Open Math›Year 8›Data analysis›Comparing data sets

Comparing data sets

Use mean, median, range and IQR to compare two data sets and draw conclusions about centre and spread.

Worked examples

Comparing means of two data sets

Straightforward

Problem

Two students recorded their daily screen time (in minutes) over five days.

Student A:

45, 60, 75, 90, 30

Student B:

50, 65, 55, 80, 55

Compare the mean screen time of each student and state who spent more time on screens on average.

Calculate the mean for Student A.

\text{Mean A} = \frac{45 + 60 + 75 + 90 + 30}{5} = \frac{300}{5} = 60 \text{ minutes}

Calculate the mean for Student B.

\text{Mean B} = \frac{50 + 65 + 55 + 80 + 55}{5} = \frac{305}{5} = 61 \text{ minutes}

Compare the two means and state the conclusion.

Student A: mean

= 60

min. Student B: mean

= 61

min. Since

61 > 60

, Student B spent more time on screens on average.

Answer

Student B had a higher mean screen time of

61

minutes, compared to Student A's mean of

60

minutes.

Comparing spread using range and IQR

Moderate

Problem

Two classes sat a maths quiz. Their sorted scores were:

Class A:

18, 22, 26, 29, 33, 36, 40

Class B:

12, 20, 26, 30, 34, 38, 44

Compare the spread of each class's scores using the range and IQR.

Calculate the range for each class.

\text{Range A} = 40 - 18 = 22

\text{Range B} = 44 - 12 = 32

Find Q1 and Q3 for Class A.

There are 7 values. The median is the 4th value:

29

. The lower half is

18, 22, 26

, so

\text{Q1} = 22

. The upper half is

33, 36, 40

, so

\text{Q3} = 36

Find Q1 and Q3 for Class B.

There are 7 values. The median is the 4th value:

30

. The lower half is

12, 20, 26

, so

\text{Q1} = 20

. The upper half is

34, 38, 44

, so

\text{Q3} = 38

Calculate the IQR for each class.

\text{IQR A} = 36 - 22 = 14

\text{IQR B} = 38 - 20 = 18

Compare and state the conclusion.

Class B has a greater range (

32 > 22

) and a greater IQR (

18 > 14

). Class B's scores are more spread out, while Class A's scores are more consistent.

Answer

Class B has greater spread: range

= 32

(vs

22

) and IQR

= 18

(vs

14

). Class A is more consistent.

Writing a comparison using two statistics

Challenging

Problem

Two athletes' 100 m sprint times (seconds) over a season are summarised below.

Athlete A: median

= 11.8

s, IQR

= 0.4

s, range

= 1.0

s
Athlete B: median

= 12.3

s, IQR

= 0.8

s, range

= 1.6

s

Compare the two athletes' performances and state which athlete is more consistent, justifying your answer with statistics.

Compare the medians to assess typical performance.

Athlete A has a lower median (

11.8

< 12.3

s). A lower sprint time means Athlete A is generally faster.

Compare the IQRs to assess consistency.

Athlete A has IQR

= 0.4

s and Athlete B has IQR

= 0.8

s. Since

0.4 < 0.8

, Athlete A's times are more tightly grouped -- Athlete A is more consistent.

Compare the ranges to confirm the conclusion about spread.

Athlete A has range

= 1.0

s and Athlete B has range

= 1.6

s. Since

1.0 < 1.6

, Athlete A also has less overall spread, confirming the conclusion.

Write a comparison statement using at least two statistics.

Athlete A has a lower median sprint time (

11.8

s vs

12.3

s) and a smaller IQR (

0.4

s vs

0.8

s), so Athlete A is both faster and more consistent than Athlete B.

Answer

Athlete A is faster (lower median:

11.8

s vs

12.3

s) and more consistent (smaller IQR:

0.4

s vs

0.8

s).

Practise

Q1·Straightforward

Data set A:

7, 11, 15, 19

. Calculate the mean of data set A.

Explanation

\text{Mean A} = \frac{7 + 11 + 15 + 19}{4} = \frac{52}{4} = 13

Q2·Straightforward

Data set B:

5, 10, 14, 19

. Calculate the mean of data set B.

Explanation

\text{Mean B} = \frac{5 + 10 + 14 + 19}{4} = \frac{48}{4} = 12

Q3·Straightforward

Data set A:

7, 11, 15, 19

has a mean of

13

. Data set B:

5, 10, 14, 19

has a mean of

12

. Which statement is correct?

Data set A has a higher mean.Data set B has a higher mean.Both data sets have the same mean.The means cannot be compared because the data sets are different.

Explanation

Data set A has a mean of

13

and data set B has a mean of

12

. Since

13 > 12

, data set A has the higher mean.

Q4·Straightforward

Data set A:

15, 18, 21, 24, 27

. Data set B:

14, 19, 22, 25, 30

. By how much is data set B's mean greater than data set A's mean?

Explanation

\text{Mean A} = \frac{15 + 18 + 21 + 24 + 27}{5} = \frac{105}{5} = 21

\text{Mean B} = \frac{14 + 19 + 22 + 25 + 30}{5} = \frac{110}{5} = 22

\text{Difference} = 22 - 21 = 1

Q5·Moderate

Data set P (sorted):

8, 12, 16, 20, 24, 28, 32

. Data set Q (sorted):

5, 11, 17, 20, 23, 29, 37

. Calculate the range of data set P.

Explanation

\text{Range P} = 32 - 8 = 24

Q6·Moderate

Data set P (sorted):

8, 12, 16, 20, 24, 28, 32

. Data set Q (sorted):

5, 11, 17, 20, 23, 29, 37

. Calculate the range of data set Q.

Explanation

\text{Range Q} = 37 - 5 = 32

Q7·Moderate

Data set P (sorted):

8, 12, 16, 20, 24, 28, 32

. Calculate the IQR of data set P.

Explanation

The median of the 7 values is the 4th value:

20

. The lower half is

8, 12, 16

, so

\text{Q1} = 12

. The upper half is

24, 28, 32

, so

\text{Q3} = 28

\text{IQR P} = 28 - 12 = 16

Q8·Moderate

Data set Q (sorted):

5, 11, 17, 20, 23, 29, 37

. Calculate the IQR of data set Q.

Explanation

The median of the 7 values is the 4th value:

20

. The lower half is

5, 11, 17

, so

\text{Q1} = 11

. The upper half is

23, 29, 37

, so

\text{Q3} = 29

\text{IQR Q} = 29 - 11 = 18

Q9·Challenging

Factory A has a median daily output of

145

units with an IQR of

18

. Factory B has a median daily output of

138

units with an IQR of

12

. Which statement correctly identifies the more consistent factory?

Factory A is more consistent because it has a higher median.Factory A is more consistent because it has a larger IQR.Factory B is more consistent because it has a smaller IQR.Factory B is more consistent because it has a lower median.

Explanation

Consistency is measured by spread, not by the size of the typical value. Factory A has IQR

= 18

and Factory B has IQR

= 12

. Since

12 < 18

, Factory B has less variability in its output and is therefore more consistent, even though its median is lower.

Q10·Challenging

Data set X (sorted):

28, 32, 36, 40, 44, 48, 52

. Data set Y (sorted):

20, 30, 36, 40, 44, 50, 62

. Calculate the IQR of data set X.

Explanation

The median is the 4th value:

40

. The lower half is

28, 32, 36

, so

\text{Q1} = 32

. The upper half is

44, 48, 52

, so

\text{Q3} = 48

\text{IQR X} = 48 - 32 = 16

Q11·Challenging

Data set X (sorted):

28, 32, 36, 40, 44, 48, 52

. Data set Y (sorted):

20, 30, 36, 40, 44, 50, 62

. Calculate the IQR of data set Y.

Explanation

The median is the 4th value:

40

. The lower half is

20, 30, 36

, so

\text{Q1} = 30

. The upper half is

44, 50, 62

, so

\text{Q3} = 50

\text{IQR Y} = 50 - 30 = 20

Q12·Challenging

Data set X has a median of

40

and an IQR of

16

. Data set Y has a median of

40

and an IQR of

20

. Which statement best compares the two data sets?

Data set X is more consistent because it has a smaller IQR, while both data sets have the same median.Data set Y is more consistent because it has a larger IQR, while both data sets have the same median.Data set X has a higher centre, so it is more consistent.The data sets cannot be compared because they have the same median.

Explanation

Both data sets have the same median (

40

), so there is no difference in the typical value. Data set X has IQR

= 16

and data set Y has IQR

= 20

. Since

16 < 20

, data set X has less spread. Its values are more tightly clustered around the median, so data set X is more consistent.