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Middle Years

11.02 Centre and spread

Worksheet
Measures of centre and range
1

Find the range of the following set of scores:

a

20, \, 19, \, 3, \, 19, \, 18, \, 3, \, 16, \, 3

b

8, \, - 5, \, - 8, \, 4, \, 2, \, 8, \, -9, \, 11

2

Consider the following set of scores: 9, \, 4, \, 14, \, 19, \, 20, \, 15, \, 12

a

Sort the scores in ascending order.

b

Find the total number of scores.

c

Find the median.

d

Find the range.

3

Find the mode of the following set of scores:

a

1, \, 1, \, 1, \, 5, \, 5, \, 9, \, 9, \, 9, \, 9, \, 10, \, 10, \, 10, \, 14, \, 14, \, 14, \, 20

b

3, \, 3, \, 6, \, 8, \, 8, \, 8, \, 8, \, 10, \, 14, \, 14, \, 14, \, 14, \, 18

4

Find the median of the following set of scores:

a

2, \, 5, \, 5, \, 7, \ 9

b

1, \, 9, \, 2, \, 4, \, 6, \, 7, \, 9

c

7, \, 8, \, 3, \, 2

d

3, \, 8, \, 13, \, 17, \, 19, \, 24, \, 26, \, 27

5

Find the mean of the following scores: \,2,\, 8, \, 17, \, 27, \, 29.

6

The range of a set of scores is 5, and the highest score is 18. Determine the lowest score in the set.

7

A group of students had a range in marks of 11 and the lowest score was 5. Determine the highest score in the group.

Compare data sets
8

Consider the set of data: 1, \, 2, \, 2, \, 4, \, 4, \, 5, \, 6, \, 6, \, 8, \, 8, \, 8, \, 9, \, 9If one score of 8 is changed to a 9, state the measure(s) of centre that would be altered.

9

Consider this set of data that represents the number of apps on six people’s phones:11, \, 12, \, 15, \, 17, \, 19, \,19If each person downloads another 7 apps, state the measure(s) of centre that would be altered.

10

The following five numbers have a mean of 11:

11, 13, 9, 13, 9

If a new number is added that is smaller than 9, describe the effect on the mean.

11

Consider the following data sets:

  • Set A: \, 5, \, 2, \, 5, \, 6, \, 6, \, 3

  • Set B: \, 26, \, 12, \, 14, \, 7, \,16

a

Which set has the lowest mean?

b

Which set has the lowest median?

12

Which data set has the lowest mode?

  • Set A: \, 87, \, 2, \, 20, \, 20, \, 8, \, 10

  • Set B: \, 11, \, 8, \, 8, \, 48, \, 2, \, 17

13

Which data set has the highest median?

  • Set A: \, 2, \, 8, \, 11, \, 17

  • Set B: \, 8, \, 20, \, 20, \, 48, \, 87

14

Determine whether the following statements are true or false:

a

Two sets of data have the same highest and lowest values. This means they must have the same mode.

b

Two sets of data that have the same highest and lowest values must have the same range.

c

If two sets of data have the same median then the data sets must themselves be the same.

d

If two sets of data have very different modes then the highest values cannot be the same.

Data displayed in frequency tables
15

State the mode of this data set:

ScoreFrequency
32
44
58
63
75
83
16

State the modal class of this data set:

ClassFrequency
30-393
40-493
50-594
60-692
70-795
80-898
17

For the following data set:

a

Find the median.

b

Find the mode.

ScoreFrequencyCumulative frequency
388
4210
5313
6518
7321
8425
18

For the following grouped data:

a

Find the median class.

b

Find the modal class.

ClassFrequency
6-108
11-152
16-205
21-253
26-304
31-353
19

Consider the data provided in the table:

a

Calculate the range.

b

State the mode.

c

Determine the median.

ScoreFrequency
6816
6941
7030
7131
7249
7329
20

Determine the mean for the following data set:

\text{Score } (x)\text{Frequency } (f)xf
4832
5630
6318
7856
8216
9872
21

Consider the frequency table:

a

Complete the table using the data set below:

2, \, 7, \, 3, \, 6, \, 3, \, 2, \, 2, \, 2, \, 5, \, 6, \, 3, \, 2, \, 2, \\ 3, \, 5, \, 7, \, 4, \, 2, \, 4, \, 3, \, 2, \, 4, \, 4, \, 6, \, 7
b

Hence find the mean, correct to two decimal places.

c

Find the median score.

\text{Score } (x)\text{Frequency } (f)xf
2
3
4
5
6
7
22

Consider the frequency table:

\text{Class}\text{Class centre } (x)\text{Frequency } (f)xf
11-1513
16-2018
21-2523
26-3028
31-3533
36-4038
a

Complete the table using the data set below:

14, \, 36, \, 17, \, 25, \, 15, \, 36, \,19, \, 29, \, 38, \, 23, \, 34, \, 18, \, 34, \\ 31, \, 36, \, 25, \, 32, \, 34, \, 39, \, 26, \, 29, \, 21, \, 37, \, 39, \, 38
b

Hence estimate the mean.

23

Consider the frequency table:

\text{Class}\text{Class centre } (x)\text{Frequency } (f)xf
21-25237
26-30288
31-35339
36-40386
41-45433
46-50482
a

Complete the frequency table.

b

Hence estimate the mean, correct to one decimal place.

24

Calculate the mean for the following data set correct to one decimal place.

\text{Class}\text{Class centre } (x)\text{Frequency } (f)xf
11-1513452
16-2018354
21-2523492
26-30286168
31-35338264
36-40388304
25

Consider the following table:

a

Estimate the mean, correct to one decimal place.

b

State the modal class.

c

Find the median class.

ScoreFrequency
1-41
5-85
9-1210
13-165
17-203
Data displayed graphically
26

Consider the following bar chart:

a

Find the range.

b

State the mode.

c

Determine the mean, correct to two decimal places.

27

Consider the following dot plot:

a

Find the total number of scores.

b

Find the median score.

c

Find the mode.

d

Find the range.

28

State the mode of the data set from the following graphical representations:

a
b
29

Find the median of the data set from the following graphical representations:

a
b
30

Find the mean for the data set from the following graphical representations, rounding your answers to one decimal place:

a
b
31

For each of the given data sets, find the following to two decimal places if necessary:

i

Mean

ii

Median

iii

Mode

iv

Range

a
Leaf
62 7
71 2 2 4 7 9
80 1 2 5 7
90 1

Key: 6\vert 2=62

\text{ }

b
Leaf
24
30 5 5 5
40 2
50 2 9 9
63 3
70 1
80 1
90 0 5

Key: 2\vert 4=24

c
d
Applications
32

A diver measures how long she can hold her breath underwater over several dives. If the median time is 2.1 minutes, this means that:

A

Most of the time she held her breath for less than 2.1 minutes.

B

The longest she held her breath is 4.2 minutes.

C

The shortest time she held her breath is 1.05 minutes.

D

Most of the time she held her breath for longer than 2.1 minutes.

E

Half the dives she was able to hold her breath longer than 2.1 minutes.

33

A real estate agent wanted to determine a typical house price in a certain area. He gathered the selling price of some houses (in dollars):327\,000, \, 376\,000, \, 424\,000, \, 439\,000, \, 444\,000, \, 469\,000, \, 472\,000, \, 475\,000, \, 485\,000, \, 496\,000

a

Calculate the mean house price.

b

What percentage of the house prices exceeded the mean?

c

Determine the median house price.

d

What percentage of house prices exceeded the median?

34

Each student in the class was asked to write down the number of siblings they had. The teacher recorded the results in the given dot plot:

a

How many students are there in the class?

b

If none of the students share the same siblings, then how many siblings are there in total?

c

Find the mean number of siblings for a student in this class, correct to one decimal place.

d

Find the mean number of children in a family for a student in this class, correct to one decimal place.

35

The given dot plot shows the number of goals scored across each of Rosey's soccer games:

a

How many games were played in total?

b

How many goals were scored in total?

c

Find the mean number of goals per game, correct to one decimal place.

36

In a study, a group of people were shown 30 names, and after one minute they were asked to recite as many names by memory as possible. The results are presented in the dot plot:

a

How many people took part in the study?

b

State the largest number of names someone remembered.

c

State the smallest number of names someone remembered.

d

Find the range of the data.

e

Find the median score.

37

A cyclist measured his heart rate immediately after finishing each event in which he competed. The results are recorded in the given stem-and-leaf plot:

a

How many events did the cyclist compete in?

b

Find his mean post event heart rate.

Leaf
162
173 8
184 5 6 9
195 5

Key: 12|3=123

38

The scores for a recent history test are shown in the stem-and-leaf plot. The maximum possible score on the test was 100.

a

How many students took the test?

b

Find the mean test score for the class.

Leaf
62 3
72 4 9
83 4 9 9
91 1 5

Key: 8|3=83

39

The size of each earthquake that occurred in a region over a three year period, measured from 0 to 9.9, is recorded in a stem-and-leaf plot:

a

How many earthquakes in total were recorded?

b

Find the mean number of earthquakes per year in the region.

c

It was found that the combined total of all earthquake sizes was 87. Find the mean size of an earthquake that occurred during the period, correct to three decimal places.

Leaf
10 0 2 3 5 6 6 7 9
23 8
33 5 7
41 2 2 3
58 9
65
73 6
87

Key: 5|2=5.2

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