Statistic

STATISTICS

FOR ENGINEERING MATHEMATICS

ANISAH AHMAD
INTAN ZARINA GHAZALI
AZURANI ABD. RAUF

JABATAN MATEMATIK, SAINS DAN
KOMPUTER
POLITEKNIK SEBERANG PERAI

STATISTICS:

Anisah Binti Ahmad
Intan Zarina Binti Ghazali

Azurani Abd Rauf
2021

Mathematics, Science and Computer Department
Politeknik Seberang Perai

©All rights reserved. No part of this publication may be translated or reproduced in
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mechanical, recording, or otherwise, without prior permission in writing from
Politeknik Seberang Perai.
i

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Perpustakaan Negara Malaysia Cataloguing-in-Publication Data

Anisah Ahmad
STATISTICS For Engineering Mathematics / Anisah Ahmad, Intan Zarina Ghazali,
Azurani Abd Rauf.
Mode of access: Internet
eISBN 978-967-0783-99-4
1. Engineering--Statistical methods.
2. Engineering mathematics.
3. Mathematical statistics.
4. Government publications--Malaysia.
5. Electronic books.
I. Title.
519.502462

ACKNOWLEDGEMENT

In The name of Allah, the Most Gracious and the Most Merciful, the
Almighty who give us the enlightenment, the truth and with regard to
Prophet Muhammad S.A.W for guiding us to straight path.
Alhamdulillah, all praises to Allah for the strengths He gave us and His
blessing in completing this book, Electrical Engineering Mathematics.
This book has been produced according to the latest syllabus of
Electrical Engineering Mathematics Polytechnic ministry of Malaysia. It
is accompanied with notes, examples and exercises. The content is
suitable for polytechnic students and lecturers as a reference and
training.
Thank you to the team of writers for all the contributions and support
in the effort to prepare this book. Lastly, we would like to express our
deepest gratitude to all staff of Mathematics, Science and Computer
Department. We are willing and humbly accept the criticism in an
effort to improve and enhance the quality of this book.

WRITER WRITER WRITER
ANISAH BINTI AHMAD INTAN ZARINA BINTI GHAZALI AZURANI BINTI RAUF

iii

PREFACE

Mathematics form an important part of daily life.
Without Mathematics, there’s nothing you can do.
Everything around you involve Mathematics.
Everything around you involve numbers. Statistics for
example, can help you interpret data in everyday life
and help you in making good decisions. Therefore,
teaching methods and materials should be taught and
produced with freshness and diversity to make them
beneficial for life. This book is designed as a reference
book for users who need information regarding
Statistics. It can help you classify various types of data,
formulas and graphs commonly used in statistics. In
addition, the book also contains many examples with
step-by-step solutions to guide you in understanding
and solving statistical questions.

TABLE OF CONTENT Pages

Chapter 1

1.0 Introduction 2
1.1 Frequency Table 4
1.2 Histogram, Frequency Polygon and Ogive 11
1.3 Data Classified
12
2.0 Central Tendency
2.1 Mean, Mode and Median using Formula 12
2.2 Mode and Median using Graph 15

3.0 Measure of Dispersion 18
3.1 Mean Deviation, Variance and Standard 19
Deviation by Formula
26
4.0 Quartiles, Deciles and Percentiles. 27
4.1 Calculate Quartile, Decile and Percentile
37
Reference

v



eBook PSP | Statistics 1

1. Introduction

WHAT IS
STATISTICS?

Statistics is the study of the

collection, analysis,

interpretation, presentation, and

organization of data. There are

various ways to interpret data.

After the data is taken or

collected, the data can be sorted

out into a table named as

frequency table. Then, from the

frequency table, we can use

histogram, frequency polygon and

ogive to present the data and

analyse it.

Statistics is defined as numerical
statements of facts in any
department of inquiry placed in
relation to each other.

Sir Arthur Lyon Bowley
~Statistician~

2 eBook PSP | Statistic

1.1. Frequency Table

Frequency refers to how often an event or a value occurs. A frequency table is a table form
of recording the frequency for each class interval that have been grouped into different class
intervals. When the collected data is large, then we can analyse it easily using frequency table.
We can make a table with a group of observations using all the terms below:

Num Terms Table 1: Definition of terms
1 n Meanings / Formula

Total number of data

2 k Number of class (Sturge’s formula)
k = 1 + 3.33 log n

3 Range Range = Largest value – Smallest value

4 C Size of class, C = Range / k

5 Class interval The range of each group of data.

6 Frequency The number of times the observation has occurred in

the data

7 Tally marks Small lines that indicate the frequency

Example 1

Based on the data given below, construct a frequency table:
Table 2: Example 1 Distribution Data

12 10 22 23 25 41
65 13 89 47 33 52
55 13 88 37 81 53
45 90 19 57 73 53
34 87 17 67 80 11

eBook PSP | Statistics 3

Solution:

STEP 1:
We use Sturge’s formula when number of class, k and size of class interval, C are not given.

, = 1 + 3.33 log Where n is the number of data
= 1 + 3.33 log 30
= 5.919 ≈

STEP 2:
Determine the range.

= ℎ − ℎ

From the given data, largest value = 90, smallest value = 10, so that;
= 90 − 10
=

STEP 3:
Calculate the size of class interval from the formula:


, = ,

80
, = 6

= 13.333 ≈

4 eBook PSP | Statistic

STEP 4:
Construct a frequency table. Ensure that the smallest value and the largest value of the given
data are grouped in the first-class interval and the last class interval respectively.

Table 2a: Example 1 Construct Frequency Table:

Smallest Class interval Tally marks Frequency
value in 8
the first 10-22 IIII III 4
class 4
interval 23-35 IIII 5
3
Largest 36-48 IIII 3
value in 3
the last 49-61 IIII
class
interval 62-74 III

75-87 III

88-100 III

1.2. Histogram, Frequency Polygon and Ogive

Histogram is a graphical display of data using bars of different heights. It is similar to bar
chart, but a histogram group’s numbers into ranges. (Frequency against Class Boundary). A
histogram has no gap between each bar and the height is proportional to the frequency

Steps for Drawing a Histogram

01 Determine the lower boundaries and upper boundaries for each class

interval

02 Choose suitable scales for the horizontal axis to represent class interval

and the vertical axis to represent frequency

03 Draw a bar for each class interval according to the frequency
value

04 Label each axis and write down the title of your histogram

eBook PSP | Statistics 5

Frequency Polygon

A frequency polygon is a line graph which is obtained by joining the midpoints at the top of
each rectangle in the histogram. They had the same purpose as histograms, but are
specifically helpful for comparing sets of data.

Steps for Drawing a Frequency Polygon

01 Draw histogram (include one class interval below the lowest class
interval and one above the highest class interval.

02 Place a point in the middle of each class interval at the height
equivalent to its frequency including the points at x-axis.

03 Connect the points (the line will touch the X-axis on both sides)

04 Label each axis and write down the title of
your frequency polygon

6 eBook PSP | Statistic

Ogive

Ogive is a cumulative frequency graph which is obtained by plotting the cumulative frequency
against the upper boundaries of each class. There are two types of ogives which are less than
ogive (normal ogive) and more than ogive.

Steps for Drawing an Ogive

01 Find Upper Boundaries and Cumulative Frequency of the data.

02 Choose suitable scales for the horizontal axis (y-axis) to represent

upper boundaries and the vertical axis (x-axis) to represent
cumulative frequency.

03 Add a class with a frequency of zero before the first class.

04 Plot the points and then join them with a smooth curve.

05 Label each axis and write down the title of your ogive.

eBook PSP | Statistics 7

Example 2

The Table 3 below shows width of the leaf (cm) collected by the students from an experiment.
The total leaves collected are 30. Construct the frequency table which contains 8 classes. Then
represent the information with a histogram and ogive.

Table 3: Example 2 Distribution Data

4.6 6.1 5.5 6.5 5.2 5.9
6.3
5.1 4.5 7.1 8.0 5.8 7.0
4.9
6.3 5.6 6.6 5.7 7.2 5.7

4.8 6.9 7.3 7.4 4.1

6.0 5.3 6.4 6.9 6.4

Solution:
o Frequency table
STEP 1:
Determine the range.

= ℎ − ℎ

From the given data, largest value = 8.0, smallest value = 4.1, so that;
= 8.0 − 4.1
= .

STEP 2:
Calculate the size of class interval from the formula:


, = ,

3.9
, = 8

= 0.4875
≈ .

8 eBook PSP | Statistic

STEP 3:
Construct a frequency table. Ensure that the smallest value and the largest value of the
given data are grouped in the first-class interval and the last class interval respectively.

Table 3a: Example2 Frequency table

Smallest Leaf Width (cm) Tally marks Frequency
value in 2
the first- 4.1-4.5 II 3
class 4
interval 4.6-5.0 III 6
6
Largest 5.1-5.5 IIII 4
value in 4
the last 5.6-6.0 IIII I 1
class
interval 6.1-6.5 IIII l

6.6-7.0 IIII

7.1-7.5 IIII

7.6-8.0 I

Solution: Table 3b: Frequency table with Class Boundaries
o Histogram
Tally marks Frequency Class Boundaries
Leaf Width (cm) II 2 4.05-4.55
4.1-4.5 III 3 4.55-5.05
4.6-5.0 IIII 4 5.05-5.55
5.1-5.5 IIII I 6 5.55-6.05
5.6-6.0 IIII l 6 6.05-6.55
6.1-6.5 IIII 4 5.55-7.05
6.6-7.0 IIII 4 7.05-7.55
7.1-7.5 I 1 7.55-8.05
7.6-8.0

eBook PSP | Statistics 9

Frequency The Width of the Leaves Taken in Science Experiment
7

6

5

4

3

2

1

0
4.05 4.55 5.05 5.55 6.05 6.55 7.05 7.55 8.05
Class Boundaries

Figure 1: Histogram

Solution: Table 3c: Frequency table with Class Boundaries
o Ogive
Tally marks Frequency Upper Cumulative Cumulative
Leaf Width Boundaries Frequency Frequency
(cm) II 2 (less than) (more than)
III 3 4.05
4.1-4.5 IIII 4 4.55 0 30
4.6-5.0 IIII I 6 5.05 2 28
5.1-5.5 IIII l 6 5.55 5 25
5.6-6.0 IIII 4 6.05 9 21
6.1-6.5 IIII 4 6.55 15 15
6.6-7.0 I 1 7.05 21 9
7.1-7.5 7.55
7.6-8.0 8.05 25 5
29 1
30 0

Cumulative Frequency10 eBook PSP | Statistic

Cumulative Frequencya) Less than ogive
The Width of the Leaves Taken in Science Experiment

35
30
25
20
15
10

5
0

4.05 4.55 5.05 5.55 6.05 6.55 7.05 7.55 8.05

Class Boundaries

Figure 2: Less Than Ogive

b) More than ogive
The Width of the Leaves Taken in Science Experiment

35
30
25
20
15
10

5
0

4.05 4.55 5.05 5.55 6.05 6.55 7.05 7.55 8.05

Class Boundaries

Figure 3: More Than Ogive

eBook PSP | Statistics 11

1.3. Data Classified

Data is often described as ungrouped or grouped. Ungrouped data which is also known as
raw data is data that has not been placed in any group or category after collection. Grouped
data is data that has been organized into groups from the raw data. The grouped data has
been bundled together in the categories. Frequency tables can be used to show this type of
data. For easy understanding, we can classify it into 3 types of data:

1) Ungrouped data without a frequency distribution.

47 35 37 32 38 39 36 34 35

2) Ungrouped data with a frequency distribution.

Table 4: Ungrouped Data with A Frequency Distribution

Score, x Frequency, f

15
2 11
39
4 15
5 10

3) Grouped data.

Table 5: Grouped Data

Marks Frequency, f
30-39 7
40-49 5
50-59 11
60-69 8
70-79 14
80-89 6
90-99 2

12 eBook PSP | Statistic

2. Central Tendency

Central tendency is a descriptive summary of a data set through a single value that reflects
the center of the data distribution. Generally, the central tendency of a data set can be
described using the following measures:

1.Mean The mean of a set of data values is the sum of all of the data values
1 divided by the number of data

2.Mode 2 The mode of a set of data values is the values that occurs most often
3.Median A set of data values is the middle value of the data set when it has been

3 arranged in ascending or descending order.

There are two ways to calculate central tendency, that is:
✓ Using formula (for all types of data – ungrouped and grouped data)
✓ Using graph (only for grouped data to find median and mode)

2.1. Mean, Mode and Median using Formula

Calculate Mean,

Table 6: Mean Formula

DATA CLASSIFIED FORMULA

Ungrouped data without a frequency ∑
distribution =

Ungrouped data with a frequency ∑
distribution = ∑
x = data value
Grouped data
f = frequency
∑

= ∑
x = mid-point

f = frequency

eBook PSP | Statistics 13

Calculate Mode

DATA CLASSIFIED Table 7: Mode Formula
FORMULA

Ungrouped data without Find the highest frequency / often number
a frequency distribution o If there is no repeated number in the set, there is no

mode.
o It is possible that a set has more than one mode.

Ungrouped data with a Look at the value of frequency and pick the highest data
frequency distribution from the table.

1. Identify Modal Class

o Modal Class = pick the highest frequency

2. Based on the modal class, insert the value into the

formula:

= 0 + ( 1 1 2)
+

Grouped data Lm0 = lower boundary of the class

containing the mode

C = size of the class interval

d1 = differences between frequency of the class

containing the mode with frequency of the class

before the mode

d2 = differences between frequency of the class

containing the mode with frequency of the class

after the mode.

14 eBook PSP | Statistic

Calculate Median

DATA CLASSIFIED Table 8: Median Formula
FORMULA

Ungrouped data without 1. Arranged in an ascending order
a frequency distribution 2. Find out the located of median at the center of data.

o If the number of data is odd, the median is the middle

value
o If the number of values in the data set is even, then

the median is the average of the two middle values.

Ungrouped data with a In general:
frequency distribution
Median = 1 (n +1)
2

1. Construct a table with cumulative frequency column
2. Determine the middle data value from the cumulative

frequency column.

1. Construct a table with cumulative frequency column

2. Identify Median class

o Median class = (refer cumulative frequency)


3. Based on the modal class, insert the value into the

formula:
(2
= + −
)

Grouped data

LM = lower boundary of the class containing the
median

N = total frequency
C = size of the class interval
F = Cumulative frequency before the class

containing the median

f m = frequency of class containing the median

eBook PSP | Statistics 15

2.2. Mode and Median using Graph

Calculate Mode

To determine mode using graph, we use histogram.
Steps:

01 Identify Modal Class

Modal Class = pick the tallest bar from the histogram

02 Join the top corners of the modal rectangle with the immediately next

corners of the adjacent rectangles. The two lines must be cutting each

03 Mark the point of the intersection of the two lines

04 Draw a perpendicular line from point onto the x-axis.

05 The point where the perpendicular will meet the x-axis will give the

mode value.

16 eBook PSP | Statistic

Example 3

Find Mode from Graph

Frequency 35
30
25 0.5 2.5 4.5 6.5 8.5 10.5 12.5
20
15 Mode =7.16 Class Boundaries
10 Figure 4: Determine Mode from Graph

5
0

0

Calculate Median Type equation here.

To determine median using graph, we use ogive.
Steps:

01 Find the value of (N= total frequency)


02 Mark it on the Cumulative frequency scale on y-axis.

03 Draw a perpendicular line from the value to ogive line and mark the point.

04 Draw a perpendicular line from the point on the ogive line to the x-axis.

05 The point where the perpendicular will meet the x-axis will give the

median value. (example: median= 122.42)

eBook PSP | Statistics 17

Example 4

The systolic blood pressure readings (mmHg) of 200
ramdomly selected polytechnic students

220

200

180

160

140 200
2 = 2 = 100

120

Cumulative Frequency100

80

60

40

20

0 104.5 119.5 134.5 149.5 164.5 179.5
89.5

= 122.42 Class Boundaries

Figure 5: Determine Median from Graph

18 eBook PSP | Statistic

3. Measure of Dispersion

The measures of central tendency are not adequate to describe data. Two data sets can have
the same mean but they can be entirely different. Thus to describe data, one needs to know
the extent of variability. This is given by the measures of dispersion. Mean Deviation, Variance
and Standard Deviation are examples of three measures that use the concept of dispersion.

Mean Deviation Variance Standard Deviation

• • The mean deviation • Variance is a numerical • Standard deviation is a
is a statistical data of value that measure of dispersion
measure that is used describes the variability of observations within
to calculate the of observations from its a data set.
average deviation arithmetic mean.
from the mean value • Symbol - σ or S
of the given data set. • Symbol - σ2 or S2 • It is the square
• It indicates on how far
• • Symbol - E root of the Variance
individuals in a group • It indicates on how
are spread out.
much observations of a
data set differs from its
mean.

Figure 6: Mean Deviation, Variance and Mode

eBook PSP | Statistics 19

3.1. Mean Deviation, Variance and Standard Deviation by Formula

Table 9: Mean Deviation, Variance and Standard Deviation Formula

Types Mean Deviation Variance Standard
Deviation

 (x − x) ( )S2 =
Ungrouped Data 2
i.e: E=
3, 5, 7, 3, 5, 4, 3 n x−x

x = data n var iance
x = data
x = mean
n = total of data x = mean
n = total of data

Ungrouped data with a  (x − x) f ( )S2 2
E = x−x f
=
frequency distribution f f

i.e x = data (score)
x = mean
Score f x = data (score)
 f = sum of
53 x = mean var iance
frequency
10 5  f = sum of

15 10 frequency

20 15

Grouped Data  (x − x) f ( )S2 2
i.e E = x−x f
=
Score f f
1-5 f
6-10 3 x = mid-point x = mid-point var iance
11-15 5 x = mean
16-20 10 x = mean
15  f = sum of
 f = sum of
frequency
frequency

20 eBook PSP | Statistic

Example 5

Ungrouped Data

3, 5, 11, 4, 8, 14, 21, 19, 5

Based on a given data, calculate Mean, Mode, Median, Mean Deviation, Variance and

Standard Deviation.
Solution:
Given n = 9
a) Mean

∑
=

3 + 5 + 11 + 4 + 8 + +14 + 21 + 19 + 5
= 9

90
= 9 = 10

b) Mode

3, 5, 11, 4, 8, 14, 21, 19, 5

The highest frequency
Therefore mode = 5

c) Median
Arrange the data values in order from the lowest value to the highest value:
3 4 5 5 8 11 14 19 21

= 1 ( + 1)

2

= 1 (9 + 1)

2

= 5 ℎ

The fifth data value, 8, is the middle value in this arrangement.
Therefore median= 8

eBook PSP | Statistics 21

d) Mean Deviation Table 10: Example 5 Mean Deviation
= 10
| − | = | − | | − | = | − |

3 7 49
5 5 25
11 1 1
4 6 36
8 2 4
14 4 16
21 11 121
19 9 81
5 5 25
∑ = 50 ∑ = 358
∑|( − )|
=

50
=9
= 5.56

e) Variance

2 = ∑( − )2


358
=9

= 39.78

f) Standard deviation
= √
= √39.78
= 6.31

22 eBook PSP | Statistic

Example 6

Ungrouped data with a frequency distribution
Calculate the mean deviation, variance, and standard deviation for the following ungrouped
data:

Table 11: Example 6
2345678

1447743

Solution: Table 11a: Solution Example 6

CF |( − )| ( − )
21 10.89
34 1 2 3.3 21.16
44 6.76
57 5 12 9.2 0.63
67
74 9 16 5.2 3.43
83 11.56
16 35 2.1 21.87
∑ = 30
20 42 4.9 2

23 28 6.8 ∑( − )
= 76.3
24 24 8.1

∑ = 159 ∑|( − )|
= 39.6

a) Mean *Median
12.5th position
= ∑
∑ data in CF

159
= 30

= 5.3

eBook PSP | Statistics 23

b) Mode
The highest frequency is 7. There are 2 data with the same frequency. Therefore, the
value of mode are 5 and 6.

c) Median

Construct a table with cumulative frequency , CF column

= 1 (24 + 1) = 12.5 – refer the data in CF column
2

Therefore The median is 5

d) Mean deviation,
∑|( − )|

= ∑
39.6

= 30
= 1.32

e) Variance,

2 = ∑( − )2
∑

76.3
= 30

= 2.54

f) Standard deviation,
= √
= √2.54
= 1.59

24 eBook PSP | Statistic

Example 7

Grouped Data

Find the mean, median and mode of the following data:

Table 12: Example 7

Score Frequency, f

10 – 14 5

15 – 19 11

20 – 24 14

25 – 29 7

30 – 34 3

Solution:

Table 12a: Solution Example 7

Score Class Frequency, f Mid- point, x fx Cumulative
Boundary, 5 Frequency,
10 – 14 12 60
15 – 19 CB 17 187 CF
20 – 24 22 308 5
25 – 29 9.5 – 14.5 27 189
30 – 34 32 96 16
14.5 – 19.5 11
 fx = 840 30
19.5 – 24.5 14
37
24.5 – 29.5 7
40
29.5 – 34.5 3

 f = 40

a) Mean

= ∑
∑

840
= 40

= 21

eBook PSP | Statistics 25

b) Mode

Mode class = (20 – 24) (highest frequency: 14)

0 = + ( 1 ) Lm0 = 19.5
d1 = 14-11=3
1+ 2 d2 =14-7=7
C =5
3
= 19.5 + (3 + 7) 5

= 21

c) Median

Median class = 40 = 20
2

Median,

= + − LM = 19.5
(2 ) n=40
F=16
= 19.5 + (4201−416) 5 fm=14
C=5

= 20.93

d) Mean deviation

Table 12b: Solution for Mean Deviation

Score Frequency, f Mid- point, |( − )| ( − )
10 – 14 5 x 45
12 405
176
15 – 19 11 17 44 14
252
20 – 24 14 22 14 363

25 – 29 7 27 42

30 – 34 3 32 33 ∑( − )
=
 f = 40 ∑|( − )|

=

26 eBook PSP | Statistic

Mean deviation,
∑|( − )|

= ∑
178

= 40
= 4.45

e) Variance

2 = ∑( − )2
∑

1210
= 40

= 30.25

f) Standard deviation

= √

= √30.25
= 5.5

4. Quartiles, Deciles and Percentiles

Quartiles, Deciles and Percentiles apply the same concept as the median in Statistics. Median
divides a set of data into two equal parts. In the same way, there are also certain other values
which divide a set of data into four, ten or hundred equal parts and that values are referred as
quartiles, deciles, and percentiles respectively

Quartile Decile Percentile

⚫ Symbol : Q ⚫ Symbol : D ⚫ Symbol : P
⚫ Divide the sorted ⚫ Divide the sorted ⚫ Divide the

data into 4 equal data into 10 sorted data into
parts. equal parts. 100 equal parts.

Figure 7: Summary of Quartiles, Deciles and Percentiles

eBook PSP | Statistics 27

4.1. Calculate Quartile, Decile and Percentile

Calculate Quartile [Q], Decile [D] and Percentile [P]

There are two methods in measuring quartile, deciles and percentile which are by using
graphical method and by formula.

General Steps:
1. Arrange the data in ascending order (for ungrouped data)
2. Recognize or determine the location of value quartile, deciles and percentile

Quartile ,Q = ( +1) k = 1,2,3 or 4 ; N = total data
Deciles ,D 4 k = 1,2,3,4,5,6,7,8,9 ; N = total data
Percentile ,P k = 1,2,3,…….., 99 ; N = total data
= ( +1)
10

= ( +1)
100

Below are the steps to find quartile, deciles and percentile by graph:
Steps:

01 Draw an ogive graph

02 Plot the location value of quartile, deciles and percentile on y-axis
03 Draw a horizontal line from y-axis so that it touch the ogive curve and

Draw vertical line from that point down to the x-axis.

04 Read the value of quartile, deciles and percentile on x – axis.

28 eBook PSP | Statistic

Example 8

The Table 13 shows the distribution age of the contestant in a marathon.
Table 13: Example 8

Age Frequency
15-24 7
25-34 13
35-44 16
45-54
55-64 10
4

By using graphical method, find the value of:
a. Quartile 2 , 2
b. Decile 7, 7
c. Percentile 25, 25

Solution:

o By graph
▪ Draw an ogive.

Find location: Q2 D7 P70

2( 50) 7( 50) 25( 50)

2 = 4 7 = 10 25 = 100
= 25 = 35 = 12.5

Answers: 43.9 28.7

(read from 38.3

graph)

eBook PSP | Statistics 29

Age of constestant

55

50

45

40

Cumulative Frequency 35

30

25

20

15

10

5

0 24.5 34.5 44.5 54.5 64.5
14.5
25=28.7 2=38.3
7 = 43.9 Class Boundaries

Figure 8: Quartile, Decile and Percentile from Graph.

o To find quartile, deciles and percentile by formula:

Table 14: Quartile, Decile and Percentile Formulae

Quartile Decile Percentile

= + − ) = + − ) = + − )C

( ( 10 (100


L = Lower Boundaries F = Cumulative frequency before the class
N =Total of Data containing the Q,D,P .

C = size of the class interval f _ K = frequency of class containing the Q,D,P

30 eBook PSP | Statistic

Example 9

The Table 15 shows the distribution age of the contestant in a marathon.
Table 15: Example 9

Age Frequency
15-24 7
25-34 13
35-44 16
45-54
55-64 10
4

By using formula, find the value of
a. Quartile 2
b. Decile 7
c. Percentile 25

Solution:
o By formula

Data Add one column (Find Cumulatif Frequency, CF)

Find Age Frequency CF
location of 15-24 7 7
the data: 25-34 13 20
35-44 16 36
45-54 46
55-64 10 50
4

eBook PSP | Statistics 31

Q2 D7 P25

2 = 2( 50) 7 = 7( 5) 25 = 25( 50)
4 10 100

= 25 = 35 = 12.5

Calculation

2 = 34.5 + 2(50) − 20) (10) 7 = 34.5 + 7(50) − 20) (10) 25 = 24.5 + 25(50) − 7) (10)

(4 ( 10 ( 100
16 16 13

= 37.6 =46.6 =28.7

Answer 37.6 43.9 28.7

32 eBook PSP | Statistic

Tutorial Exercise

1. Identify the mode from the given data.
989 , 892 , 714 , 668 , 1000 , 668 , 998 , 777 , 899

2. Determine the variance and standard deviation for the set of data
8, 10, 9, 7, 4, 6, 2, 8

3. Given a set number 8 , 6 , 16 , 12 , 13 , 5 , 4 dan 8. Find mean deviation for the set of
number.

4. Given that the mean of the data ( − ), ( − ), ( + ) + + ( − ) is 5.
Find the value of . Then, find
a. Median
b. Variance
c. Standard deviation .

5. Given the mean of the Data ( − ), ( + ), ( + ), ( − ), ( + ) is 12.2
a. Find the value of .
b. Then find the variance and standard deviation of the data.

eBook PSP | Statistics 33

6. The recorded data is the number of residents living in each unit of an apartment. Give
your answer to 2 decimal places.

Class Frequency
2 3
3 2
4 5
5 7
6 6
7 4
8 3

Calculate:
a. Mean
b. Mode
c. Median
d. Mean deviation
e. Variance
f. Standard deviation
g. Percentile 10
h. Decile 4
i. Quartile 3

34 eBook PSP | Statistic

7. The thickness of 200 samples of steel plate are measured and the results ( in

milimetress) are as follow:

Thickness (mm) Frequency

6.2-6.4 14

6.5-6.7 46

6.8-7.0 54

7.1-7.3 42

7.4-7.6 38

7.7-7.9 6

From the data above, construct a histogram and find the mode.
8. The tables shows the police reports of the accident that are occurred .

Numbers of accident /day Days
0-4 5
5-9 37
10-14 87
15-19
20-24 121
25-29 77
30-34 42
35-39 21
10

Calculate:
a. Mean
b. Mode
c. Median
d. Decile 7
e. Percentile 25
f. Quartile 2

eBook PSP | Statistics 35

9. The length of 20 pieces of wood are measured in cm and the result are as follow.

11 12 16 13 15

21 22 12 25 12

21 19 17 20 24

14 8 9 20 11

a. Construct a frequency table using a class interval of 4 beginning with the class 7-
10

b. By using a scale of 2 cm to 4 cm for the length of woods an x – axis and 2 cm to 2
pieces of woods on the y- axis , draw ogive for the data.

c. Find the variances and standard deviation of the data

10. A survey is conducted in a supermarket on a sample of 150 customers who have

purchased a brand of perfume reveals the following age distribution as following

table.

Age ( years ) Number of customer

15 – 19 9

20 – 24 16

25 – 29 27

30 – 34 44

35 – 39 42

40 – 44 10

45 – 49 2

Total 150

Draw a histogram to represent the above data. Then find the mode.

36 eBook PSP | Statistic

Answers:

1. 668
2. Variance = 6.1875 , standard deviation = 2.487
3. Mean deviation , E= 15.75
4. a. p =3

b. median =0
c. s 2 = 66 , s = 8.12
5. a. x=5
b. variance =74.96 , standard deviation =8.658
6. a. Mean =5.17
b. Mode =5
c. Median =5
d. Mean deviation=1.74
e. Variance=4.03
f. Standard deviation= 2.01
g. Percentile 10 =2
h. Decile 4 =5
i. Quartile 3= 6

7. 6.87
8. a. Mean = 18.1

b. mode = 16.68
c. median = 17.43
d. D7=21.45
e. P25=12.83
f. Q2=17.43
9. c. variance=4.81
10. Mode = 33.97

eBook PSP | Statistics 37

REFERENCES

Bird,J.(2017). Higher Engineering Mathematics ( 7 th Edition). UK. Routledge
E.Walope,R.H Myers,R.(2016) Probability & Statistics for Engineers & Scientists,Mylab Statistics
Update ( 9th Edition) . UK Pearson.
Bird,J.(2017). Higher Engineering Mathematics 8th Edition . Routledge

https://www.thefreedictionary.com/mean+deviation
https://keydifferences.com/difference-between-variance-and-standard-deviation.html
https://mathlibra.com/calculation-of-percentiles-for-ungrouped-data/
https://www.onlinemath4all.com/deciles.html
https://youtu.be/FFYvNrRGVOo

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