Practice Comparative Statistics in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Comparative statistics involves using statistical measures to compare two or more groups, data sets, or distributions.

Is A bigger/better/different than B? By how much? Is the difference real?

Showing a random 20 of 50 problems.

Example 1

medium
Test A and B both have mean 5050. A's SD is 22; B's SD is 2020. A student scores 5454 on each. On which test is this score more impressive?

Example 2

easy
Two distributions have means 5050 and 5555 but overlap heavily. What does heavy overlap suggest about the difference?

Example 3

medium
A headline: 'Town A's cancer rate is double Town B's!' A has 44 cases / 20002000; B has 22 cases / 20002000. Why is this comparison fragile?

Example 4

easy
Box plots for two classes show: Class A median=70, IQR=20. Class B median=75, IQR=5. Which class has better performance? Which is more consistent?

Example 5

challenge
Explain why a difference can be statistically significant yet have a confidence interval that includes practically trivial values, using a large-sample example.

Example 6

easy
Boys' median height =150= 150cm, girls' =148= 148cm in a class. State the comparison in one sentence.

Example 7

challenge
Hospital A reports a 10%10\% mortality rate; Hospital B reports 5%5\%. Before concluding B is better, list two confounders that could explain the gap.

Example 8

easy
A study finds a 0.10.1-point GPA difference is 'statistically significant' with 50,00050{,}000 students. Is it necessarily practically important?

Example 9

easy
True or false: two groups with the same mean must have the same spread.

Example 10

hard
True or false: when comparing two groups, comparing only the means can hide important differences in spread.

Example 11

easy
Class A mean test score =78= 78. Class B mean =84= 84. Which class scored higher on average and by how much?

Example 12

easy
Group X has range 1010. Group Y has range 3030. Which group is more variable?

Example 13

medium
Group A: mean 8080, SD 44. Group B: mean 7676, SD 44. Compute Cohen's d (mean difference over pooled SD ≈4\approx 4).

Example 14

medium
A new teaching method raises mean scores from 7070 to 7272 (SD ≈15\approx 15). Is this a large effect? Compute the effect size and interpret.

Example 15

medium
Group P (n=20n=20): mean =50= 50, SD =5= 5. Group Q (n=20n=20): mean =55= 55, SD =5= 5. Compute Cohen's dd for the mean difference.

Example 16

hard
True or false: the smaller the p-value in a two-sample test, the larger the effect size must be.

Example 17

hard
School A: 30%30\% of students score above 8080. School B: 45%45\% score above 8080. Sample sizes are both 200200. Compute the absolute and relative differences in the proportion scoring high.

Example 18

easy
Sample A: {4,5,6,7,8}\{4, 5, 6, 7, 8\}. Sample B: {1,3,6,9,11}\{1, 3, 6, 9, 11\}. Both have mean 66. Which has the larger range?

Example 19

easy
To decide if A is 'really' different from B, is eyeballing a bar chart enough?

Example 20

easy
Boys' median height =152= 152 cm; girls' median =148= 148 cm. State the comparison in one sentence.
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