson 11 ~30 min Unit 4 · Data & Probability +85 XP

Comparing Data Sets

Go beyond averages — compare two data sets using measures of centre AND spread to draw meaningful, evidence-based conclusions.

Today's hook: Two basketball teams both average 72 points per game. But Team A scores between 60–84 and Team B scores between 68–76. Which team is more consistent? Comparing data sets goes beyond just comparing means.
0/5QUESTS

Printable Worksheets

Print or save as PDF — or build a custom worksheet from any module's questions.

Build Apply Master Build custom
Think First
warm-up

Before you read on — two classes both have a mean score of 65% on a test. Class A's scores range from 40% to 90%. Class B's scores range from 58% to 72%. Which class performed more consistently? Write your thinking.

Record your answer in your workbook.
1
The Big Idea
+5 XP

To compare two data sets fairly, you must compare measures of centre (mean or median) AND measures of spread (range). A smaller range means more consistent data. Same mean does not mean same performance.

Team A and Team B both average 72 points. But Team A has range 24 (60 to 84) while Team B has range 8 (68 to 76). Team B is far more consistent even though the means are identical. Centre tells you the typical value; spread tells you how reliable that typical value is.

Centre vs Spread Team A Mean: 72 Range: 24 Team B Mean: 72 Range: 8 Same mean → different spread Team B more consistent
Compare: Centre (mean/median) AND Spread (range)
Centre = typical value
Mean or median tells you the middle of the data.
Spread = consistency
Smaller range = more consistent, more predictable.
Always compare both
Comparing only means gives an incomplete picture.
2
What You'll Master
objectives

Know

  • The definitions of centre (mean, median) and spread (range)
  • How to read a back-to-back stem-and-leaf plot
  • That consistency is measured by spread, not centre

Understand

  • Why comparing only means is not sufficient
  • How a smaller range indicates more consistent data
  • Why sample size affects how you interpret comparisons

Can Do

  • Calculate and compare mean, median and range for two data sets
  • Draw and read a back-to-back stem-and-leaf plot
  • Write a conclusion comparing two data sets in context
3
Words You Need
vocabulary
CentreA measure of the typical or middle value in a data set (mean or median).
SpreadHow far apart the data values are from each other; measured by the range.
RangeThe difference between the highest and lowest values: range = max − min.
MeanThe sum of all values divided by the number of values.
ConsistentHaving little variation; data clustered close together (small range).
Back-to-back stem plotA display showing two data sets sharing a common stem, with leaves on either side.
4
Spot the Trap
heads-up

Wrong: "Both groups have the same mean, so their data is the same." Same mean says nothing about spread — the data could be completely different.

Right: Compare mean and range. Two sets can have identical means but very different spreads, telling very different stories.

Wrong: "Group A is better than Group B." Better in what way? Higher mean? Smaller range? Always specify what you mean.

Right: "Group A has a higher mean score, but Group B is more consistent with a smaller range." Name the measure and the direction.

5
Comparing Centres
+5 XP

When comparing two data sets, calculate the mean (sum ÷ count) and the median (middle value when ordered) for each set. The group with the higher mean or median typically performs better on average.

Group A scores: 12, 15, 18, 20, 25. Mean = (12+15+18+20+25)÷5 = 90÷5 = 18. Median = 18 (middle value).
Group B scores: 10, 16, 18, 22, 24. Mean = 90÷5 = 18. Median = 18. Same centre — but the spread will be different!

Mean & Median Group A 12 15 18 20 25 Mean = 18 Median = 18 Group B 10 16 18 22 24 Mean = 18 Median = 18 Identical centres! Now compare the spread...
Mean = sum ÷ count  |  Median = middle value (ordered)
Order first for median
Always sort the data before finding the median.
Mean uses all values
Outliers pull the mean but not the median.
Centre is a summary
It hides information — always follow up with spread.
6
Comparing Spread
+5 XP

The range is the simplest measure of spread. Range = maximum − minimum. A smaller range means the data is clustered closer together — the group is more consistent. A large range means the values are spread out — the group is more variable.

Group A: 12, 15, 18, 20, 25. Range = 25 − 12 = 13.
Group B: 10, 16, 18, 22, 24. Range = 24 − 10 = 14.
If Group A were 5, 15, 18, 20, 30 → range = 25 — much less consistent despite the same mean. A single outlier can blow out the range.

Range = max − min 5 30 Range = 25 (large) 12 25 Range = 13 (smaller) Smaller range = more consistent
Range = max − min  |  Smaller range = more consistent
Range is sensitive to outliers
One extreme value can make the range very large.
State the context
Say what a smaller range means for this real-world situation.
Bigger isn't always better
For consistency, you want a smaller range, not a larger one.
7
Back-to-Back Stem Plots
+5 XP

A back-to-back stem-and-leaf plot places two data sets side by side, sharing a common stem in the middle. The left group's leaves read outward to the left (right to left from the stem). This makes visual comparison of centre and spread very easy.

Team A scores: 61, 68, 72, 75, 78, 84. Team B scores: 69, 71, 73, 74, 76, 77. The stems are 6, 7, 8. Reading Team B's leaves, they cluster in the 70s row. Team A's values are spread across three rows, showing greater variability.

Back-to-Back Stem Plot Team A Stem Team B 6 8 1 9 7 8 5 2 1 3 4 6 7 8 4 Team A: spread across rows 6, 7, 8 Team B: clustered in the 70s
Left leaves read right-to-left  |  Shared stem in middle  |  Always add a key
Left leaves go outward
Read the left group's leaves from the stem outward (right to left).
Order the leaves
Sort leaves in each row so the smallest is closest to the stem.
Include a key
Always write a key so readers know what each leaf represents.

Watch Me Solve It · 3 examples

Watch Me Solve It · Compare two data sets
+15 XP per step
Q1
PROBLEM
Class A: 55, 60, 65, 70, 75. Class B: 40, 60, 65, 70, 90. Compare the two classes using mean and range.
  1. 1
    Calculate the mean for each class
    Class A: (55+60+65+70+75) ÷ 5 = 325 ÷ 5 = 65
    Class B: (40+60+65+70+90) ÷ 5 = 325 ÷ 5 = 65
    Both classes have the same mean. Centre alone doesn't tell the full story.
  2. 2
    Calculate the range for each class
    Class A: 75 − 55 = 20
    Class B: 90 − 40 = 50
    Class A's range is much smaller: 20 vs 50. The spread is very different!
  3. 3
    Write a comparison in context
    Both classes have the same mean score of 65. However, Class A is more consistent with a range of 20, while Class B is less consistent with a range of 50.
    Always state the measure, the direction, and what it means in context.
AnswerSame mean (65). Class A more consistent: range 20 vs 50.
Watch Me Solve It · Draw a back-to-back stem plot
+15 XP per step
Q2
PROBLEM
Group X: 23, 27, 31, 35, 38. Group Y: 21, 24, 33, 36, 39. Draw a back-to-back stem-and-leaf plot.
  1. 1
    Identify the stems (tens digits)
    Stems: 2 (for 20s), 3 (for 30s)
    Both groups have values in the 20s and 30s, so the stems are 2 and 3.
  2. 2
    Fill in the leaves for each group
    Group X | Stem | Group Y
      7 3 |  2  | 1 4
    8 5 1 |  3  | 3 6 9
    Group X leaves go left (read outward from stem). Group Y leaves go right.
  3. 3
    Add a key
    Key: Group X: 3|2 means 23. Group Y: 2|1 means 21.
    A key is essential so readers can interpret the leaves correctly.
AnswerStems: 2, 3. X leaves read right-to-left. Key: 3|2 = 23.
Watch Me Solve It · Write a conclusion in context
+15 XP per step
Q3
PROBLEM
Runner A: mean time = 58 s, range = 4 s. Runner B: mean time = 61 s, range = 1 s. Compare the runners. Who would you pick for a relay team and why?
  1. 1
    Compare the centres (mean times)
    Runner A mean = 58 s (faster on average). Runner B mean = 61 s.
    Lower time is better for a runner. Runner A is faster on average.
  2. 2
    Compare the spread (range)
    Runner A range = 4 s. Runner B range = 1 s.
    Runner B is far more consistent — their time barely changes each run.
  3. 3
    Write a conclusion with context
    Runner A is faster on average (mean 58 s vs 61 s). Runner B is more consistent (range 1 s vs 4 s). For a relay where you need reliability, Runner B is the safer choice.
    Context determines which measure matters most. Always relate back to the real-world question.
AnswerRunner A faster (mean); Runner B more consistent (range). Context determines choice.
9
Common Pitfalls
heads-up
Comparing data sets with different sizes
If Group A has 5 scores and Group B has 20 scores, comparing their ranges directly may be unfair. A larger sample is more likely to include extreme values, producing a wider range even if the groups are equally consistent.
Fix: Acknowledge the sample sizes when comparing. "Group B has 20 students, so its wider range may partly reflect the larger sample size."
Using range when there is an outlier
If most scores are 60–70 but one score is 10, the range is 60 — but this single outlier distorts the spread. The range no longer represents the typical variation in the data.
Fix: Note the outlier separately. "Excluding the outlier of 10, the range is only 10, suggesting the data is actually quite consistent."
Not stating context in conclusions
"Group A has a smaller range" is mathematically correct but incomplete. You must say what this means in the real-world context of the question.
Fix: Every conclusion must include: the measure used, the direction (larger/smaller), and what this means in context (e.g., "more consistent test scores").
Copy Into Your Books

Centre

  • Mean = sum ÷ count
  • Median = middle value when ordered
  • Higher mean = higher typical performance

Spread

  • Range = max − min
  • Smaller range = more consistent
  • Larger range = more variable

Back-to-Back Stem Plot

  • Shared stem in the middle
  • Left group reads outward (right to left)
  • Always include a key

Conclusion Formula

  • Group ___ has a higher mean, so it performs better on average.
  • Group ___ has a smaller range, so it is more consistent.
  • Always relate to real-world context.

How are you completing this lesson?

Brain Trainer · 4 problems

D
Brain Trainer · Comparing Data
4 problems

Four problems to sharpen your data comparison skills. Work each, then reveal the answer.

  1. 1 Group A: mean = 15, range = 8. Group B: mean = 15, range = 2. Which group is more consistent?

    Group B is more consistent. Both groups have the same mean (15), but Group B has a much smaller range (2 vs 8), meaning their values cluster much closer together.Group B: smaller range = more consistent

  2. 2 A back-to-back stem plot has stems 4 and 5. Group X leaves on stem 5: 2, 4, 6. Group Y leaves on stem 5: 0, 1, 9. Which group has the higher median for the 50s values?

    Group X values in the 50s: 52, 54, 56. Median = 54. Group Y values in the 50s: 50, 51, 59. Median = 51. Group X has the higher median.Group X median (50s) = 54 > Group Y median (50s) = 51

  3. 3 Why is comparing only means not enough when comparing two data sets?

    Two data sets can have the same mean but completely different spreads. One could be very consistent (all values close together) while the other is highly variable. The mean hides this important difference.Same mean + different range = very different data sets

  4. 4 What does "consistent performance" mean for a sports team?

    A team performs consistently when their scores don't vary much from game to game. Statistically, this means a small range — the difference between their best and worst results is small. A consistent team is reliable and predictable.Consistent = small range of scores across games
Complete in your workbook.

Quick Check · 5 questions

1
Data set: 14, 18, 22, 25, 28. What is the range?
+10 XP
2
Which statement about range is correct?
+10 XP
3
Group A: 10, 12, 14, 16, 18. Group B: 6, 10, 14, 18, 22. Which is true?
+10 XP
4
In a back-to-back stem plot, how are the left group's leaves read?
+10 XP
5
Find the median of: 21, 12, 18, 24, 15.
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
Apply Medium 3 MARKS

Q6. Team Red scores: 45, 52, 60, 63, 70. Team Blue scores: 55, 57, 60, 63, 65. Calculate the mean and range for each team. Which team is more consistent? Explain.

Answer in your workbook.
Understand Easy 2 MARKS

Q7. A back-to-back stem plot shows: Left side (Group X), stem 3: leaves 8, 5, 2 (reading outward). What are the actual data values for Group X on stem 3? List them from smallest to largest.

Answer in your workbook.
Reason Hard 4 MARKS

Q8. Two call centres handle customer complaints. Centre A: mean wait time = 4.2 min, range = 6 min. Centre B: mean wait time = 5.0 min, range = 1 min. A manager says "Centre A is better because it has a lower mean." Do you agree? Give a full explanation using both mean and range.

Answer in your workbook.
Comprehensive Answers

Quick Check

1. B — Range = 28 − 14 = 14.

2. C — A smaller range means more consistent data.

3. A — Both means = 14. Group A range = 8; Group B range = 16. Group A more consistent.

4. D — Left group leaves read right to left, outward from the stem.

5. B — Ordered: 12, 15, 18, 21, 24. Median = 18.

Show Your Working Model Answers

Q6 (3 marks): Red: mean = (45+52+60+63+70)÷5 = 290÷5 = 58; range = 70−45 = 25 [1]. Blue: mean = (55+57+60+63+65)÷5 = 300÷5 = 60; range = 65−55 = 10 [1]. Team Blue is more consistent because its range (10) is much smaller than Team Red's (25), meaning Blue's scores cluster much closer together [1].

Q7 (2 marks): Leaves 8, 5, 2 on stem 3 represent 38, 35, 32 [1]. Ordered smallest to largest: 32, 35, 38 [1].

Q8 (4 marks): Partially agree [1]. Centre A does have a lower mean (4.2 vs 5.0 min), suggesting faster average service [1]. However, Centre B is far more consistent, with a range of only 1 min vs Centre A's 6 min [1]. From a customer experience perspective, Centre B is more reliable — customers always wait close to 5 min, while Centre A customers might wait anywhere from about 1 to 7 minutes [1].

Stretch Challenge · +25 XP, +10 coins

The Clever Coach

A coach must choose one player for a cricket team. Player A's last 6 scores: 0, 15, 45, 60, 72, 84. Player B's last 6 scores: 30, 35, 38, 42, 45, 50. Calculate mean and range for each. Which player would you choose, and does the purpose of the selection matter (once-off match of the century vs regular season)? Justify your answer fully.

Reveal solution

Player A: mean = (0+15+45+60+72+84)÷6 = 276÷6 = 46, range = 84−0 = 84. Player B: mean = (30+35+38+42+45+50)÷6 = 240÷6 = 40, range = 50−30 = 20. Player A has a higher mean (better on average) but is unpredictable (range 84 — could score 0!). Player B is more consistent. For a once-off high-stakes match, Player A is risky. For reliable regular season performance, Player B is the safer choice.

R
Quick Review

Centre

Mean = sum ÷ count; Median = middle value ordered

Spread

Range = max − min; smaller = more consistent

Back-to-back plot

Shared stem; left leaves read right to left outward

Compare both

Always state centre AND spread in conclusions

Context matters

Relate your conclusion to the real-world situation

Sample size

Larger samples may show wider range — acknowledge this

Interactive: Data Comparison Explorer

Adjust values in two data sets and watch mean and range update live. See how centre and spread can change independently.

Your Badges

0 of 6
First Steps
3-Day Streak
3 in a Row
Lesson Ace
Stretch Seeker
Daily Warrior
DAILY CHALLENGE · TODAY
The Consistency Sprint
Name a real-world situation where consistency (small range) matters more than a high mean. Explain in under 60 seconds for double coins!
NEXT UP · 0% UNLOCKED
Lesson 12 — Interpreting Graphs
0/5
Finish this lesson's Practice MCs to unlock!

Mark lesson as complete

Tick when you've finished Learn, Practice and the Stretch. Earns +85 XP and +25 coins.

Unit overview · Maths subject page · Unit quiz