Comparative Statistics Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Comparative Statistics.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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?

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Comparative statistics uses summary measures to ask whether two or more groups truly differ, and by how much.

Common stuck point: The procedure for comparative statistics is the easy part; the trap is comparing only the means and ignoring the spread. Asking "Am I weighing two or more groups against each other rather than describing a single group?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I weighing two or more groups against each other rather than describing a single group?

Worked Examples

Example 1

easy

Compare two groups on a test: Group A (

n=30

): mean=75, SD=10. Group B (

n=30

): mean=82, SD=8. Calculate the difference in means and comment on whether group B performs better.

Answer

Group B scores 7 points higher on average with Cohen's d ≈ 0.77 (medium-large effect).

First step

Difference in means:

\bar{x}_B - \bar{x}_A = 82 - 75 = 7

points

Full solution

2
Group B's mean is 7 points higher
3
But consider variability: SD_A=10 and SD_B=8; the groups overlap substantially
4
Cohen's d (effect size): $d = \frac{7}{\sqrt{(10^2+8^2)/2}} = \frac{7}{\sqrt{82}} \approx \frac{7}{9.06} \approx 0.77$ — medium-large effect

Comparing groups requires reporting both the difference in centers and the effect size (like Cohen's d). A 7-point difference might be meaningful or trivial depending on the variability. Effect size standardizes the difference for meaningful interpretation.

Example 2

medium

Three products have customer satisfaction scores. Product A: mean=4.2, SD=0.5 (n=50). Product B: mean=3.8, SD=1.2 (n=50). Explain which product is preferable and why SD matters.

Example 3

medium

Group P (

n=20

): mean

= 50

, SD

= 5

. Group Q (

n=20

): mean

= 55

, SD

= 5

. Compute Cohen's

d

for the mean difference.

Example 4

medium

Brand X bulbs: mean life

1{,}200

h, SD

200

h. Brand Y bulbs: mean life

1{,}100

h, SD

50

h. Which brand has higher average life and which is more reliable?

Example 5

medium

Math class A mean

= 70

, SD

= 10

. Math class B mean

= 70

, SD

= 5

. Both classes have the same average; which class has more students near the mean, and why?

Example 6

hard

School A:

30\%

of students score above

80

. School B:

45\%

score above

80

. Sample sizes are both

200

. Compute the absolute and relative differences in the proportion scoring high.

Example 7

hard

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

Example 8

challenge

Hospital A reports a

10\%

mortality rate; Hospital B reports

5\%

. Before concluding B is better, list two confounders that could explain the gap.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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 2

hard

Men: mean height = 70", SD = 3". Women: mean height = 64", SD = 2.5". A person is 67" tall. Calculate their z-score in each distribution and determine which group they are more extreme in.

Example 3

easy

Group A mean

= 80

, group B mean

= 75

. Which group scored higher on average, and by how much?

Example 4

easy

Two classes have the same mean of

70

, but class A's scores range

65

–

75

and B's range

40

–

100

. Which is more variable?

Example 5

easy

A study finds a

0.1

-point GPA difference is 'statistically significant' with

50{,}000

students. Is it necessarily practically important?

Example 6

easy

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

Example 7

easy

Two distributions have means

50

and

55

but overlap heavily. What does heavy overlap suggest about the difference?

Example 8

easy

Boys' median height

= 150

cm, girls'

= 148

cm in a class. State the comparison in one sentence.

Example 9

easy

Which single number summarizes how far apart two group means are in standard-deviation units?

Example 10

easy

Sample A (

n=5

) mean

= 100

; sample B (

n=500

) mean

= 100

. Which mean is a more reliable estimate?

Example 11

medium

Group A: mean

80

, SD

4

. Group B: mean

76

, SD

4

. Compute Cohen's d (mean difference over pooled SD

\approx 4

Example 12

medium

A new teaching method raises mean scores from

70

72

(SD

\approx 15

). Is this a large effect? Compute the effect size and interpret.

Example 13

medium

Two regions report unemployment of

5\%

and

7\%

. The difference seems small, but explain when a

2

-point gap could be highly meaningful.

Example 14

medium

Test A and B both have mean

50

. A's SD is

2

; B's SD is

20

. A student scores

54

on each. On which test is this score more impressive?

Example 15

medium

A headline: 'Town A's cancer rate is double Town B's!' A has

4

cases /

2000

; B has

2

cases /

2000

. Why is this comparison fragile?

Example 16

medium

Two delivery services average

30

minutes, but service X ranges

28

–

32

and Y ranges

10

–

50

. Which would you choose for a time-critical delivery and why?

Example 17

medium

A drug trial: treatment group recovery

70\%

(

n=200

), control

60\%

(

n=200

). The

10

-point gap — is it likely real or chance? What determines this?

Example 18

medium

Group A:

\{2,4,6\}

, group B:

\{1,5,9\}

. Both have mean

4

. Compute each range to compare variability.

Example 19

medium

Test scores: class A mean

75

(SD

5

), class B mean

75

(SD

15

). A student scoring

85

is in which class's top region more clearly?

Example 20

challenge

Two groups have identical means and identical variances but you suspect they differ. Name one distributional feature that a comparison of means and variances would MISS, and give an example.

Example 21

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 22

challenge

Construct an example where group A beats group B on the mean but B beats A on the median, and explain what asymmetry causes this.

Example 23

easy

Class A mean test score

= 78

. Class B mean

= 84

. Which class scored higher on average and by how much?

Example 24

easy

Boys' median height

= 152

cm; girls' median

= 148

cm. State the comparison in one sentence.

Example 25

easy

Two box plots: Plot 1 IQR

= 8

; Plot 2 IQR

= 20

. Which dataset is more spread out around the middle half?

Example 26

easy

Sample A:

\{4, 5, 6, 7, 8\}

. Sample B:

\{1, 3, 6, 9, 11\}

. Both have mean

6

. Which has the larger range?

Example 27

medium

A diet group lost mean

4.0

kg (SD

= 2.0

). Control lost mean

1.0

kg (SD

= 2.0

). Estimate Cohen's

d

Example 28

medium

A student scores

80

on Test A (class mean

= 75

, SD

= 5

) and

82

on Test B (class mean

= 70

, SD

= 12

). On which test did the student perform better, relative to peers?

Example 29

medium

True or false: a statistically significant 0.1-point GPA difference detected with 100{,}000 students is automatically practically important.

Example 30

medium

Boys' mean shoe size

= 8.0

, SD

= 1.5

. Girls' mean shoe size

= 6.5

, SD

= 1.5

. Compute Cohen's

d

Example 31

medium

Two surveys with

n = 100

each. Survey 1:

52\%

support. Survey 2:

48\%

support. Margin of error for each is

\pm 5\%

. Can we conclude they differ?

Example 32

medium

A drug raises mean recovery rate from

60\%

66\%

. Compute the relative improvement.

Example 33

hard

Two five-number summaries: A: min

20

, Q1

40

, median

50

, Q3

60

, max

80

. B: min

35

, Q1

48

, median

52

, Q3

58

, max

65

. Which has a narrower IQR, and which has a wider range?

Example 34

hard

Sample A (

n = 5

) mean

= 100

; Sample B (

n = 5{,}000

) mean

= 100

. Which mean is a more reliable estimate of its population mean, and why?

Example 35

hard

Group 1 mean

= 12

(SD

= 2

n = 16

). Group 2 mean

= 14

(SD

= 2

n = 16

). Compute the standard error of the difference in means.

Example 36

hard

From the previous question (mean diff

= 2

\text{SE}_{\text{diff}} \approx 0.707

), compute the

t

-statistic for the difference in means.

Example 37

hard

Sample A min/max are

5

and

25

; sample B min/max are

14

and

16

. Without further info, which sample is more variable?

Example 38

hard

In a study, women's mean reaction time

= 240

ms (SD

= 30

); men's mean

= 260

ms (SD

= 30

). Cohen's

d

for the difference is approximately ___.

← Back to Comparative Statistics Practice Mode

Related Concepts

Mean Standard Deviation Signal vs Noise

Background Knowledge

These ideas may be useful before you work through the harder examples.

meanstandard deviation