BA201 Engineering Mathematic UNIT2 - Measures of Central Tendency

Prepared by : NURIZATY MUHAMAD NOR Page 1

B3001/UNIT2/1

Unit

2

MEASURES OF

CENTRAL

TENDENCY

Understanding differences between

mean, mode and median with formula

method and graphical method.

On completion of this unit, the students

should be able to :

1. Solve mean, mode and median

for ungrouped data and grouped

data using formula method.

2. Solve mean, mode and median

for ungrouped data and grouped

data using graphical method.

General Objective

am

Specific Objectives

khusus


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2.0 INTRODUCTION

Measure of central tendency or measure of location is a value which is

representative of a set of data. The role of central tendency is to determine the central

value of a set of data. The various types of measures of location are mean, mode and

median.

2.1 UNGROUPED DATA

2.1.1 Mean

Mean of a set of data is the sum of the values of all observations divided by the total

number of observation.

If the data represent a sample, the mean is defined as follows :

Mean, = N

x

where,

x = sum of values of all data N = total number of data

Example 2.1:

Find the mean of the following data :

5, 7, 7, 9, 4, 6, 5, 10, 12 and 8

Solution :

Mean, = N

x

= 10

8 12 10 56 4 9 7 7 5

= 7.3

INPUT


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Example 2.2

Data in Table 2.1 shows the test marks that scored by a student in Mathematics 3.

Table 2.1

Test Marks

1

2

3

4

5

80

100

60

55

75

Solution :

Mean = testsofnumber total

marks of sum

= 5

370

= 74

2.1.1.1 Frequency Table For Ungrouped Data

You must understand what a frequency table is. Raw data can be summarized in

a frequency distribution table. Frequency distribution table for ungrouped data shows the

number of observations or frequencies for each data. This method can apply if the raw

data is large.

So, the student gets 74 as the mean of

his mathematics 3 tests.


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Example 2.3:

The following data shows the marks scored by 35 students in mid-term exam.

80 45 40 75 85 45 50

80 60 60 40 50 65 80

60 40 60 70 80 60 50

65 45 75 80 75 50 45

40 60 65 70 60 45 50

Calculate the mean of the marks.

Solution :

The marks can be summarized in frequency distribution table for ungrouped data as

follows:

Table 2.2 : Frequency distribution table for ungrouped data

Marks (x) Frequency (no. of students) ( f ) Total (fx)

80

45

40

60

50

65

70

75

85

5

5

4

7

5

3

2

3

1

400

225

160

420

250

195

140

225

85

Total 35 2100

Min, = f

fx

= 35

2100

= 60


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This is because all data have the same frequency

2.1.2 Mode

Mode is the value that occurs with the highest frequency in a set data of

data.There are two or more values of mode if there are two or more data that have the

same highest frequency. A set of data also do not have a mode if a data given have the

same frequency.

Example 2.4:

Find the mode of the following data :

a) 10, 20, 50, 30, 20, 40, 60 and 20

b) 10, 20, 50, 30, 20, 40, 60 and 50

c) 10, 20, 50, 30, 20, 40, 30 and 50

d) 10, 20, 50, 30, 20, 40 and 60

e) 10, 20, 50, 30, 20, 10, 30 and 50

Solution:

a) mode, M0 = 20

b) M0 = 20 dan 50

c) M0 = 20, 30 dan 50

d) M0 = 20

e) M0 = None

Example 2.5 :

Find the mode in table 2.2 above.

Solution :

Mode is 60 marks, because it has with the highest frequency.


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2.1.3 Median (M)

Median is the centre value of a set of data after the data is arranged in ascending

or decending order.

Example 2.6:

For a set of data: 3, 6, 11, 4, 8, 14, 21, find the median.

Solution :

3, 4, 6, 8, 11, 14, 21

So, median = 8

Example2.7:

For a set of data 61, 65, 68, 78, 79, 84, 90, 91, determine the median.

Solution :

Arrange the data in ascending order

61, 65, 68, 78, 79, 84, 90, 91

Because there are two numbers in the centre of a set of data,

median = 2

7978

= 78.5

3 numbers 3 numbers

Arrange the numbers in

ascending order

All sets of data have median but not

all of them have mode


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ACTIVITY 2a

2a.1 Calculate the mean for a set of data:

a) 40, 65, 75, 70, 75, 80

b) 400, 450, 350, 300, 380

c) 250, 350, 300, 200, 700

2a.2 Find the median for these samples of data:

a) 7, 3, 4, 2, 1, 5, 6, 8

b) 30, 35, 38, 37, 40, 45, 33

c) 355, 370, 365, 340, 360

2a.3 Determine the mode for these samples of data :

a) 30, 40, 50, 60, 30, 40, 30

b) 3, 3, 3, 4, 4, 5, 6, 7, 3, 4

c) 1, 3, 5, 7, 9, 11, 13, 15, 17


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FEEDBACK 2a

2a.1 a) 67.5

b) 376

c) 360

2a.2 a) 4.5

b) 37

d) 360

2a.3 a) 30

b) 3

c) None


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2.2 GROUPED DATA

2.2.1 Mean

Mean for grouped data can be calculated using the formula as follows :

Mean, = f

fx

where, (sigma) = total f = frequency

x = class midpoint =2

limlim itUpperitLower

Example 2.8:

Find the mean for grouped data in table 2.3below.

Table 2.3

Class Frequency, f

0 - 4

5 - 9

10 - 14

15 - 19

30

51

10

10

Solution :

First, you must complete the frequency table as in table 2.4

Table 2.4

Class Midpoint, x Frequency, f fx

0 - 4

5 - 9

10 - 14

15 - 19

2

7

12

17

30

51

10

10

60

357

120

170

INPUT


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101f 707fx

So,

Min, = f

fx

101

707

= 7

2.2.2 Median

Median is the center value of a set of data that is arranged in order. For grouped

data, the median class should be determined first before calculating the median by (N/2).

Median is given by the following formula :

Cfm

FN

Lm

2

Where, L = lower boundary of the median class

N = total number of frequency

F = cumulative frequency before median class

fm = frequency of median class

C = size of median class

Example 2.9:

Table 2.5 shows the weight of 100 polytechnic students. Calculate the median using

formula.

Table 2.5 : Weight of 100 polytechnic students.

Weight (kg) Frequency

25 - 49

50 - 74

75 - 99

100 - 129

15

25

30

20

2

34

2

1915

= 17


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125 - 149 10

Solution :

Table 2.6 : Frequency distribution table of 100 polytechnic students

Weight

(kg) frequency, f

Cumulative

frequency

25 - 49

50 - 74

75 - 99

100 - 124

125 - 149

15

25

30

20

10

15

40

70

90

100

Step 1 : Determine median class

Total number of frequency, N = 100

Median class is given by Tn/2 = T 100/2 = T50

Therefore median class is the class 75 -99 kg

Step 2 : Using formula

L = lower boundary of the median class = 74.5

N = total number of frequency = 100

F = cumulative frequency before median class = 40

fm = frequency of median class = 30

C = size of median class= 99.5 - 74.5 = 25

Cfm

FN

Lmed

2

2530

40505.74

med

= 82.8 kg


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2.2.2.1 Estimating Median of Grouped Data From Ogive

An ogive is also known as a cumulative frequency graph.

Step 1 : Construct a table with the values of upper boundaries and cumulative

frequencies.

Step 2 : Plot an ogive graph of cumulative frequency against upper boundaries.

The median can be determined from the ogive based on the Tn/2

Example 2.10:

Table 2.7 shows the weight of a group of students.

Berat (kg) 41 - 45 46 - 50 51 - 55 56 - 60 61 - 65 66 - 70 71 - 75

Kekerapan 2 5 7 12 9 3 2

Table 2.7 : Weight of a group of student

Draw an ogive for the given data and from the ogive estimate the median of the weight

of the students.

Solution :

Table 2.8 : Frequency distribution table

Weight

(kg)

Frequency Cumulative

Frequency

Upper

boundaries

41 - 45 2 2 45.5

46 - 50 5 7 50.5

51 - 55 7 14 55.5

56 - 60 12 26 60.5

61 - 65 9 35 65.5

66 - 70 3 38 70.5

71 - 75 2 40 75.5


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Figure 2.1: A less than ogive of weight of a group of student.

Median,m = Tn/2 = T 40/2 = T20

= 58.5 kg ( From the graph)

2.2.3 Mode

2.2.3.1 Formula Method

Mode for grouped data can be determined by the following method :

cdd

dLM M

21

10 0

dimana,

= Lower boundary of the mode class

= frequency of the mode class frequency of the class before

d2 = frequency of the mode class frequency of the class after

= the width of the mode class

0 M L

1 d

c


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Example 2.10

Determine the mode in table 2.9:

Table 2.9

Class Frequency

118 - 126 3

127 - 135 5

136 - 144 9

145 - 153 12

154 - 162 5

163 - 171 4

172 - 180 2

Solution :

Mode class : 145 - 153

= 144.5

120Mf , 910

M

f , 510

Mf

So, 391211 00

MM

ffd

751212 00

MM

ffd

9c

Mod, 973

35.1440

M

= 147.2

0 M L

Mode

class 12

0Mf

910

Mf

510

Mf

cdd

dLM M

21

10 0


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2.2.3.2 Estimating Mode From Histogram

A histogram is constructed based on the frequency distribution. The values of

data (lower boundaries) are plotted on the horizontal axis while the frequencies are

plotted on the vertical axis.

Example 2.11:

Construct a histogram and determine the mode from table 2.10.

Markah Kekerapan

31 40 5

41 50 7

51 60 14

61 70 10

71 80 4

Table 2.10 : The marks of a group of students in Mathematics Test.

Solution

Mode = 56.8

mode class

Frequency

Mode

class


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ACTIVITY 2b

2b.1 The monthly school expenses is shown in the following data. Find the mean,

mode and median using formula and graphical method.

Expenses (RM) Frequency

11 21 3

22 32 7

33 43 10

44 54 14

55 65 9

66 76 6

77 - 87 1

n = 50

2b.2 The weight of 50 students in kilograms are given in the following table.

Determine the mean, mode and median using formula and graphical method.

Weight (kg) Frequency

45 49 9

50 54 6

55 59 16

60 64 10

marks


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65 69 7

70 74 2

75 - 79 0

ANSWERS 2b

2b.1 Mean = 47.02

Median = 47.4

Mode = 48.38

2b.2 Mean = 57.5

Median= 57.6

Mode = 57.6


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PRACTICES

2-a. The following table shows the height of 50 students in a class. Determine the mean,

mode and median.

Height

(cm) 146 150 151 155 156 160 161 - 165 166 - 170 171 - 175

No. of

Students 5 10 14 11 6 4

2-b. The following table shows the marks scored by 54 students in a Geography Test.

Determine the mean, mode and median for the data.

Marks 0 -19 20 - 39 40 - 59 60 -79 80 -99

No. of Students 3 8 16 10 8

2-c. A sample of wire produced by a factory in 1 week is shown in the table below..

Determine the mean, mode and median.

Diameter(mm) 0.95 0.97 0.98 1.00 1.01 1.03 1.04 1.06 1.07 1.09

Frequency 7 12 15 11 5

2-d. i) Find the mean of data : 4, 5, 78, 75, 70.


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ii) Find the median of data : 50, 60, 70, 90, 110, 130, 150, 80, 120.

iii) Find the mode of data : 10, 30, 50, 70, 90, 110, 130, 150, 170.

ANSWERS

2-a. Mean=159.5cm, median=159.1cm, mode class=156 160

2-b. Mean=54.8 marks, median=54.5 marks, mode class=40 59

2-c. Mean=1.02mm, median=1.02mm, mode class=1.01 1.03

2-d i) 46.8

ii) 90

iii) None

Documents

BA201 Engineering Mathematic UNIT2 - Measures of Central Tendency