# My Additional Mathematics Modules - Form 4 - Statistics (Version 2012) by nklye

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```									 Additional
Mathematics
Form 4
Topic: 7

DECISIVE
(Version 2012)

by

NgKL
(M.Ed.,B.Sc.Hons.,Dip.Ed.,Dip.Edu.Mgt.&Lship,Cert.NPQH.)
edmet-nklpunya.blogspot.com                                                                                        2
7.1     MEASURES OF CENTRAL TENDENCY
IMPORTANT POINTS:

Ungrouped Data
Ungrouped Data                                                                Grouped Data
(in a Frequency Table)

Data sets which are not grouped into       Data sets which are not grouped     Data sets which are grouped into
classes.                                   into classes but are presented in   classes and presented in Frequency
Frequency Table.                    Table.
Example:                                   Example:                            Example:
Number of          Number of         Number of            Number of
The masses of six pupils in kilogram:
Books Read          Students         Books Read            Students
50, 52, 59, 60, 53, 59.
0                  5                0–1                   11
1                  6                2-3                   12
2                  8                4-5                   15
3                  4                6-7                    8
4                  2                8-9                    7
Modal Class = The class with
Mode = The value which is repeated         Mode = The value of data which      highest frequency.
the most number of times in a set of       has the highest frequency.          Mode is obtained from the highest
data.                                                                          bar of a histogram with the
procedure as shown below.
Example:
frequency         Modal class
Mode of the masses of six pupils in
kilogram:; 50, 52, 59, 60, 53, 59.

Mode = 59

mode

Mean, x =
_
x                                 _
Mean, x =
 fx                   Mean, x =
_
 fx
N                                      f                                   f
 x = sum of all the values of             x = value of data                   x = class mid-point
data.                               f = frequency                       f = frequency
N = number of values of data.
Median, m = the value in the middle        Median, m = the value in the
position of a set of data after the data   middle position of a set of data                          N      
are arranged in ascending order.           after the data are arranged in                             F
       
Median, m = Lm +  2            c
fm 
ascending order.
Example:                                                                                                    
       
The median of the masses of six                                                                             
pupils in kilogram:                                                            Lm = lower boundary of the
median class.
50, 52, 53, 59, 59, 60.                                                 N = sum of frequency.
F = cumulative frequency of the
53  59                                                             class before the median class.
Median =                56                                                  fm = frequency of the median class
2
c = size of the median class.
edmet-nklpunya.blogspot.com                                                                                                     3

EFFECTS OF UNIFORM CHANGES IN A SET OF DATA ON THE MODE, MEAN AND MEDIAN:

1. When a constant number k is added or subtracted to each data in a set, then
* the new mode = original mode  k
* the new mean = original mean  k
* the new median = original median  k

2. When a constant number k is multiplied to each data in a set, then
* the new mode = k x original mode.
* the new mean = k x original mean.
* the new median = k x original median.

Exercise 7.1

1. Find the mode, mean and median of the following sets of ungrouped data.

(a) 9, 5, 3, 3, 7, 13, 9                                                  (b) 2, 8, 11, 9, 6, 5, 12, 11

(c) 3, 4, 11, 3, 10, 11, 2, 3, 7                                          (d) −3, −2, 1, 4, 5, 9

2.     Find the mode, mean and median of the following sets of ungrouped data.

Total
Pocket money (RM), x             25        30    35      40   45     50                         mode =
∑
(a)
Number of Students, f             2         4    4       6    5       5
mean, x =
Cumulative frequency, F

fx                                                                                   median =

(b)    Marks, x        No. Pupils, f         fx             F
(c)    No. Goals, x       No. Players, f        fx     F
0                  3
3                 12
1                  8
4                 10
2                  6
5                  9
3                  4
4                  3                                                    6                  7

5                  1                                                    7                  5
∑f               ∑fx=

mode=                mean, x =             median=                   mode=               mean, x =          median=
edmet-nklpunya.blogspot.com                                                                                         4
No. of
Score, x     No. pupils, f           fx             F            Marks, x                     fx          F
(d)                                                              (e)               Sudents, f
8              4                                                13           6
9              8                                                14           8
12              11                                               15          12
15              10                                               16          10
20              5                                                17           5
21              2                                                18           3

mode=            mean, x =                median=                  mode=         mean, x =         median=

3. Determine (i) the modal class of each of the following grouped data.

Height / cm,           No.pupils,
(a)                                             LB        UB
x                     f
(i) Modal class =
141 – 145                 7
(ii) On a graph paper and by using a scale of
146 – 150                 9                                 2 cm to 5 unit on x-axis and 2 cm to 2 units
on frequency, f axis, draw a histogram of the
151 – 155                16                                 grouped data. Hence, from the graph, estimate
the mode of the data.
156 – 160                 6

161 – 165                 2
edmet-nklpunya.blogspot.com                                                                                             5

(a)     Marks, x        No.pupils, f      LB           UB
(i) Modal class =
20 – 29                2
(ii) On a graph paper and by using a scale of
30 – 39                5                                       2 cm to 5 unit on x-axis and 2 cm to 2 units
on frequency, f axis, draw a histogram of the
40 – 49                7                                       grouped data. Hence, from the graph, estimate
the mode of the data.
50 – 59               10

60 – 69                6

4. Find the mean of each grouped data of the following.

Height / cm,      No. pupils, f   Mid-point, x                  fx
(a)
141 – 145              7
146 – 150              9
151 – 155              16
156 – 160              6
161 – 165              2
∑f =                                ∑fx=
edmet-nklpunya.blogspot.com                                                                               6

Number of
Marks                          x                 fx
(b)                  pupils, f
20 -29              2
30 – 39               4
40 – 49               5
50 – 59               10
60 − 69               6
70 − 79               3
∑f =                         ∑fx=

No. of
Mass / kg
(c)                     pupils, f
30 – 39               8
40 – 49              10
50 – 59               7
60 – 69              15
70 − 79              10

(d) The table below shows the duration of telephone calls received in an office on a certain day for 40
calls. Determine the mean of the duration of calls.

Duration of        No. of
Calls / minutes     Calls, f
1–3                    2
4–6                    4
7 – 10                  5
11 – 13                 10
14 – 17                  6

5. For each of the following sets of data, without drawing an ogive, calculate the median of the set
of data.

Number of
(a)   Height / cm,
pupils, f
141 – 145                  7
146 – 150                  9
151 – 155                  16
156 – 160                  6
161 – 165                  2
edmet-nklpunya.blogspot.com                                                                                 7

No. of
(b)   Marks, x
pupils, f
20 – 29               2
30 – 39               4
40 – 49               5
50 – 59               10
60 – 69               6
70 − 799               3

Mass /            Number of
(c)
kg                pupils
30 – 39                  8
40 – 49                  10
50 – 59                  8
60 – 69                  14
70 − 79                  10

(d) The table below shows the duration of telephone calls received in an office on a certain day for 40 calls.
Without drawing an ogive, determine the median of the duration of calls.

Duration
Number
of Calls /
of Calls
min
2–3                  9

4–5                  12

6–7                  10

8–9                  7

10 – 11               2
edmet-nklpunya.blogspot.com                                                                                             8

7.2 OGIVE
    An ogive is a statistical graph which is drawn of cumulative frequency of a set of grouped data against its
frequency class of upper boundary.
    An ogive can be used to estimate the median, m, first quartile, Q1 and third quartile, Q3 of the grouped data.

Cumulative frequency, F

3N
N = Sum of frequency
4
N
Q1 = First quartile
2
m = Median
N
Q3 = Third quartile
4

Upper boundary
Q1m Q3

    To draw an ogive, a Cumulative Frequency & Upper Boundary table has to be built.
    A class with zero frequency and its upper boundary also need to be created.
Example:
Frequency,           Cumulative frequency,
Mass / kg                                                             Upper boundary
f                         F
20 – 29                  0                         0                        29.5
30 – 39                   8                           8                     39.5
40 – 49                   10                         18                     49.5
50 – 59                   8                          26                     59.5
60 – 69                   14                         40                     69.5
70 – 79                   10                         50                     79.5

A graph is then plotted with its cumulative frequency against upper boundary to give an ogive.
edmet-nklpunya.blogspot.com                                                                                 9

Exercise 7.2

1. The table below shows marks scored by 30 pupils in a test. Draw an ogive, hence determine the
median, m, first quartile, Q1, and third quartile, Q3 of the test.

Number of
Marks                          F               UB
pupils, f

20 – 29               2
30 – 39               4
40 – 49               5
50 – 59               10
60 – 69               6
70 – 79               3

Exercise 7.3 – Effect of Uniform Chances in a Set of Data on the Mode, Mean and Median

1. The mode, mean and median of a set of numbers are 6, 8.5 and 7.8 respectively. Determine the new mode,
mean and median if each of the numbers in the set is;
(i) added by 3 and then divided by 2.
(ii) subtracted by 5 and then multiplied by 4.
edmet-nklpunya.blogspot.com                                                                                 10

2. The mode, mean and median of a set of data are 32.5, 30 and 31.5 respectively. Find the new mode, mean
and median if each value in the data is;
(i) added by 3 and then multiplied by ½.,
(ii) subtracted by 1.2.

3. A set of data with 6 numbers has a mean of 21. When a new number is added to the set, the mean
becomes 20. Find the value of the number added.

7.3       MEASURE OF DISPERSION

Ungrouped Data
Ungrouped Data                                                            Grouped Data
(in a Frequency Table)
Range = midpoint of the higest
Range = largest value –              Range = largest value –
class – midpoint of
smallest value of data.              smallest value of data.
the lowest class.
Inter quartile range                 Inter quartile range                 Inter quartile range
= Q3 − Q1                            = Q3 − Q1                            = Q3 − Q1
Variance,                                                                 Variance,
x                         Variance,
2

 fx
_                                                                 2
_
2 =                − x2
 fx
2
_
 =
2
 x2
N                          =                  x                     f
2                       2

where;                                     f
x    2                                                                   where;
= sum of square of the     where;
f = frequency.
values of data.                f = frequency.                        x = class midpoint.
N = number of value of data        x = value of data.                    x = mean
x = mean                           x = mean

Standard deviation,                  Standard deviation,                  Standard deviation,

 =
x    2

x
_
2          =
 fx    2

x
_
2
 =
 fx 2

x
_
2

N                                 f                                   f
edmet-nklpunya.blogspot.com                                                                              11

Effects of uniform changes in a set of data on the range, inter quartile range, variance and standard
deviation.
1. When a constant number k is added or subtracted to each data in a set, then
* the new range, interquartile range, variance and standard deviation = original range range, interquartile
range, variance and standard deviation
respectively.
2. When a constant number k is multiplied to each data in a set, then
* the new range = k x original range.
* the new interquartile range = k x original interquartile range..
* the new variance = k2 x original variance.
* the new standard deviation = k x original standard deviation.

Exercise 7.3(a)

1. Find the range and inter quartile range of each set of the following data.

(a) 46, 35, 41, 40, 32, 38, 44, 40                          (b) 17, 4, 6, 10, 12, 12

2. Find the range and inter quarter range of each of the following data.

(c ) 22, 20, 25, 19, 24                                     (b) 3, 12, 8, 4, 10, 6, 7

3. Find the range and inter quartile range of each set of the following data.

No. of
(a)   Score                     F
Pupils, f
1           3
2           6
3           12
4           20
5           18
6           11
edmet-nklpunya.blogspot.com                                                                                            12

No. of      No. of
(b)    book        pupils
0              10
1              14
2              20
3              26
4              18
5              12

(c)   Mass /       No. of
kg           pupils

50              2
51              3
52              10
53              20
54              8
55              7

5. The table below shows the number of chicken sold over a period of 60 days.

No. of chickens,        No. of days,
x                    f

11 – 15               11
16 – 20               16
21 – 25               19
26 – 30                8
31 − 35                6

(a) Find the range of incomes of the workers.

(b) Calculate the first quartile, Q1,, the third quartile, Q3 and the inter quartile range.

(c) Draw an ogive, hence determine the first quartile, Q1,,third quartile, Q3 and the inter quartile range from
the ogive.
edmet-nklpunya.blogspot.com                                                            13

Exercise 7.3(b):
1. Find the mean, variance and standard deviation of each set of the following data.

(a) 9, 5, 3, 3, 7, 13, 9

(b) 2, 8, 11, 9, 6, 5, 12, 11

(c) 3, 4, 11, 3, 10, 11, 2, 3, 7

2. Find the mean, variance and standard deviation of each of the following data.

Score,     No. of
(a)       x        pupils, f
1           3
2           6
3           12
4           20
5           18
6           11

No. of      No. of
(b)       book        pupils
0            10
1            14
2            20
3            26
4            18
5            12
edmet-nklpunya.blogspot.com                                                                                     14

(c)   Mass /        No. of
kg            pupils

50             2
51             3
52            10
53            20
54             8
55             7

No.
No. of
(d)    of
family
children
0             1
1             2
2             8
3             2
4             1
5             1

Exercise 7.3(c):
1. The table below shows the duration of telephone calls received in an office on a certain day for 40 calls.
Find the mean, variance and standard deviation of the duration of calls.

Duration of      Number
Calls / min      of Calls
2–3                    9
4–5                 12
6–7                 10
8–9                 7
10– 11               2
edmet-nklpunya.blogspot.com                                                                                         15

2. The table below shows marks scored by 30 pupils in a test. Find the mean, variance and standard deviation
of the test.

Number of
Marks
pupils
20 – 29               2
30 – 39               4
40 – 49               5
50 – 59               10
60 - 69               6
70 - 79               3

4. The table below shows the lengths of 60 mature long beans in a field study. Find the mean, variance
and standard deviation of the lengths of the beans.

Number
Length / cm
of Beans
10 – 14               8
15 – 19                  15
20 – 24                  19
25 – 29                  13
30– 34                  5

Exercise 7.3(d) – Effect of Uniform Chances in a Set of Data on the Measures of Dispersions

1. The range and the variance of a set of data are 12 and 13 respectively. Each value in the set of data
is multiplied by 3 and then subtracted by 5. Find
(a) the new range,
(b) the new variance

2. A set of data has a range of 30, an inter quartile range of 5 and a standard deviation of 8. Each value in the
set of the data is divided by 4 and then added by 3. Find
(a) the new range,
(b) the new inter quartile range,
(c) the new standard deviation.
edmet-nklpunya.blogspot.com                                                                                       16

Exercise 7.4: Problem Solving I

1. Given the mode and the mean of the following set of data, 9, p, 14, q, 33, q are 33 and 20 respectively.
Determine the values of p and q.

2. The median of the set data 4, 5, 6, 8, k, 9, is 7. Determine the value of k.

3. A set of data has seven numbers. Its mean is 9. If a number p is added to the set, the new mean is 12. What is
the possible value of p?

4. A set of data x1, x2, x3, x4, x5 has a mean of 10 and a variance of 4. A value of x 6 is added to the set of data,
the mean remains unchanged. Determine
(a) the value of x6,
(b) the variance of the new set of data.

5. A set of data consists of 6 numbers. The sum of the numbers is 39 and the sum of the squares is 271.
(a) Find the mean and variance of the set of data.
(b) If a number 5 is taken out from the set of data, find the new mean and standard deviation of the new data.
edmet-nklpunya.blogspot.com                                                                                17

Past SPM Papers

1. The diagram below is a histogram which represents the distribution of the marks obtained by 40 pupils
in a test.

Number of Pupils

14

12

10

8

6

4

2

0                                                             Marks
0.5        10.5    20.5      30.5      40.5      50.5

(a)    Without using an ogive, calculate the median mark.                            [3 marks]

(b)    Calculate the standard deviation of the distribution.                       [4 marks]
(SPM 2005/SectionA/Paper 2)
edmet-nklpunya.blogspot.com                                                                                    18

2. A set of data consists of 10 numbers. The sum of the numbers is 150 and the sum of the squares
of the numbers is 2472.
(a) Find the mean and variance of the 10 numbers.                                                 [3 marks]
(b) Another number is added to the set of data and the mean is increased by 1. Find
(i) the value of this number,
(ii) the standard deviation of the set of 11 numbers.                                         [4 marks]
(SPM 2004/SectionA/Paper 2)

3. A set of examination marks x1, x2, x3, x4, x5, x6 has a mean of 5 and a standard deviation of 1.5.
(a) Find
(i) the sum of the marks, x,
(ii) the sum of the squares of the marks, x2.                                                   [3 marks]

(b) Each mark is multiplied by 2 then 3 is added to it. Find, for the new set of marks,
(i) the mean,
(ii) the variance.                                                                              [4 marks]
(SPM 2003/Section A/Paper 2)
edmet-nklpunya.blogspot.com                                                                                      19

4. The positive integers consists of 2, 5 and m. The variance for this set of integers is 14. Find the value of m.
[4 marks]
(SPM 2006/Paper 1)

5. A set of data consists of five numbers. The sum of the numbers is 60 and the sum of the squares
of the numbers is 800.
Find, for the five numbers
(a) the mean,
(b) the standard deviation.                                                                       [3 marks]
(SPM2007/Paper 1)

6. Table 1 shows the cumulative frequency distribution for the scores of 32 students in a competition.
Score                    < 10       < 20      < 30       < 40       < 50
Number of students        4          10         20        28         32
Table 1

(a) Based on Table 1, copy and complete Table 2.

Score                   0–9       10 – 19    20 – 29    30 – 39    40 – 49

Number of students

Table 2
(b) Without drawing an ogive, find the interquartile range of the distribution.
[5 marks]
(SPM2007/Section A/Paper 2)
edmet-nklpunya.blogspot.com                                                                                  20

7. Table 1 shows the frequency distribution of the scores of a group of pupils in a game.

Score         Number of pupils
10 – 19               1
20 – 29               2
30 – 39               8
40 – 49              12
50 – 59               k
60 – 69               1

(a) It is given that the median score of the distribution is 42.
Calculate the value of k.                                                                    [3 marks]

(b) Use the graph paper provided to answer this question.
Using a scale of 2 cm to 10 cm scores on the horizontal axis and 2 cm to 2 pupils on the vertical axis,
draw a histogram to represent the frequency distribution of the scores.
Find the mode score.                                                                        [4 marks]

(c) What is the mode score if the score of each pupil is increased by 5?
[1 mark]
(SPM2006/Section A/Paper 2)

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