5B Lesson Worksheet 11.3
(Refer to Book 5B P.11.20)
Objective: To draw and interpret box-and-whisker diagram, and use it to compare the distributions of
different sets of data.
Box-and-whisker Diagrams
25% 25%
A box-and-whisker diagram (or box plot) is a statistical diagram
25%
25%
used for showing the smallest datum, the 1st quartile, the median,
the 3rd quartile and the largest datum of a set of data.
Instant Example 1
Consider the data 5, 9, 11, 12, 15, 20, 23, 27, 30.
smallest
median
largest
datum 1st quartile
3rd quartile datum
Instant Practice 1
The following are the lengths (in cm) of 6 pencils.
(a) Find the 1st quartile, the median and the 3rd
9, 11, 12, 13, 13, 13.5
(a) Find the 1st quartile, the median and the 3rd
quartile of the set of data.
(b) Draw a box-and-whisker diagram to represent
(b) Draw a box-and-whisker diagram to represent
the data.
(a) 1st quartile =
the data.
9 + 11
= 10
2
(a) 1st quartile = (
) cm
(
)+(
)
Median =
cm =
2
3rd quartile =
Median = 15
3rd quartile =
23 + 27
= 25
2
(b)
(b)
5
1.
quartile of the lengths of the pencils.
10
15
20
25
30
9
2.
Consider the data 2, 7, 9, 9, 13, 15, 17, 18.
(a) Find the 1st quartile, the median and the
10
11
12
Length (cm)
13
14
The following are the hourly wages (in $) of
10 workers.
3rd quartile of the set of data.
36, 37, 38, 40, 44, 46, 48, 48, 51, 53
(b) Draw a box-and-whisker diagram to
(a) Find the 1st quartile, the median and the 3rd
represent the data.
(a) 1st quartile =
(
quartile of the set of data.
)+(
2
)
(b) Draw a box-and-whisker diagram to represent
=
the data.
Median =
3rd quartile =
(b)
0
5
10
15
20
P.1
Ex 11B: 2−4
Instant Example 2
Instant Practice 2
In the box-and-whisker diagram, find the median, In the box-and-whisker diagram, find the median,
the range and the inter-quartile range.
12
14
16
18
20
the range and the inter-quartile range.
22
Median = 16
Inter-quartile range
Largest datum = 24
3
24
4
5
Median = (
6
7
Capacity (L)
∴ Range = 24 − 12 = 12
Range
Q1 = 14
Q3 = 20
∴ Inter-quartile range = 20 − 14 = 6
)
Smallest datum = (
∴
)
Q3 = (
)
)
)−(
Range = [(
Q1 = (
∴
9
)L
Largest datum = (
Smallest datum = 12
8
Inter-quartile range = [(
)] (
)=
) − ( )] ( ) =
In each of the following box-and-whisker diagrams, find the median, the range and the inter-quartile range.
Ex 11B: 5−8
[Nos. 3–4]
3.
4.
40
50
60
70
80
90
10
15
20
Weight (kg)
25
30
Median =
Largest datum = (
Smallest datum = (
)
)
Level Up Question
5.
The box-and-whisker diagrams show the time taken (in min) by Group A
two groups of students to finish a typing test. The inter-quartile
range of the time taken by group B students is 14 min.
(a) Find the value of a.
(b) By considering the inter-quartile ranges, which group has a
smaller dispersion in the time taken? Explain your answer.
P.2
Group B
23 26 28 30
37
a 45 48
Time (min)
5B Lesson Worksheet 11.4
(Refer to Book 5B P.11.30)
Objective: To find the standard deviation of ungrouped and grouped data.
[In this worksheet, give the answers correct to 3 significant figures if necessary.]
Standard Deviation
(a) For a set of ungrouped data x1, x2, …, xn with mean x ,
( x1 − x ) 2 + ( x2 − x ) 2 +    + ( xn − x ) 2
n
(b) For a set of grouped data with class marks x1, x2, …, xn, and corresponding frequencies f1, f2, …, fn,
standard deviation () =
if the mean is x ,
f1 ( x1 − x ) 2 + f 2 ( x2 − x ) 2 +    + f n ( xn − x ) 2
f1 + f 2 +    + f n
standard deviation () =
(Note: The square of standard deviation is called variance.)
Instant Example 1
Instant Practice 1
Find the mean and the standard deviation of the data Find the mean and the standard deviation of the data
1, 3, 5, 9, 12.
1 m, 7 m, 11 m, 13 m, 17 m, 23 m.
1 + 3 + 5 + 9 + 12
5
=6
Mean =
Mean =
=
)+(
)+(
)+(
( )
)+(
)+(
)
m
=
Standard deviation
Standard deviation
(1 − 6) + (3 − 6) + (5 − 6) + (9 − 6) + (12 − 6)
5
2
(
2
2
2
2
[( ) − ( )]2 + [( ) − ( )]2 + [( ) − ( )]2 +
=
=4
[( ) − ( )]2 + [( ) − ( )]2 + [( ) − ( )]2
( )
m
=
Find the mean and the standard deviation of each of the following sets of data. [Nos. 1–2]
1.
2.
7, 11, 15, 19
Mean =
(
)+(
)+(
( )
)+(
)
Standard deviation
=
[(
)−(
)]2 + [(
)−(
[(
)−(
)]2 + [( ) − (
( )
)]2 +
)]2
P.2
4 s, 10 s, 11 s, 12 s, 17 s, 18 s
Ex 11C: 1−5
Instant Example 2
Instant Practice 2
The table shows the hourly charges (in $) of some The table shows the consumption of sugar (in g) by
car parks.
some students every day.
Hourly charge ($) 21−30 31−40 41−50 51−60
Consumption (g) 11−15 16−20 21−25 26−30
Class mark ($)
25.5
35.5
45.5
55.5
Class mark (g)
Frequency
9
12
3
6
Frequency
4
16
12
8
Find the mean and the standard deviation of the (a) Complete the above table.
(b) Find the mean and the standard deviation of the
hourly charges of the car parks.
Mean
9  25.5 + 12  35.5 + 3  45.5 + 6  55.5
=$
9 + 12 + 3 + 6
= $37.5
consumption of sugar by the students every day.
(
(
(b) Mean =
(
Standard deviation
)+(
)+(
)+(
)(
)(
)+(
)+
)
g
)
=
9(25.5 − 37.5) 2 + 12(35.5 − 37.5) 2 +
3(45.5 − 37.5) + 6(55.5 − 37.5)
9 + 12 + 3 + 6
= $10.8, cor. to 3 sig. fig.
2
Standard deviation
2
=$
=
(
)(
)2 + (
(
)(
(
)2 + (
)+( )+(
=
3.
)(
)(
)+(
)(
)2 +
)(
)+(
)2
)
g
, cor. to 3 sig. fig.
The table shows the daily data usage (in MB) of Daily data
101−150 151−200 201−250 251−300
usage (MB)
45 smartphone users.
Class mark
(a) Complete the table.
(MB)
(b) Find the mean and the standard deviation of the Frequency
13
12
5
daily data usage of the smartphone users.
Ex 11C: 7, 8
(b) Mean
=
MB
Level Up Question
4.
The following shows the time taken (in s) by Jimmy to swim 100 m in 7 trials.
62, 60, 58, 57, 66, 64, 60
(a) Find the mean and the standard deviation of the time taken by Jimmy.
(b) If the standard deviation of the time taken by Sandy to swim 100 m in 7 trials is 2.86 s, who has a
larger dispersion in the time taken? Explain your answer.
P.2