While calculating the mean, median, mode and range can tell us a lot about a data set, these calculations can also be very powerful in comparing and contrasting two different data sets.

We also can benefit from examining the shape of the distribution of two sets of data when comparing them.

Example:

The number of minutes spent exercising per day for $10$10 days is recorded for two people who have just signed up for a new gym membership.

Person A: $45$45 $50$50 $50$50 $55$55 $55$55 $60$60 $60$60 $65$65 $65$65 $65$65

Person B: $20$20 $30$30 $45$45 $55$55 $60$60 $60$60 $65$65 $70$70 $70$70 $70$70

(a) Calculate the mean, median, mode and range for Person A

mode = $65$65

median = $57.5$57.5

mean = $57$57

range = $20$20

(b) Calculate the mean, median, mode and range for Person B

mode = $70$70

median = $60$60

mean = $54.5$54.5

range = $50$50

Person A

(d) Which statistical measure supports your answer to part (c)?

The range. The smaller range for Person A indicates that the number of minutes they exercise each day is more consistent than that of Person B.

The range for Person B is more than double that of Person A, indicating more inconsistency in their exercise routine.

(e) Which person seems to train more overall?

Person B

(f) Which statistical measure(s) supports your answer to (e)?

The mode and median for Person B are both larger than for Person A.

While the mean for Person B is slightly lower than Person A, this is due to the negative skew of their data.

Overall, the larger mode and median for Person B indicates that they exercise for longer overall.

Worked Examples

Question 1

The beaks of two groups of bird are measured, in mm, to determine whether they might be of the same species.

**Length of beaks of two groups of birds (in mm.)**
Group 1	$33$33	$39$39	$31$31	$27$27	$22$22	$37$37	$30$30	$24$24	$24$24	$28$28
Group 2	$29$29	$44$44	$45$45	$34$34	$31$31	$44$44	$44$44	$33$33	$37$37	$34$34

Calculate the range for Group 1.
Calculate the range for Group 2.
Calculate the mean for Group 1. Give your answer as a decimal.
Calculate the mean for Group 2. Give your answer as a decimal.
Choose the most appropriate statement that describes the set of data.
Although the ranges are similar, the mean values are significantly different indicating that these two groups of birds are of the same species.
A
Although the ranges are similar, the mean values are significantly different indicating that these two groups of birds are not of the same species.
B
Although the mean values are similar, the ranges are significantly different indicating that these two groups of birds are not of the same species.
C
Although the mean values are similar, the ranges are significantly different indicating that these two groups of birds are of the same species.
D

question 2

The box plots drawn below show the number of repetitions of a $70$70 kg bar that two weightlifters can lift. They both record their repetitions over $30$30 days.

Which weightlifter has the more consistent results?
Weightlifter A.
A
Weightlifter B.
B
What statistical evidence supports your answer?
The mean.
A
The range.
B
The mode.
C
The graph is positively skewed.
D
Which statistic is the same for each weightlifter?
The median.
A
The mean.
B
The mode.
C
Which weightlifter can do the most repetitions of $70$70 kg?
Weightlifter A.
A
Weightlifter B.
B

Question 3

$Derek$ planted some tomato seeds and two seeds sprouted. He puts Seedling A outside in the sun and Seedling B inside in the kitchen. The heights of the seedlings over time, in cm, are graphed below:

Loading Graph...

How many centimetres did Seedling A grow in the $second$ week?

Complete the table below showing the increases in growth for both seedlings:

	Week 1	Week 2	Week 3	Week 4	Week 5	Week 6	Week 7
Seedling A	$3$3 cm	$2$2 cm	$\editable{}$ cm	$\editable{}$ cm	$2$2 cm	$\editable{}$ cm	$\editable{}$ cm
Seedling B	$\editable{}$ cm	$\editable{}$ cm	$\editable{}$ cm	$\editable{}$ cm	$\editable{}$ cm	$\editable{}$ cm	$\editable{}$ cm

Which seedling appears to be growing more slowly?
Seedling A.
A
Seedling B.
B
Which conditions seem to give optimum growth for a tomato seedling?
Growing inside in the kitchen.
A
Growing outside in the sun.
B

Outcomes

12D.D.1.5

Interpret statistical summaries to describe the characteristics of a one-variable data set and to compare two related one-variable data sets; describe how statistical summaries can be used to misrepresent one-variable data; and make inferences, and make and justify conclusions, from statistical summaries of one-variable data orally and in writing, using convincing arguments

Comparing data