-
additionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticszefryadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticsadditionalmathematicsstatisticshjklzxcvbnmqwert
STATISTICS
Name
........................................................................................
-
Statistics
[email protected]
CHAPTER 7 : STATISTICS
7.1 MEASURES OF CENTRAL TENDENCY (i) Ungrouped data data that is
not grouped into classes (ii) Grouped data - data that is grouped
in certain classes
7.1.1 Calculating the mean of ungrouped data
N
xx
Example 1:
Find the mean of 58, 67, 45, 73 and 77
Example 2:
Number 1 2 3 4 5
Frequency 5 8 4 6 2
Find the mean of the number.
7.1.2 Determining mode of ungrouped data
Mode is the value which appears the most number of times in a
set of data (value that has the
highest frequency)
Example 1: Determine the mode for the following sets of
number.
(a) 2, 5, 6, 2, 6, 7, 2, 4, 8, 2 Answer: ..
(b) 5, 5, 8, 10, 4, 4 Answer: ..
(c) 2, 2, 3, 4, 4 , 4, 6, 6 Answer: ..
Note: It is possible that a set of data either has more than one
mode or has no mode.
Where
x - mean of the set data. x values in the set of data N total
number of data
-
Statistics
[email protected]
Example 2: Determine the mode of the number of pens a student
has.
Number of pens 1 2 3 4 5
Number of students 6 7 5 3 3
Answer: ..
7.1.3 Determining the median of ungrouped data
When the values in set of data are arranged in either ascending
or descending order, the value that
lies in the middle is the median.
Example 1: 3, 4, 5, 6, 7, 8, 9
Example 2: 21, 20, 19, 18, 17, 16, 15, 14
Activity 1:
1. Calculate the mean, mode and median for the following sets of
data.
(i) 1, 4, 5, 8, 9, 8, 8, 7, 4 (ii) 5, 8, 12, 10, 5, 3, 7, 5, 20,
10
2. (i) Find the mean of 6, 8, 4, 9 and 11
(ii) Find the value of x if the mean of 4, 5, 6, 7, 11 and x is
7.
3. Find the mode of each following sets of data.
(i) 8, 6, 10, 8, 5
(ii) 2, 2, 5, 5, 11, 11
Median
4 numbers 4 numbers
Median =
-
Statistics
[email protected]
4. Find the mean, mode and the median of the following data
Time ( hours ) 12 13 14 15 16
Number of cars 3 5 10 6 6
5. The following frequency distribution table shows the score of
a group of students in a
quiz. Find the mean , mode and median
Score 5 6 7 8 9 10
Number of students 5 6 4 3 8 4
7.1.4 Determine modal class of grouped data from the frequency
distribution table.
Modal class of a set of data is the value of class which occurs
most frequently. The value of mode
can be obtained by drawing the histogram
Example 1:
The following table below shows the mark for 50 students in
their Additional Mathematics test.
Find the modal class for the students.
Mark 10-19 20-29 30-39 40-49 50-59 60-69 70-79
Number of students 3 6 13 10 7 7 4
7.1.5 Find mode from a histogram
Note: In drawing a histogram, class boundaries are used.
HISTOGRAM
Plot: frequency against boundary of the class
Frequency
Boundary of the class
-
Statistics
[email protected]
Example 1:
The following table below shows the mark for 50 students in
their Additional Mathematics test. .
Mark 10-19 20-29 30-39 40-49 50-59 60-69 70-79
Number of students 3 6 13 10 7 7 4
Table 3
Complete the following table.
Mark Frequency , f Lower class boundary Upper class boundary
10 19 3
From the table, draw a histogram, hence determine the mode from
the histogram.
Calculate mean of grouped data
f
fxx where f is the frequency for each class
x is the corresponding class midpoint
Note :
Class mid point = 2
limitupper limit lower
Example 1:
1. The table below shows the marks obtained by 30 students in a
Mathematics test.
Marks 30 - 39 40 - 49 50 - 59 60 -69
Number of students 4 8 12 6
Find the mean for the marks.
-
Statistics
[email protected]
Solution :
7.1.7 Calculating median of grouped data from the cumulative
frequency distribution table
Median, m = L + Cf
FN
m
2
1
, where
Note: Size of class interval = upper class boundary lower class
boundary
Activity :
The following table shows the mark for 40 students in their
Additional Mathematics test. Find
the median by using the formula
Mark 10-19 20-29 30-39 40-49 50-59 60-69 70-79
Number of students 3 5 7 9 10 4 2
TABLE 1
Marks Number of students, f Class mark ,x fx
f fx
L = Lower boundary of the class in which the median
lies.
N = total frequency
C = Size of class interval
F = cumulative frequency before the class in which the
median lies.
fm = frequency of the class in which the median lies
-
Statistics
[email protected]
Solution:
Mark Frequency , f Cumulative
frequency, F
10 19 3
7.1.8 Estimate median of grouped data from an ogive
OGIVE
Plot: cumulative frequency against upper class boundary
Cumulative
frequency
Upper class boundary
-
Statistics
[email protected]
Refer to table 1
Construct a cumulative frequency table and then draw cumulative
frequency curve (ogive)
From the graph, find the median weight.
Mark Frequency , f Upper class
Boundary
Cumulative
frequency, F
Activity 2:
1. The following table shows the marks obtained by 30 students
in Mathematics test. Find the mean for the data.
Mark 10-19 20-29 30-39 40-49 50-59 60-69 70-79
Number of students 3 6 13 10 7 7 4
Solution :
Mark Number of students, f Class mark ,x fx
-
Statistics
[email protected]
2. The data below shows the scores obtained by 30 students in a
game.
0 3 3 6 8 9 9 10 10 11
5 0 7 6 9 10 12 17 5 4
2 5 8 8 10 11 7 8 12 11
Using 3 scores as a size of class interval, construct a
frequency distribution table for its data.
Then find
(a) the mean (b) median (c) modal class
Note : Students have to draw the frequency distribution table as
below.
Number of scores x f fx Cumulative
frequency
3. The table below shows the age of factory workers in 1995.
Age(year) 20-25 26-31 32-37 38-43 44-49 50-55 56-61
Number of workers 5 24 16 20 13 12 10
(i) Draw a histogram and hence estimate the mode of the data.
(ii) Find the median without using an ogive.
-
Statistics
[email protected]
7.1.9 Determine the effects on mode, median and mean for a set
of data when
(a) Every value of the data is change uniformly
Measures of
central
tendency
Added by k Subtracted by k Multiplied by k Divided by k
New mean Original mean + k Original mean - k k(Original
mean)
k
mean original
New mode Original mode + k Original mode - k K(Original
mode)
k
mode original
New median Original median+ k Original median - k K(Original
median)
k
median original
1. The mean, mode and median of a set data are 7.4, 9 and 8
respectively. Find the new
mean, mode and median if every value of the set data is
a) divided by 2 b) subtracted by 4 c) multiplied by 3 d) added
by 3
2. Find the mean of 12, 14, 16, 18, 20. By using the result,
find the mean of 8 ,10 ,12 ,14,16
7.1 MEASURES OF DISPERSION
7.2.1 Finding the measures of dispersion of ungrouped data
Formulae:
(i) Range of ungrouped data = largest value smallest value (ii)
Interquartile range = Upper quartile lower quartile
= 3Q 1Q
(iii) Variance, 2 :
2
2)(
N
xxi 2
2
2 xN
xor
, where
N
xx
(iv) Standard deviation, : = iancevar
-
Statistics
[email protected]
Example :
For the given ungrouped set of data, find
(i) range (ii) interquartile range (iii) variance (iv) standard
deviation
(a) 3 , 6 , 8 , 12 , 15 , 16
(b) 3 , 5 , 6 , 7 , 8 , 4 , 8 , 9 , 10 , 12 , 14
-
Statistics
[email protected]
Activity 3:
1. Find range, interquartile range, variance and standard
deviation for each set of the
following data.
(a) 10 , 7 , 19 , 13 , 14 , 10
(b) 4 , 3 , 2 , 7 , 9 , 10 , 12 , 6 , 15
(d) 4 , 12 , 15 , 10 , 7 , 6 , 1
2. Given that the mean of set data 5 , 7 , x , 11 , 12 is 9.
a. Find the value of x. b. Find the variance and the standard
deviation of the data.
7.2.2 Finding measures of dispersion of grouped data
Example:
For the following table,
Find (i) range
(ii) interquartile range using ogive
Age ( year ) Number of workers Upper class boundary Cumulative
frequency
25 29 16
30 34 20
35 39 24
40 44 20
45 49 14
50 54 6
Solution:
-
Statistics
[email protected]
7.2.3 Determining the variance and standard deviation of grouped
data.
Formulae :
N
xxfiance
2
2)(
var or 2 = 22
xf
fx
where
f
fxmeanx
standard deviation = 222)(
var xf
fxor
N
xxfiance
-
Statistics
[email protected]
Example 1:
From the following table, calculate the variance and standard
deviation.
Mass ( kg ) Number of
students
46 50 3
51 55 7
56 60 10
61 65 5
66 70 6
71-75 9
Solution:
Example 2:
The heights of 100 men to the nearest cm are recorded as
follows. Calculate the mean ( giving
your answer to one decimal place). Calculate the standard
deviation for the height of the men.
Height , x(cm) Frequency, f
159 - 160 2
161 162 14
163 164 28
165 166 26
167 168 19
169 170 9
171 - 172 2
-
Statistics
[email protected]
Solution:
Class f x fx 2x fx2
Activity 4:
1. The following table shows the number of durians sold in 50
consecutive days.
Number of durians Number of days
0 8 10
9 17 15
18 26 20
27 - 35 5
Calculate the variance and standard deviation
-
Statistics
[email protected]
2. The table below shows the distribution of marks of 120
students in a Physics test.
Marks Number of
students
20 29 2
30 39 14
40 49 35
50 59 50
60 69 17
70 - 79 2
Calculate
a. mean b. median c. standard deviation
for this distribution.
7.2.4 Determining the effects on measures of dispersion when
some values in a set of data are changed.
A. If every value of the data is changed uniformly
Measures Added by k Subtracted by k Multiplied by k Divided by
k
New
Variance, 2
2
(original
variance)
2
(original variance) )( 22 k
2
2
k
New
Standard
Deviation,
(original
standard
deviation)
(original standard
deviation)
)(k k
original
New Range
Original range Original range
range) original(k
k
range original
New
Interquartile
range
Original
interquartile
range
Original
interquartile range
k(original
interquartile
range) k
ileinterquart original
-
Statistics
[email protected]
B. If there are extreme values in the set of data
Range
Extreme values in a set of data will significantly increase the
range of the set of data.
Interquartile range
Extreme values in a set of data will have little or no effect on
the interquartile range.
Variance and Standard Deviation
Extreme values also significantly increase the value of standard
deviation and variance but
standard deviation is affected to a smaller degree as compare to
variance.
C. If certain values are added or removed
When a value is added or removed from a set of data, the effect
on the measures of dispersion is
uncertain. In general, the range and the interquartile range are
less affected as compared to the
variance and the standard deviation. Variance and standard
deviation are more significantly
affected when the added or removed value has a greater
difference from the mean.
Activity:
(a) The interquartile range and the standard deviation of a set
of data are 5 and 2.5. Find the new
interquartile range and standard deviation if every value of the
data is divided by 2 followed
by an addition of 10.
(b) Given a set of data 4, 5, 8, 12, 14, 18, 20. Determine the
standard deviation of the set of data.
Explain how the standard deviation will change if a value of 100
is added to the set of data.
Activity 5:
1 The following table shows the results in five Additional
Mathematics Test obtained by two
students A and B.
Students A 71 76 80 83 90
Students B 40 67 95 98 100
(a) Find the mean and the standard deviation of the result for
each students.
(b) If a student with a more consistent performance is selected,
which students would
be selected? Explain the reason for your selection.
2 Find the mean and standard deviation for the following set of
numbers: 2, 3, 5, 8, 10.
Hence, using this result find the mean and standard deviation
for
(a) five numbers : 7, 8, 10, 13, 15
(b) five numbers : 10, 15, 25, 40, 50
(c ) ten numbers : 2, 2, 3, 3, 5, 5, 8, 8, 10, 10
-
Statistics
[email protected]
3 The mean for 7, 14, 2x, x, 16, 9, 10 and 2 is 8. Find the
value of
(a) x
(b) the standard deviation
(c ) standard deviation if each of the number is added by 2.
Enrichment
SPM 2006
1. A set of positive integers consists of 2 , 5 and m. The
variance for this set of integers is 14. Find the value of m. [ 3
marks ]
2. Table 1 below 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
Table 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 to answer this question
Using a scale of 2 cm to 10 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 ]
SPM 2005
3. The mean of four numbers is m .The sum of the squares of the
numbers is 100 and the
standard deviation is 3k. Express m in terms of k. [ 3 marks
]
4. Diagram 2 is a histogram which represents the distribution of
the marks obtained by 40 pupils in a test.
-
Statistics
[email protected]
(a) Without using an ogive, calculate the median mark. [ 3 marks
]
(b) Calculate the standard deviation of the distribution. [ 4
marks ]
SPM 2004
5. A set of data consist of 10 numbers. The sum of the numbers
is 150 and the sum of the squares of the numbers is 2 472.
(a) Find the mean and variance of the 10 numbers.
(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 2003
6. 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, 2x [ 3 marks ] (b) Each mark is multiplied by 2 and then 3
is added to it. Find for the new set of
marks,
(i) The mean,
(ii) The variance [ 4 marks ]
7. The mean for 20, 5 + x, 5 + 4x and 5 are 10. Calculate the
value of (a) x
(b) median
Upper class boundary
14
12
10
8
6
4
2
0 0.5 10.5 20.5 30.5 40.5 50.5
frequency
Diagram 2
-
Statistics
[email protected]
(c) standard deviation
8. The table shows the mark obtained by a group of students in a
competition.
Mark 1 2 3 4 5
Number of
students 3 7 5 4
1
Find
(a) mean , mode and median (b) standard deviation
9.
Score 0 1 2 3 4 5 6
Frequency 3 2 2x 5 x + 4 4 3
The table above shows the score obtained by 30 students in a
test.
(a) Find the value of x (b) Find score mode (c) If we represent
the table by pie chart, calculate the angle sector for the students
who
get score more than 3.
SPM 2001
10. (a) Given that 4 positive integers have mean 9. If a number,
y, is added , the mean
become 10. Find the value of y. [ 2 marks ]
(b) Find the standard deviation for the set of number 5, 6, 6,
4, 7
[ 3 marks ]
11.
Mark Number of students
6 10 12
11 15 20
16 20 27
21 25 16
26 30 13
31 35 10
36 40 2
Table 2
Table 2 shows the frequency distribution table for the marks
obtained by 100 students.
(a) Calculate the variance. [ 3 marks ]
(b) Using the graph paper provided to solve this question
Construct a cumulative frequency distribution table and draw an
ogive. Hence, find the
percentage of the number of students who get mark between 6 and
24. [7 marks ]
-
Statistics
[email protected]
SPM 2000
12.
Mark < 10 < 20 < 30 < 40 < 50 < 60 < 70
< 80
Number of
students 2 8 21 42 68 87 98 100
The table above shows the mark obtained by 100 students in a
test.
(a) Using the table above, copy and complete the following
table.
Mark 0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79
frequency 2
[ 2 marks ]
(b) Without drawing an ogive, estimate the interquartile range
for the data.
[ 4 marks ]
SPM 1998
13. The mean of set data 2, k, 3k, 8, 12 and 18 in increasing
order is m. If every number in
the set data is subtracted by 2, the median for the new data is
8
5m. Find
(a) the value of m and k [ 4 marks ] (b) Variance for the new
data. [ 2 marks ]
14. Set X consist of 50 score, in a certain match, has mean 8
and standard deviation 3.
(i) Calculate x and 2x (ii) The sum of a number, 180 scores with
mean 6 and the sum of the square of a
number, 1200, are removed from set X. Evaluate the mean and
variance for left
data in set X. [ 7 marks ]
SPM 1996
15. The mean for the list of number x 2, x + 4, 2x + 5, 2x 1, x
+ 7 and x 3 is 7. Find
(a) The value of x [ 1 marks ]
(b) Variance [ 2 marks ]
-
Statistics
[email protected]
SPM 1995
16.
Number of class Number of students
6 35
5 36
4 30
The table shows the number of students in a class. Find
(i) the mean (ii) the standard deviation, for the number of
students in every class.
(b)
Age Number of people
1 - 20 50
21 - 40 79
41 - 60 47
61 - 80 14
81 - 100 10
(Solution using graph method is not allowed)
The table above shows the age of 200 people in a village.
Calculate
(i) median (ii) Upper quartile, of the age of the people. [ 6
marks ]
17. (a) Given a list of number: 3, 6, 3, 8. Find the standard
deviation of the number.
[ 2 marks ]
(b) Find a set of five possible positive integer with mode 3,
median 4 and mean 5.
SPM 1994
18. Set A is a set that consist of 10 numbers. The sum of the
number is 150 and the sum of the
square of the marks are 2890.
(i) Find the mean and variance. (ii) If one number is added to
the set of numbers in Set A, (in case mean is unchanged),
find the standard deviation of the 11 numbers.
19. Find the interquartile range for the following data:
10, 24, 11, 18, 8, 26, 19, 13, 17, 16, 25
-
Statistics
[email protected]
SPM 1993 P1
20.
Mark 1 - 20 21 - 40 41 - 60 61 - 80 81 100
Number of
students 5 8 12 11
4
The table shows the mark obtained by a group of students in a
Monthly Test.
(a) By using graph paper , draw a histogram and hence determine
the mode of the data. (b) Without drawing an ogive, find the
median. (c) Calculate mean mark.
Answer :Enrichment 1
1 11
2 (a) k = 4
(b) mode = 43
(c) 48
3 2925 km
12. (a) (b) 21.78
4 (a) 24.07 (b) 11.74 10 (a) m = 8, k = 2 (b) 322
5 (a) 15 , 2.222
(b) (i) k = 26 (ii) 5.494
11 (i) 3650,400 2xx (ii) New variance = 1.5
6 (a) (i) 30 (ii) 163.5
(b) (i) 13 (ii) 9
12 (a) x=4 (b) 192
7 (a) x = 1 (b) 7.5 (c) 5.958 13 (a) (i) 34 (ii) 2.449
(b) (i) 33.16 (ii) 49.44
8. (a) 2.65, 2, 2.5 (b) 1.108 14 (a) 2.121 (b) 3,3,4,6,9 or
3,3,4,5,10 or 3,3,4,7,8
9. (a) x= 3 (b) 4 (c) 168o 16 (a) (i) 15, 64 (ii) 7.628
10. (a) y = 14 (b) 1.020 17 13
11. (a) 61.26 (b) 70% 18 (a) 56.5 (b) 52.17 (c) 51
