1 Use the percentage ogive shown below to find:
a the 1st quartile b the 3rd quartile
c the 1st decile d the 45th percentile
e the 86th percentile f the 21st percentile.
2 Use the ogive shown below to find:
a the 5th decile b the 3rd quartile
c the 9th decile d the 2nd pentile
e the 28th percentile f the 85th percentile.
3 Find the mode and median of each of the following sets of data.
a 4, 5, 9, 4, 8, 6, 7
b 11, 15, 14, 19, 15, 16, 14
c 2, 0, 5, 6, 8, 4, 9
d 5, 9, 8, 4, 7, 9, 4, 10
80
60
40
20
0
Cumulati
v
e percentage
100
x
30 40 50 60 70 80 90 100 110 120
10 20
x
30 40 50 60
100
75
50
25
0
Cumulati
v
e frequenc
y
70 125
25 35 45 55 65 75
New QMaths 11B CD-ROM
Additional exercise 2.3 Finding data positions and centres continued
4 The shoe sizes of a group of Year 11 students were recorded:
8 9 5 6 8 7 6 8 12 9 8 7 6 10 8 6 7 5 8
Find the mode and the median shoe sizes.
5 Find the modal class and median of the discrete data in the table.
6 Find the mean of each of the following sets of numbers.
a 1, 2, 4, 2, 5, 3, 2, 5
b 5, 6, 9, 8, 5, 7, 6, 9, 8, 5, 7
c 10, 12, 14, 15, 18, 17, 11, 13
7 Find the mean, median and mode (if possible) of each set of measurements using a graphics calculator.
a 25 cm, 32 cm, 46 cm, 18 cm, 35 cm
b 34°C, 32°C, 28°C, 33°C, 21°C, 291°C
c 115 kg, 120 kg, 140 kg, 122 kg, 138 kg, 110 kg, 142 kg, 108 kg
8 Use a graphics calculator to find the mean and median of the following data.
9 The weekly wages (in $) of a group of university graduates in their first year of employment were recorded as follows:
825 885 913 815 988 1000 1125 1093 1063 1015 875
968 863 983 1023 875 825 1020 855 843 893 938
908 928 958 975 1065 1175 985 1043 1000 1113 1055
898 1153 835 820 878 820 963 905 893 938 900
938 950 953 920 978 833 863 990 1130 973 1140
a Construct a cumulative frequency distribution table for this data.
b Use this table to draw a percentage ogive for the data.
c From the ogive, calculate the median.
d A graduate is referred to as in the ‘top 10%’ for wages. What is her minimum wage?
e Find the other deciles.
x 3 4 5 6 7 8 9
f 3 4 5 7 9 2 1
Class f
0–4 7
5–9 8
10–14 12
15–19 11
20–24 5
25–29 1
10 The ogive shows times taken by 200 senior high school students to complete 10 push-ups.
a Calculate the cumulative frequency for the 83rd percentile.
b Find the 83rd percentile. c Find P15.
d Find the 3rd, 4th and 9th deciles. e Find the quartiles.
f What is the median?
11 A particularly dangerous stretch of road was studied. The ages of drivers involved in serious accidents were collected as follows:
20 18 21 38 19 20 68 22 51 43 25 28 17 23 28 57 20 18 49 69
60 23 18 21 67 72 18 24 32 66 21 19 18 37 69 22 23 54 73 19
20 59 68 18 17 45 73 67 17 25 21 32 41 49 64 71 22 24 21 67
a Gather this data together in a cumulative frequency distribution table.
b Construct a percentage ogive from the table you have completed.
c Estimate the quartiles for this distribution.
d Find the median age.
e Why are there no drivers under 15?
f Find D4, P38 and P56.
12 The effective life of a particular make and model of car spring was tested by the manufacturer, with the results shown in the table.
Time taken (s) Times for push-ups
20 40 60 80
80
60
40
20
0
Cumulati
v
e frequenc
y
100 100
120 140 160 180 200
Life of spring (hours) f
0–299 2
New QMaths 11B CD-ROM
Additional exercise 2.3 Finding data positions and centres continued
13 The results of an opinion poll of people’s voting preferences were as follows:
Liberal 38% Labor 45%
Democrats 2% Nationals 3%
Greens 9% Others 3%
What is the modal political party?
14 A number 16 chicken is supposed to weigh 1.6 kg. The weights (kilograms) of some number 16 chickens were recorded by a quality controller as follows:
1.6 1.5 1.4 1.9 1.7 1.6 1.6 1.4 1.3 1.5 1.5 1.7
1.5 1.6 1.8 1.7 1.4 1.2 1.4 1.6 1.5 1.6 1.5 1.4
a Put this data into a frequency distribution table.
b Find the mode of the data.
c Find the median chicken size.
15 A new housing estate that was built close to a school was surveyed by the Department of Education to find out how many children were in each family. The results were as follows:
0 2 5 0 3 4 0 1 0 3 0 2 2 3 0 3 4 1
0 2 2 0 3 1 0 2 0 2 0 1 4 3 0 1 0
a Is the variable discrete or continuous?
b Draw a histogram using this data.
c What is the modal number of children?
d What is the median?
e Find the mean.
f Would the mean, mode or the median be the most useful for the planners in the Department of Education? Why/why not?
16 The scores after the first round of a golf tournament are shown below, with the number of players on each score shown in brackets:
69 (1), 70 (3), 71 (2), 72 (8), 73 (15), 74 (9), 75 (2), 76 (3), 77 (1), 78 (1), 80 (2), 81 (1) a Is the data discrete or continuous?
b Find the mean, median and mode of the scores.
17 The heights of 100 seedlings were measured to the nearest centimetre. The results are shown in the table.
Find the median height.
Height (cm) f
9.5–10.5 5
10.5–11.5 8
11.5–12.5 17
12.5–13.5 22
13.5–14.5 31
14.5–15.5 11
15.5–16.5 6
1 a 53.5 b 80 c 41 d 61.5 e 90 f 50.5
2 a 48.3 b 54 c 61 d 46.7 e 44.2 f 58
3 a Mode = 4, median = 6 b Mode = 14 and 15, median = 15
c No mode, median = 5 d Mode = 4 and 9, median = 7.5
e Mode = 210, median = 210 f Mode = 25.4, median = 24.1
g Mode = 109, median = 106
4 Mode = size 8, median = size 8
5 Modal class 10–14, median = 13 (discrete)
6 a 3 b 6.82 c 13.75
7 a Mean = 31.2 cm, median = 32 cm, no mode (some calculators give 46)
b Mean = 73.2°C, median = 32.5°C, no mode (some calculators give 291)
With 291°C discarded as non-compliant: mean = 29.6°C, median = 32°C, no mode (some calculators give 21, 28, 32, 34 as modes).
c Mean = 124.4 kg, median = 121 kg, no mode (some calculators give 142)
8 Mean = 5.8, median = 6
9 a b
c $945 d $1110
e D1= $830, D2= $860, D3= $890, D4= $920, D5= $945, D6= $970, D7= $995, D8= $1040, D9= $1110
10 a 166 students b 74 s c 26 s d D3= 40 s, D4= 48 s, D9= 80 s
e Q1= 36 s, Q2= 54 s, Q3= 69 s f 54 s
11 a b
Weekly wage ($) f c.f. % c.f. 800–839 840–879 880–919 920–959 960–999 1000–1039 1040–1079 1080–1119 1120–1159 1160–1199 7 7 8 8 9 5 4 2 4 1 7 14 22 30 39 44 48 50 54 55 12.7 25.5 40 54.5 70.9 80 87.3 90.9 98.2 100
Age (years) f c.f. % c.f. 10–20
20–30 30–40 40–50
12 21 4 5 12 33 37 42 20 55 61.7 70 Percentage frequency 100 80 60 40 20 0
800 840 880
Weekly salary ($)
920 96010001040 1080 112011601200 90
70
50
30
10
New graduate salaries
v
e frequenc
y100 80
60
New QMaths 11B CD-ROM
Additional exercise 2.3 Answers continued
12 a
b P5= 640, P17= 940, P83= 1630 hours
c D1= 800, D2= 975, D3= 1080, D4= 1190, D5= 1280, D6= 1380, D7= 1480, D8= 1590, D9= 1720 hours
d Q1= 1030, Q2= 1280, Q3= 1530 hours e P10
13 Labor
14 a b 1.5 and 1.6 kg c 1.5 kg
15 a Discrete b
c 0 children d 1 child e About 1.5 children
f The median would be more useful than the mode, as it is a better indicator of the number of children likely to attend the school. The mean might be better still.
16 a Discrete b Mean ≈ 73.6, mode = 73, median = 73
17 13.4 cm (continuous) Weight (kg) f
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
1 1 5 6 6 3 1 1
Cumulati
v
e frequenc
y
200 160 120 80 40 0
300 600 900
Life (hours) 1200 1500 1800 240
2100 Life of car springs
Frequenc
y 10
8 6 4 2 0
0 1
Number of children
2 3 4 5
12 14
Children in families
