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Primary Mathematics: Comparing Data Groups, Consistency and Variability | Worked Learning Guide

Comparing two groups is not the same as comparing two single numbers. A fair comparison looks at what is typical, how much the values vary, and whether both groups are being measured on the same basis.

Two classes can have the same mean but very different consistency. Two sports teams can have the same range but different typical scores. Two survey groups can have different totals, making percentages more useful than raw counts. Statistical comparison requires more than asking which number is bigger.

This guide is an upper-Primary enrichment and transition bridge. It extends mean, median, range, proportions and graph reading into group-to-group reasoning.

Fair basis · Compare center · Compare spread · Consistency · Counts and proportions · Write careful conclusions · 24 questions · Worked solutions

1. Compare like with like

Class A has 12 pupils who walk to school. Class B has 18.

If Class A has 20 pupils and Class B has 40, raw counts do not show the larger proportion.

A: 12/20 = 60%.

B: 18/40 = 45%.

Class B has more walkers by count, but Class A has a larger proportion.

2. Use the same units and time period

One class reports weekly reading time in hours; another reports daily reading time in minutes.

Convert to a common basis before comparing.

A fair comparison needs aligned units, definitions and time periods.

3. Compare typical values

Class A scores: 68,69,70,71,72.

Class B scores: 60,65,70,75,80.

Both means and medians are70.

A conclusion such as “both groups have the same typical score by mean and median” is supported.

4. Different centers can be clear even with overlap

A: 10,11,12,13,14.

B: 13,14,15,16,17.

Mean A=12; mean B=15.

The groups overlap at13 and14, yet B has the higher center overall.

5. Equal center does not mean equal spread

A: 68,69,70,71,72. Range=4.

B: 50,60,70,80,90. Range=40.

Both centers are70, but A is far more tightly clustered.

6. Equal range does not mean equal distribution

A: 1,5,9.

B: 1,1,9.

Both range8.

But A has middle value5, while B has median1. The interior distribution differs.

7. Consistency is about variability

If two archers have similar average scores, the one whose scores vary less is more consistent.

Consistency questions often require spread, not just center.

Worked example

A: 8,9,9,10,9. Range2.

B: 5,9,13,7,11. Range8.

Both means=9. A is more consistent by range.

8. A higher average can come with lower consistency

A: 8,8,9,9,10. Mean8.8, range2.

B: 4,9,10,14,13. Mean10, range10.

B has higher average but much greater variation.

Which group is “better” depends on the decision: higher typical result or greater consistency?

9. Compare medians when extreme values distort means

A: 10,11,12,13,50.

B: 10,11,12,13,14.

Medians both12.

Means differ sharply because A contains one extreme high value.

If the question asks for a typical middle performance, median may be more representative.

10. Compare group sizes before using counts

Survey A: 30 of50 prefer option X.

Survey B: 45 of100 prefer option X.

A proportion60%; B45%.

Raw count 45 is larger, but support is more common in Survey A.

11. Use percentages for unequal group sizes

Percentages place groups on a common 100-part scale.

This is especially useful for categorical comparisons across groups of different sizes.

12. Small groups can produce unstable-looking percentages

Group A: 3 of4 pupils =75%.

Group B: 30 of40 pupils =75%.

Percentages match, but the second estimate is based on many more observations.

A group of four can change by25 percentage points when one response changes.

13. One outlier can affect groups differently

A group of five is more strongly affected by one extreme value than a group of fifty if the other values are similar.

Group size belongs in the interpretation.

14. Use paired comparisons only when the pairing is meaningful

Pre-test and post-test results for the same pupils can be compared pupil by pupil.

Two unrelated classes do not have a natural one-to-one pairing unless the study design creates one.

15. Compare change from a baseline

Class A mean rises from60 to70: +10.

Class B mean rises from80 to88: +8.

A has larger absolute gain.

Relative gain A=10/60≈16.7%; B=8/80=10%.

Both absolute and relative comparisons support A in this example.

16. Same final value can hide different starting points

Two pupils finish at80. One started60, the other75.

Final performance is equal; improvement is not.

State which quantity is being compared.

17. Overlapping groups can still differ statistically

Group A may have values10–20 and Group B values15–25.

The overlap15–20 does not mean the distributions are identical.

Compare centers and spread, not whether every value is unique to one group.

18. Avoid claiming every individual follows the group summary

If Group B has a higher mean, it does not follow that every B value exceeds every A value.

Group-level summaries do not automatically describe each individual case.

19. Match the conclusion to the evidence

Too strong: “Class A is always more consistent.”

Better: “In these five recorded trials, Class A had the smaller range and therefore showed less variation by this measure.”

The second conclusion names the dataset and the measure used.

20. Use more than one statistic when the decision needs it

A school choosing a timing system may care about both average journey time and consistency.

One group could be faster on average but more variable.

Center and spread answer different parts of the decision.

21. Graphs can compare groups visually

Aligned dot plots, line plots or bar charts can make differences in center and spread visible.

Both graphs should use the same scale if visual comparison is intended.

22. Different graph scales can manufacture apparent differences

If Group A graph spans0–100 and Group B graph spans40–60, small B differences may look much larger.

Always inspect axes before comparing visual spread.

23. Common group-comparison errors

• Comparing raw counts when group sizes differ.
• Using different units or time periods.
• Comparing means while ignoring very different spreads.
• Calling one high value representative of a group.
• Assuming higher group mean means every individual is higher.
• Ignoring sample size.
• Comparing graphs with different scales by eye.
• Using stronger language than the data justify.

24. A group-comparison protocol

1. Align: Same variable, unit and time period?
2. Size: Are group sizes equal or should proportions be used?
3. Center: Compare mean/median/mode as appropriate.
4. Spread: Compare range and visible variability.
5. Shape: Look for clusters, gaps and extremes.
6. Question: Is the decision about typical level, consistency, improvement or frequency?
7. Conclude: State exactly what the data support.

25. Practice: 24 original questions

Questions 1–8: Center and spread

1. Find mean of68,69,70,71,72.
2. Find mean of50,60,70,80,90.
3. Compare ranges of Questions1 and2.
4. Which group is more consistent by range?
5. A=10,11,12,13,14; B=13,14,15,16,17. Compare means.
6. A=8,9,9,10,9; B=5,9,13,7,11. Compare means and ranges.
7. A=10,11,12,13,50; B=10,11,12,13,14. Compare medians.
8. Explain why equal median does not imply equal spread.

Questions 9–16: Counts and proportions

9. Class A:12 of20 walk. Class B:18 of40 walk. Compare proportions.
10. Survey A:30 of50 support X. Survey B:45 of100. Which proportion is larger?
11. 3 of4 and30 of40 both equal what percentage?
12. Why is one response change more influential in a group of4 than a group of40?
13. Class A mean rises60→70. Class B80→88. Compare absolute gains.
14. Compare percentage gains for Question13.
15. Two pupils finish at80 but start60 and75. Compare improvements.
16. Explain why a higher group mean does not mean every individual value is higher.

Questions 17–24: Interpretation

17. Create two five-value groups with same mean but different ranges.
18. Create two groups with same range but different medians.
19. Write one conclusion for a group with smaller range but equal mean.
20. Write one conclusion for a group with higher mean but larger range.
21. Explain why aligned graph scales matter.
22. Give one example where median may be more useful than mean.
23. Give one example where percentages are better than raw counts.
24. Create a two-group comparison question and state whether center, spread or proportion is most important.

26. Worked solutions

1. 70. 2. 70. 3. 4 versus40. 4. First group.

5. Means12 and15. 6. Both means9; ranges2 and8. 7. Both medians12. 8. Values may be arranged very differently around the same middle value.

9. 60% versus45%; Class A larger. 10. 60% versus45%; Survey A larger. 11. 75%. 12. One of4 changes the proportion by25 percentage points; one of40 changes by2.5 points.

13. +10 versus+8; A larger. 14. About16.7% versus10%; A larger. 15. Improvements20 and5. 16. Distributions can overlap; means summarize groups rather than order every individual pair.

17. Example:10,10,10,10,10 and2,6,10,14,18 both mean10; ranges0 and16. 18. Example1,5,9 and1,1,9 both range8; medians5 and1. 19. “The groups have the same mean, but Group A is more consistent by range.” 20. “Group B has the higher average but greater variability; which is preferable depends on the decision.”

21. Different scales distort visual comparisons of spread and change. 22. Example: one extreme value pulls the mean away from the middle of most observations. 23. Example: comparing support in classes of20 and40 pupils. 24. Answers vary; statistic choice should match the posed question.

27. Comparison laboratory: speed versus consistency

Travel times in minutes:

Route A: 18,19,20,21,22.

Route B: 10,15,20,25,30.

Both mean20.

Route A range4; Route B range20.

If the goal is predictable journey time, Route A is more consistent. If both have the same mean, there is no average-time advantage in this dataset.

28. Parent and tutor guide

When comparing groups, ask the learner to name the decision first: typical level, consistency, improvement, or share of a group. This determines which statistic deserves attention.

Use unequal group sizes to teach why proportions can be fairer than raw counts.

29. Mastery receipt

• I align units and definitions before comparing.
• I use proportions when group sizes differ.
• I compare center and spread separately.
• I understand consistency as low variability.
• I know equal center does not imply equal distribution.
• I distinguish final value from improvement.
• I read group graphs on common scales.
• I write conclusions no stronger than the evidence.

Sources and scope

Illustrative Mathematics — Is It Center or Is It Variability? provides a useful distinction between questions about typical level and questions about consistency or spread.

For Singapore Primary curriculum scope, use the MOE Primary Mathematics Syllabus P1–P6, updated October 2025. This guide is an upper-Primary enrichment/transition bridge.

Continue the Statistical Thinking collection

• Statistical Questions, Data Collection and Variability
• Distributions, Center, Spread and Typical Values
• Data Claims, Sampling, Misleading Graphs and Evidence
• BTT Primary Mathematics Learning Hub

The Quiet Return

A fair group comparison asks two questions before declaring a winner: where is the group centered, and how much do its values vary?