FlipWorkbook DUM30172

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

dV = dV × dr
dt dr dt

dV

dr = dt = 10π = − 5 cm s −1
dt dV 32
4π(4)2

dr

Example 2.3.5 :
Boyle’s Lan connect the pressure P, units of a gas with its volume, V units. In
the formula pv = m , where m is a constant

a. Find dp in terms of m and V.
dv

b. If p is increased at a constant rate of 1 m unit per hour, at the
3

distant when p = 1 m units . Calculate the rate of change in the
9

volume of the gas.

Solution :

a. pv = m

p = m = mv-1
v

dp = −mV −2 = − m
dV V2

b. dp = 1 m unit h−1, p = 1m
dt 3 9

∴1m=m⇒ V =9
9v

dp = dp × dV
dt dV dt

Therefore the rate change in the volume of the gas is

dV = dp = 1m = 1 = −27
dt dt 3 3
dp m −  1 2
dV − V2 3

The rate of change in the volume of the gas is -27 m3 h-1

20

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

EXERCISE 2.3

1. Given that y = 2x2 − x , when x increase at the rate of 0.3 cm/s.
Find the rate of change of y when x = 2 .

2. Given that y = 4x2 − x , when x decrease at the rate of 0.2 cm/s.
Find the rate of change of y when x = 2 .

3. The area of a circle increase at the uniform rate of 4 πcm 2 s −1 . Calculate the rate of
5

increase of the radius of this circle where its radius is 8 cm.

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DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

4. The area of a square increase at the rate of 20 cm2s −1 .
Find the rate of change in the length of its side when the area is 25 cm2.

5. Given that V = πr3 and r cm decrease at a rate of 0.5 c m / s .
Find the rate of change of V when r = 3 cm .

6. The volume of a sphere with radius r cm increase at a rate of 20 cm3 / s . Find the
rate of r when r = 1 cm .

22

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

7. The side of a cube decrease at the rate of 0.2 cm / s . Find the rate of change of the
volume of cube when the side is 5 cm.

Answers: 1. 2.1, 2. -3, 3. 1 cms-1, 4. 2 cms-1, 5. -42.417 cm3s-1, 6. 1.591 cms-1, 7. -15 cm3s-1

20

23

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

After completing the unit, students should be able to:
• Solve problems involving small increment.

UNIT
LEARNING

2.4 SMALL CHANGE
2.4.1 SMALL CHANGES IN QUANTITIES

If δy is a small change in y as the result of a small change δx in x, we have
the approximation

dy ≈ δy
dx δx
δy ≈ dy × δx

dx

This result can be used to find the small changes in y (δy ), provided that δx

is small. The smaller the value of δx the more accurate the approximation.

To solve problems on small changes we have to write the small changes that
has to be calculate in the form of mathematical

Example 2.4.1.1 : dy and hence , find the change in value
Given the curve y = 2x2 + 3x. Find dx

of y when x increase from 2 to 2.01

Solution :

y = 2x2 + 3x

dy = 4x + 3
dx

Given x = 2 and δx = 2.01 – 2 = 0.01

δy ≈ dy × δx
dx

≈ [4x + 3] x 0.01

≈ [4(2) + 3] x 0.01

≈ 0.11

24

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

Example 2.4.1.2:
If pv = 15 , find the approximate change in p when v change from 15 to 14.6.

Solution :

pv = 15 v = 15

p = 15 = 15v −1 δv = 14.6 – 15 = -0.4
v
δp ≈ dp x δv
dp = −15v −2 = − 15 dx
dv v2

dp ≈ δp
dv δv

≈ − 15 x (−0.4)
v2

≈ − 15 X(− 0.4)
(15)2

≈ 0.027

Example 2.4.1.3 :
The radius of a circle increase from 4 cm to 4.002 cm. Find the corresponding
change in the area of the circle.

Solution : r=4 δr = 4.002 – 4 = 0.002
A = π r2
dA = 2 π r δA ≈ dA × δr
dr dr
dA ≈ δA
dr δr
≈ 2 π r x 0.002
≈ 2 π (4) x (0.002)
≈ 0.016 π cm

25

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

Example 2.4.1.4 :
The base of a closed rectangular box is a square of side x cm and its
height is 3x cm . If the volume of box increasing from 81 cm3 to 83 cm3 . Find

the corresponding change in x.

Solution :

V = x × x × 3x = 3x3 x
x
V = 81 ⇒ 3x3 = 81
x = 3 cm 3x

dv = 9x2
dx
dv ≈ δv
dx δx

δx ≈ δv = 2 ≈ 0.0247
dv 9(3)2

dx

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DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

2.4.2 APPROXIMATION

The approximation value of y is given by:

y New = y original + δy

= y original + dy × δx
dx

Example 2.4.2.1:

Given y = x3, find dy when x = 3. Hence find the approximation value of
dx

(3.002)3.

Solution :

y = x3 therefore y = (3)3 = 27

dy = 3x2 x = 3 and δx = 3.002 – 3 = 0.002
dx

when x = 3, dy = 3(3)2 = 27
dx

ynew = yoriginal + δy

(3.002)3 = 27 + (27 x 0.002)
= 27.054

27

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

Example 2.4.2.2:

Given y = 1 , find dy when x = 6 hence find the approximation value of
x2 dx

1 . Correct to 2 significant figures.

(5.998 )2

Solution :

y = 1 = x−2 ⇒ yoriginal = 1 = 1 = 1
x2 x2 62 36

dy = −2x − 3 = − 2 given x = 6 and δx = 5.998 – 6 = -0.002
dx x3

when x = 6, dy = − 2 =− 2 = −0.009
dx 216
(6)3

ynew = yoriginal + δy

1 =1 + (− 0.009 × −0.002) = 0.0279 ≈ 0.03
36
(5.998)2

Example 2.4.2.3 :

Given y 1 dy when x = 9. Hence find the approximation value of
dx
= 2 x 2 , find

2(9.01)12

Solution :

1

y = 2x 2

dy = −1 given x = 9 and δx = 9.01 – 9 = 0.01

dx x2

when x = 9, dy = 1 = 1
dx 3
(9)21

ynew = yoriginal + δy

2(9.01)2 = 6 +  1 × 0.01 = 6.0033

3 

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DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

EXERCISE 2.4

1. Given the curve y = x3 +5x2 . Find dy , hence, find the corresponding change in y
dx
when x decreases from 3 to 2.998 .

2. The radius of circle increase from 4 to 4.02 . Find the approximation change in its
perimeter.

3. The height of a cone is fixed at 8cm . If the radius of the base of the cone increases
from 5 to 5.25 cm, find the approximation increase in the volume of the cone.

Answers: 1. 3x2 + 10 x , -0.114, 2. 0.13, 3. 20.9

29

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

4. The volume v cm3 of a liquid in a container is given by v = h(5 + 6h), where h cm is

the depth of the liquid. Find
dv
a) dh in terms of h.

b) the approximation change in the volume when depth decreases from

7 to 6.98 cm.

Answers: a) 5 + 12h, b) -1.78

5. Given y = 2 x 2 . Find the value of dy when x = 4 . Hence find the approximation
dx

value of : a) 2(4.01)2

b) 2(3.98)2

Answers: a) 32.16, b) 31.68

30

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

6. Determine the approximate value of the following.

a) 1
3 125.02

b) 5 31.99

c) (0.98)41

Answers: a) 0.19999, b) 1.999875, c) 0.995

31

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

UNIT After completing the unit, students should be able to:
LEARNING • Determine velocity and acceleration by using first order
OUTCOMES differentiation and second order differentiation.

• Solve problems involving motion of a straight lint

2.5 MOTION OF A STRAIGHT LINE
2.5.1 Definitions
One of the applications of rates of change is those that involved motion in a straight line. If a
particle P, moves in a straight line and its position is given by the displacement function

s(t), t ≥ 0, then ,The velocity of P, at time t, is given by
v(t) = s′(t)

{the derivative of the displacement function}
and the acceleration of P, at time t, is given by

a(t) = v′(t) = s′′(t)

{the derivative of the velocity function}

Note : s(0), v(0) and a(0) give us the position, velocity and acceleration of the particle at

time t=0, and these are called the initial conditions.
We can use sign diagrams to interpret
• where the particle is located relative to O
• the direction of motion and where a change of direction occurs,
when the particles speed is increasing/decreasing

Table 2.1 : Signs of s(t )

s(t) Interpretation

= 0 P is at O
> 0 P is located to the right of O
< 0 P is located to the left of O

Table 2.2: Signs of v(t )

v(t) Interpretation

= 0 P is instantaneously at rest
> 0 P is moving to the right
< 0 P is located to the left

32

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

Example :

The position of a particle is given by s = f (t) = t3 − 6t 2 + 9t , where t is measured in

t seconds and s in metres. Find the acceleration at time t .
What is the acceleration after time 3 s ?

Solution:

s = f (t) = t3 − 6t2 + 9t

→v(t) = ds = 3t 2 −12t + 9
dt

→ a(t) = d 2s = dv = 6t −12
dt 2 dt

a(3) = 6(3) −12 = 18 −12 = 6

Interpretation: The acceleration after time 3 sec

is 6 m/s2

Example :

A particle moves in a straight line with position, relatives to some origin 0 given by
s(t) = t 2 − 5t + 3 where t is the time in seconds.

(a) Find an expression for the particle’s velocity
and acceleration

(b) Find the initial condition and describe the
motion at this instant

(c) Describe the motion of the particle at
t = 2 seconds

Solution:

(a) s(t) = t 2 − 5t + 3

s′(t) = v(t) = 2t − 5 cms−1

v′(t) = a(t) = 2 cms−2

(b) When t = 0
s(0) = 3cm

v(0) = −5 cms-1

a(0) = 2 cms-2

Particle is 3cm to the right of origin 0 , moving to the left with a velocity of 5cms-1 and the
velocity is increasing at a rate of 2cms-1 ,each second.

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DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 2: APPLICATION OF DIFFERENTIATION

(c) When t = 2
1. Find the acsc(e2l)er=at2io2 n−a5n(2d)v+e3lo=cit−y3ocfma lorry which moves in a straight line if

s = 3t 4 + 7tv5 (+2)2t=+28(2.) − 5 = −1cms-1

a(2) = 2cms-2
Particle is 3cm to the left of 0 , moving to the left with a velocity of 1cms-1
and
2. An objtehcet vmeolovceitiynias isntcrareigahstinligneatwaithrapteosoitfio2ncmgivs-e1n,ebaychs(st)ec=otn2d−. 5t + 2 cm from
the origin, t ≥ 0 , t in seconds.
(a) Find an expression for the particle’s velocity and acceleration
(b) Find the initial condition and describe the motion at this instant
(c) Describe the motion of the particle at t = 2 seconds

3. A particle moves along a straight line such that the displacement, s metres, from
a fixed point O is given by s = 5 t3 −15t 2 + 70 where t is the time in seconds
3
after passing through O . Find

(a) the velocity and the acceleration of the particle,
(b) the maximum velocity of the particle,
(c) the value of t when the acceleration is zero.

34

UNIT 3 DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY
STATISTICS AND
PROBABILITY

INTRODUCTION

OVERVIEW

Statistics is a mathematical body of science that pertains to the collection, analysis, interpretation
or explanation, and presentation of data, or as a branch of mathematics. The early writings on
statistical inference date back to Arab mathematicians and cryptographers, during the Islamic
Golden Age between the 8th and 13th centuries. Al-Khalil (717–786) wrote the Book of
Cryptographic Messages, which contains the first use of permutations and combinations, to list all
possible Arabic words with and without vowels. In his book, Manuscript on Deciphering
Cryptographic Messages, Al-Kindi gave a detailed description of how to use frequency analysis to
decipher encrypted messages. Al-Kindi also made the earliest known use of statistical inference,
while he and later Arab cryptographers developed the early statistical methods
for decoding encrypted messages. Ibn Adlan (1187–1268) later made an important contribution, on
the use of sample size in frequency analysis. Today, statistics is widely employed in government,
business, and natural and social sciences.

CONTENTS CONTENTS

3.1 Definitions and Terminologies
3.2 Data in Tabular Form
3.3 Data in Graphical Form
3.4 Measure of Central Tendency
3.5 Measure of Dispersion
3.6 Concepts of Probability
3.7 Processing Data by Appropriate Tools

1

UNIT LEARNING DUM 30172 ENGINEERING MATHEMATICS 3
OUTCOMES UNIT 3: STATISTICS AND PROBABILITY

After completing the unit, students should be able to:
• Differentiate between discrete and continuous, grouped and
ungrouped data

3.1 DEFINITIONS AND TERMINOLOGIES

STATISTICS
Statistics is the study of making sense of data. Statistics represents scientific procedures and methods
for collecting, organizing, summarizing, presenting, and analysing data. Statistics help us to organize
numerical information in the form of tables, graphs, and charts. Statistics also helps us to understand
statistical techniques and underlying decisions that effect our lives and well-being and then make
informed decisions.

Example: Real life examples of statistics

Manufacturing Investing

Statistics is often used in manufacturing to Investors use statistics and probability to assess
monitor the efficiency of different processes. how likely it is that a certain investment will pay off.

Example: Example:

Manufacturing engineers may collect a random A given investor might determine that there is a 5%

sample of widgets from a certain assembly line chance that the stock of company A will increase

and track how many of the widgets are 100x during the upcoming year. Based on this

defective. They may then perform a normal probability, they’ll decide how much of their portfolio

distribution test to determine if the proportion of to invest in the stock.

widgets that are defective is lower than a certain

value that is considered acceptable.

Medical Studies Sales Tracking

Statistics is regularly used in medical studies to Retail companies often use descriptive statistics like

understand how different factors are related. the mean, median, mode, standard deviation to

track the sales behavior of certain products.

Example:

Medical professions often use correlation to Example:

analyze how factors like weight, height, smoking The results of descriptive statistics will give

habits, exercise habits, and diet are related. companies an idea of how many products they can

If a certain diet and overall weight is found to be expect to sell during different time periods and

negatively correlated, a medical professional allows them to know how much they should keep in

may recommend the diet to an individual who inventory.

needs to lose weight.

2

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

POPULATIONS
Populations is the entire group that you want to study. It can mean a group containing elements of
anything you want to study , such as people, objects, events, organizations, animals, items, etc.

SAMPLE
Sample is the part of population that you will collect data from.

Example : Populations vs Sample Sample
Population

Advertisements for IT jobs in the Malaysia The top 50 search results for advertisements for
IT jobs in the Malaysia on January , 2022

Undergraduate students in the Malaysia 300 undergraduate students from UiTM who
volunteer for psychology research study

Engineering students in TVET Mara 250 students from Kolej Kemahiran Tinggi Mara
Kuantan

Students of Kolej Kemahiran Tinggi Mara Kuantan 25 students from DKA, DKI, DKN, DKP and DKQ

DATA
Data is a collection of information or facts, such as numbers, words, measurement, observations, etc.

Data can be categorized into two types:

Types of data Description

Anything that can be counted or measured. It refers to numerical data. There are

Quantitative data two types of quantitative data:
 Discreate data: data are obtained by counting and only whole numbers
are possible. Such as, the number of stamps sold by a post office in equal

periods of time.
 Continuous data: data are obtained by measurement. It is not fixed but

has a range of data. Such as, the heights of a group of people.

Qualitative data It is descriptive, referring to things that can be observed but not measured. Such
as colors or emotions.

3

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

There are two types of data:

Types of data Examples

The data that is gathered for the first time during a study or experiment. It is not
sorted into categories, classified, or otherwise grouped. It is basically a list of
numbers.

Raw data or Example:
ungrouped data
i. The height of the six students is 152 cm, 179 cm, 180 cm, 149 cm,168 cm

and 172 cm.

ii. The scores of 60 students are shown in the following table:

Score 45 6 7 8

Numbers of students 5 4 15 23 13

Data that has been bundled together in categories.

Example:

Grouped data i. The mass of 50 mangoes is shown in the following table:

Mass(g) 42-48 49-55 56-62

Numbers of mangoes 10 19 21

4

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

TUTORIAL 3.1

1. Data are obtained on the topics given below. State whether the data are quantitative or

qualitative data.

No Data Quantitative Qualitative
data data

1 The numbers of website visitors

2 The post on social media about certain news

3 The revenue of the project in a year

2. Data are obtained on the topics given below. State whether the data are discreate or

continuous data.

No Data Discreate Continuous
data data

1 The numbers of days on which rain falls in a month for
each month of the year

2 The mileage travelled by each of a number of salesmen

3 The time that each of a batch of similar batteries lasts

4 The amount of money spent by each of several families on
food

5 The amount of petrol produced daily , for each of 31 days,
by refinery.

6 The amount of coal produced daily by each of 15 miners

7 The number of bottles of milk delivered daily by each of 20
milkmen

8 The size of 10 samples of rivets produced by a machine

9 The number of people visiting an exhibition on each of 5
days

10 The time taken by each of 12 athletes to run 100 meters.

11 The number of defective items produced in each of 10 one-
hour periods by a machine.

5

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

3. State whether the data given below are ungrouped data or grouped data.

No Data Ungrouped Grouped
data data

1 The data shows the distances (in meters, m) achieved by a
student in a long jump event: 3.85m, 3.33m, 3.45m, 3.73m.

The data shows the scores obtained by a group of participants

2 of certain course

Scores 1 2 3 4

Frequency 14 6 28 3

3 The data shows the numbers of goals scored by Ahmad in a
football games: 2, 4, 1, 2, 3.

The data shows the marks obtained by a group of students in

4 Test 1 61-70 71-80 81-90 91-100
Marks

Frequency 4 6 18 5

Data show the masses (in kg) of 80 castings
5 Mass (kg) 5.5-5.9 6.0-6.4 6.5-6.9 7.0-7.4 7.5-7.9

Frequency 4 13 21 23 19

Data show the lengths (in mm) of 50 copper plugs

6 Lengths (mm) 15-19 20-24 25-29 30-34 35-39

Frequency 9 15 5 8 13

The time (in hours) spent on swimming practices by 20 students

7 in a week 1 2 3 45
Mass (kg)

Frequency 3 9 3 3 2

6

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

UNIT LEARNING After completing the unit, students should be able to:
OUTCOMES • Represent a set of data in tabular form and interpret them.

• Analyze and interpret data in tabular form

3.2 DATA IN TABULAR FORM

TERMINOLOGIES The numbers of occasions of which any value occurs.
Frequency distribution Two types of frequency distribution are ungrouped data
and grouped data
Cumulative frequency distribution The sum of the frequency for a class and the frequencies
Class interval before that class.
Lower class limit The difference between the upper-class limit and the lower-
Upper class limit class limit.
Lower class boundary
Lowest value of the class interval
Upper class boundary
Class midpoint Highest value of the class interval
Size of class
The average of the lower limit of the class interval and the
upper limit of the previous class interval.

The average of the upper limit of the class interval and the
lower limit of the next class interval.

Average of the upper- and lower-class limits

The difference between upper boundary and lower boundary

Example: Ungrouped data frequency distribution table.

Given below are marks obtained by 20 students in Mathematics Quiz. Construct a frequency

distribution table.

19 17 19 15 12 19 12 15 19 15

15 19 12 15 19 12 17 15 19 17

Solution:

Marks obtained Tally Frequency
12 IIII 4
15 IIII I 6
17 III 3
19 IIII II 7

7

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

Example: Grouped data frequency and cumulative frequency distribution table

Table below shows the grades of 38 students in a Mathematics Test.
67 98 99 73 96 72 88 63 87 75 84 86 75 64 85 73 71 85 64
92 84 73 63 62 59 54 86 73 64 93 72 50 74 84 75 61 73 77
a. If the number of classes required is 5, determine the class interval.
b. Construct a frequency and cumulative frequency distribution table for the set of data above.

Solution:
a. The highest value is 99 , The lowest value is 50 , Class required is 5

b. Tally Frequency Cumulative frequency
Marks obtained III 3 3
50 - 59 8 11
60 - 69 IIII III 13 24
70 - 79 IIII IIII III 9 33
80 - 89 5 38
90 - 99 IIII IIII
IIII

Example:

The data shows the frequency distribution for the masses in kilogram of 12 ingots. Determine the lower-

class limit, upper class limit, midpoint, lower class boundary, upper class boundary of a data. Then, find

the size of the class intervals.

Mass (in kg) Frequency Midpoint Lower class Upper class Cumulative

boundary boundary frequency

7.1 - 7.3 3 3

7.4 - 7.6 9 7.50 7.35 7.65 12

Lower Upper
class limit class limit

Size of class interval = Upper class boundary – Lower class boundary
= 7.35 -7.05
= 0.3
8

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

TUTORIAL 3.2
1. The mass in kilograms, correct to the nearest one-tenth of a kilogram of 60 bars of metal are as
shown. Form a frequency distribution of about 8 classes for these data. Then construct a cumulative
frequency distribution table.

39.8 40.3 40.6 40.0 39.6
39.6 40.2 40.3 40.4 39.8
40.2 40.3 39.9 39.9 40.0
40.1 40.0 40.1 40.1 40.2
39.7 40.4 39.9 40.1 39.9
39.5 40.0 39.8 39.5 39.9
40.1 40.0 39.7 40.4 39.3
40.7 39.9 40.2 39.9 40.0
40.1 39.7 40.5 40.5 39.9
40.8 40.0 40.2 40.0 39.9
39.8 39.7 39.5 40.1 40.2
40.6 40.1 39.7 40.2 40.3

9

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

2. The time taken in hours to the failure of 50 specimens of a metal subjected to
fatigue failure tests are as shown. From a frequency distribution, having about 8
classes for these data. Then construct a cumulative frequency distribution table:

28 22 23 20 12 24 37 28 21 25
21 14 30 23 27 13 23 7 26 19
24 22 26 3 21 24 28 40 27 24
20 25 23 26 47 21 29 26 22 33
27 9 13 35 20 16 20 25 18 22

10

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

3. The information given below refers to the value of resistance in ohms of a batch of
48 resistors of similar value. Form a frequency distribution for the data , having
about 6 classes . Then construct a cumulative frequency distribution table:
21.0 22.4 22.8 21.5 22.6 21.1 21.6 22.3 22.9 20.5
21.8 22.2 21.0 21.7 22.5 20.7 23.2 22.9 21.7 21.4
22.1 22.2 22.3 21.3 22.1 21.8 22.0 22.7 21.7 21.9
21.1 22.6 21.4 22.4 22.3 20.9 22.8 21.2 22.7 21.6
22.2 21.6 21.3 22.1 21.5 22.0 23.4 21.2

11

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

UNIT LEARNING After completing the unit, students should be able to:
OUTCOMES • Represent a set of data in graphical form.

• Analyze and interpret data in graphical form

3.3 DATA IN GRAPHICAL FORM

3.3.1 UNGROUPED DATA
Graphical representation is a way of analysing numerical data. It exhibits the relation between
data, ideas, information, and concepts in a diagram. It is easy to understand. There are
different types of graphical representation for ungrouped data. Some of them as follows:

Bar Chart

Horizontal bar chart A bar chart is the simplest way to represent a
set of ungrouped data. It is also known as a
bar graph. A bar chart is a graphical display
of data using rectangular bars of different
heights. The height of a bar represents the
frequency of the corresponding observation.
The space between two consecutive bars
must be the same. The bars ca be either
horizontal or vertical.

A bar chart should have:
 Title
 Axes
 Labels
 Intervals
 Bars

Vertical bar chart

12

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

Pie Chart

A pie chart is a circular graph that uses ‘pie
slices’ to show the relative sizes of data.
Various observations of the data are
represented by the sectors of the circle. The
whole circle represents the sum of the values
of all the components, with the total angle
formed at the center adding up to . The
angle at the center corresponding to a
particular observation component is given by:

If the values of the observations/ components
are expressed in percentage, then the central
angle corresponding to a particular
observation/ component is given by:

Example:

The number of issues of tools or materials from a store in a factory is observed for

seven, one-hour periods in a day, and the results of the survey are as follows:

Period 1 2 345 67

Numbers of issues 34 17 9 5 27 13 6

Present these data on a vertical bar chart.

Solution:

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DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

Example:

The distance in miles travelled by four salesmen in a week are as shown below.

Salesmen PQRS

Distance travelled miles 413 264 597 143

Use a horizontal bar chart to represent these data diagrammatically.

Solution:

Example:

The table below shows the types of medals won by the Malaysian Contingent at the SEA Games.

Medal Gold Silver Bronze

Percentage 20% 45% 35%

a) Calculate the angle of the sector for each type of medal
b) Construct a pie chart to illustrate the above data

Solution:

Angle of sector (gold) = 20% of 360° 126°
20 × 360°
= 100
72°
= 45% of 360°
Angle Of sector (silver) = 45 × 360°
100
= 162°
360°– 72°– 162° =
=
Angle of sector (bronze) = Types Of Medals

Bronze, 126° Gold, 72°

Silver, 162°
14

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

TUTORIAL 3.3 (a)

1. Build the frequency distribution table. Present this data on a vertical and a horizontal bar chart.

Volume (ml) Temperature (C)
100 20
150 35
200 15
250 30
300 28
350 13

15

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

2. The pie chart shows the monthly expenditure of En. Wan Sabri’s family
Monthly Expenditure

Miscellaneous Savings, 30°

Monthly Installment, 80°

Education , 90°

Food, 120°

a) Which category represents the biggest expenditure?
b) What fraction of the expenditure is for savings?
c) What percentage of the expenditure is for miscellaneous items?

16

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

3.3.2 GROUPED DATA

Grouped data is a data that is organized and arranged into a different classes or categories. There are
different types of graphical representation for grouped data. Some of them as follows:

Histogram

A histogram is constructed based on the
frequency distribution. There are some
features that must be considered when
constructing a histogram. The values of data
are plotted on the horizontal axis while the
frequencies are plotted on the vertical axis.
On the horizontal axis, each bar is labelled
from the lower-class boundary to the upper-
class boundary of each class. Do not leave
any space between the bars.

Steps to draw a histogram based on the
frequency table of a grouped data as follows:
1. Find the lower and upper boundary of

each class interval
2. Select a suitable vertical and horizontal

axis.
3. Draw a rectangle to represent each of the

class intervals with its width representing
the class and its height representing the
frequency
Frequency polygon

It is formed by joining the center points of the
tops of the rectangles of the frequency
histogram obtained with straight lines and
extended to include the two frequency columns
on the sides.

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DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

Steps to draw a frequency polygon as follows:
1. Find the midpoint of each class interval
2. Connect the midpoint of each class
interval.
3. Add one class with frequency 0 before
the first class and after the last class.
4. Can be drawn from a histogram.

Ogive

An ogive is also known as a cumulative
frequency graph. There are some features
that must be considered when drawing an
ogive. Data may also be presented on a
cumulative frequency distribution curve or
ogive. The cumulative frequency distribution
is a table giving values of upper-class
boundaries. Plot the graph of cumulative
frequency against the upper boundary of
each class. Draw a smooth curve that passes
through each point that is plotted. An ogive
can be drawn either as a less than ogive or a
more than ogive.

Example:

The following table shows the distribution of marks of 40 students in a Mathematics test.

Construct a histogram and frequency polygon for the above data.

Marks Numbers of students

30-39 6

40-49 8

50-59 12

60-69 8

70-79 5

80-89 1

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DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

Solution: Numbers of students Lower class boundary Upper class boundary
6 29.5 39.5
a. Histogram 8 39.5 49.5
12 49.5 59.5
Marks 8 59.5 69.5
30-39 5 69.5 79.5
40-49 1 79.5 89.5
50-59
60-69
70-79
80-89

A histogram for the distribution of marks of 40 students in a mathematics test

b. Frequency polygon

Marks Midpoint Numbers of Lower class Upper class
students boundary boundary
30-39 34.5 6
40-49 44.5 8 29.5 39.5
50-59 54.5 12 39.5 49.5
60-69 64.5 8 49.5 59.5
70-79 74.5 5 59.5 69.5
80-89 84.5 1 69.5 79.5
79.5 89.5

The frequency polygon for the distribution of marks of 40 students in a mathematics test
19

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

Example:

The masses of a sample of 90 apples are shown in the following table. Draw an ogive to present the

distribution of the masses of the oranges.

Mass(g) Number of apples
150-159 5
160-169 18
170-179 26
180-189 16
190-199 10
200-209 9
210-219 6

Solution: Number of apples Upper class Cumulative
boundary frequency
Mass(g) 0
5 149.5 0
140-149 18 159.5 5
150-159 26 169.5 23
160-169 16 179.5 49
170-179 10 189.5 65
180-189 9 199.5 75
190-199 6 209.5 84
200-209 219.5 90
210-219

An ogive for the distribution of the masses of the apples.
20

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

TUTORIAL 3.3 (b)

1. From the below frequency table, draw a histogram

Marks Scored 40 – 49 50 – 59 60 – 69 70 – 79 80 – 89 90 - 99
8 6
Frequency 4 5 7 10

21

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

2. The mass of 40 concrete slab were weighed (in kg):
77 83 79 74 84 72 81 78 68 78
80 71 91 62 77 86 87 72 80 77
76 86 67 71 67 83 94 64 82 74
66 75 76 82 78 88 66 79 74 64

a. Divide the set of values into seven equal width classes
b. Form the frequency distribution table and draw the histogram and frequency polygon
c. Form a cumulative frequency distribution and draw the corresponding ogive.

22

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

3. The diameter in millimeters of a reel of wire is measured in 48 places and the results are as shown.
2.10 2.29 2.32 2.21 2.14 2.22 2.28 2.18 2.17 2.20
2.23 2.13 2.26 2.10 2.21 2.17 2.28 2.15 2.16 2.25
2.23 2.11 2.27 2.34 2.24 2.05 2.29 2.18 2.24 2.16
2.15 2.22 2.14 2.27 2.09 2.21 2.11 2.17 2.22 2.19
2.12 2.20 2.23 2.07 2.13 2.26 2.16 2.12
a. Form a frequency distribution of diameters having about 6 classes.
b. Draw a histogram depicting the data
c. Form a cumulative frequency distribution table
d. Draw an ogive for the data.

23

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

4. The time taken in hours to the failure of 50 specimens of a metal subjected to
fatigue failure tests are as shown.

28 22 23 20 12 24 37 28 21 25
21 14 30 23 27 13 23 7 26 19
24 22 26 3 21 24 28 40 27 24
20 25 23 26 47 21 29 26 22 33
27 9 13 35 20 16 20 25 18 22

a. Form a frequency distribution of diameters having about 6 classes.
b. Draw a histogram depicting the data
c. Draw an ogive for the data.

24

UNIT LEARNING DUM 30172 ENGINEERING MATHEMATICS 3
OUTCOMES UNIT 3: STATISTICS AND PROBABILITY

After completing the unit, students should be able to:
• Calculate the mean, mode and median of ungrouped data and
grouped data.
• Analyze and interpret the measure of central tendency

3.4 MEASURE OF CENTRAL TENDENCY

Measure of central tendency

(Mean, Median and Mode)

Mean : Ungrouped data ∑ ∑x = x (raw data), x = xf
average of Grouped data ∑n
all numbers Ungrouped data with frequency table

f

x = ∑ xf
∑f

x is the mid-class value

a) Ranking the set-in ascending order of magnitude
b) Selecting the value of the middle number for sets containing an odd number

of values, or finding the value of the average of the two middles values for
sets containing an even number.

Median- a) Formula
the value
Grouped data Where;
of the
middle LB = Lower boundary
term. C = class width
FB = cumulative frequency before the class median

f m = frequency of class median

b) From the ogive
Ungrouped data Most commonly occurring value in a set of data.

a) Formula b) From the histogram

Mode Where. A B
L is the lower boundary
Grouped data value l u
l is AC, u is BD and
C D
c is the class width D
D

L mode

25

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

i. MEAN

Example:

Find the mean of the following set of data 24, 34, 25, 13

Solution:

x = ∑x
n

x = 24 + 34 + 25 +13 = 24
4

Example:

Given the following frequency distribution table, determine the mean,

Variable x 15 16 17 18
Frequency 5 4 8 10

Solution:

x = ∑ xf = (15 × 5) + (16 × 4) + (17 × 8) + (18 ×10)
∑f
5 + 4 + 8 + 10

= 16.85

Example:

The following are the scores of Form 4 Bestari students in their first Mathematics Quiz. Find the
mean of the score.

Scores 2 - 4 5 – 7 8 - 10

Frequency 13 20 12

Solution: Frequency Midpoint fx
Scores (f) (x) 3 x 13 = 39
2–4 6 x 20 = 120
13 2+4 =3 9 x 12 = 108
2 ∑fx = 267
5 – 7 20
5+7 =6
8 – 10 12 2

8 + 10 = 9
2

Total ∑N = 45

Mean = ∑ fx = 267 = 5.93
∑N 45

26

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

ii. MEDIAN

Example:

Given a set of data, find the median

5 7 8 4 5 10 8 9 7 5

Solution:
Rewriting the observation is ascending order of magnitude we have,

4 5 5 5 7 7 8 8 9 10

As there are even numbers of observations, the median is the average of the fifth and sixth
values that is 7.

Example:

Given a set of data, find the median 45
37 8

Solution:
Rewriting the observation is ascending order of magnitude we have,

34 578

As there are odd numbers of observations, the median is the middle values that is 5.

Example: From the ogive, estimate the median

Cumulative Frequency Median for grouped data - the value
in the middle of the set when the
4 data is arranged in numerical order.
0 Therefore, the median divides the
total number of data into two equal
3 parts. The total number of data can
5 be divided into four equal parts, or
quartiles.
3
0 The first quartile is a value that is
one-quarter from the total data
2 which has the value less than it.
5 The third quartile is a number
2 which is three-quarters from the
0 total data which has the value less
than it. The interquartile range is
1 the difference between the third
5 quartile and the first quartile.
1
0

5

0 59.5 69.5 79.5 89.5 99.5
49.5
Time
(seconds)

Solution: ½ of 40 students = ½ x 40 = 20

Median = 73.5

27

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

iii. MODE

Example:
Find the mode of the data below

a) 1 3 3 5 6 7 7 7 9

b) 1 1 3 3 5 5 7 7

c) 1 3 3 3 5 6 7 7 7 9

Solution: b) no mode c) mode = 3 and 7
a) mode = 7

Example:

Time (Class interval) Frequency
118-122
123-127 1
128-132 2
133-137 2
138-142 4
143-147 6
148-152 8
5
Determine the mode.

Solution:

Time (Class interval) Frequency

118-122 1
123-127 2
128-132 2
133-137 4
138-142 6
143-147 8
148-152 5

The class having the highest frequency is the sixth class with boundaries 143-147.

28

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

TUTORIAL 3.4

1. Determine the mean, median and mode for the sets given.
a. {3, 8, 10, 7, 5, 14, 2, 9, 8}
b. {26, 31, 21, 29, 32, 26, 25, 28}
c. {4.72, 4.71, 4.74, 4.73, 4.72. 4.71, 4,73, 4,72}
d. {73.8, 126.4, 40.7, 141.7, 28.5, 237.4, 157.9}

29

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

2. Data below shows the amount of tomato sauce (in ml) for 10 bottles that was produced by a manufacturer
within 1 hour.
118 115 121 122 121 119 120 123 115 118
Based on the above data, calculate
a) Mean
b) Median
c) Mode

30

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

3. The number of matches in a box for 60 boxes of a matches are as shown in the table below

No of matches in a box 45 46 47 48 49 50

No of boxes 3 11 15 18 9 4

Based on the above data, calculate
a) Mean
b) Median
c) Mode

31

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

4. The frequency distribution given below refers to the heights in centimeters on 100 people.

Heights (cm) Frequency
150-156 5
157-163 18
164-170 20
171-177 27
178-184 22
185-191 8

Determine:
a) Mean

b) Median

c) Mode

32

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

5. The gain of 90 similar transistors is measured and the results are as shown

Transistors Frequency
83.5-85.5 6
86.5-88.5 39
89.5-91.5 27
92.5-94.5 15
95.5-97.5 3

Determine:
a) Mean

b) Median

c) Mode

33

UNIT LEARNING DUM 30172 ENGINEERING MATHEMATICS 3
OUTCOMES UNIT 3: STATISTICS AND PROBABILITY

After completing the unit, students should be able to:
• Calculate the variance and standard deviation of ungrouped and
grouped data.

3.5 MEASURE OF DISPERSION

Measure of dispersion

( Range, Variance and Standard Deviation)

Range Ungrouped data Difference between the highest and lowest values in the set of observations
Variance
Grouped data Difference between midpoint of the highest and the lowest class

∑σ 2 = (x − x)2
or
n

Ungrouped data

Grouped data

σ = ∑ (x − x)2 or
n

Standard Ungrouped data
deviation

Grouped data

Example:
Given 6, 4, 8, 3, 4. Find the range of the data set.

Solution:
The range = 8 – 3 = 5.

34

DUM 30172 ENGINEERING MATHEMATICS 3
UNIT 3: STATISTICS AND PROBABILITY

Example:
Given 6, 4, 8, 3, 4. Find the variance and standard deviation.

Solution:

The standard deviation is

Example:

The time x, in hours, spent on swimming practices by 20 students in a week was recorded in the data
below.

Time (x) 123456
Number of students 354332

Find the variance of the distribution

Solution: f fx x2 fx2
x 3 3 1 3
1 5 10 4 20
2 4 12 9 36
3 3 12 16 48
4 3 15 25 75
5 36 72
∑fx2 = 254
62 12

∑f = 20 ∑fx = 64

Mean, x = 64 = 3.2 hours
20

( )∑∑σ 2 =
fx 2 − x 2
f

= 254 − (3.2)2
20

= 12.7 – 10.24 = 2.46 hours

35