EBOOK STATISTICS

POLYTECHNIC SERIES

INT RODUCT ION TO

FIRST EDITION

NORSHAFIZA BINTI MOHAMAD KHAIRUDDIN



POLYTECHNIC SERIES

INT RODUCT ION TO

FIRST EDITION

NORSHAFIZA BINTI MOHAMAD KHAIRUDDIN

POLITEKNIK SULTAN IDRIS SHAH
KEMENTERIAN PENGAJIAN TINGGI

Published by:
Politeknik Sultan Idris Shah
Sg.Lang, 45100 Sg. Ayer Tawar

Selangor
No. Tel : 03 3280 6200
No. Fax : 03 3280 6400
Laman web : http://psis.mypolycc.edu.my





ABSTRACT iiii
PREFACE iv
TABLE OF CONTENTS v
SUBTOPIC 1.1 PAGE 1

SUBTOPIC 1.2 PAGE 8
PAGE 21
SUBTOPIC 1.3 PAGE 55

SUBTOPIC 1.4
REFERENCE

SUBTOPIC 1.1

WHAT IS
STATISTICS?

Statistics is a branch of applied mathematics
that involves the collecting, analyzing,

interpreting and presenting empirical data

The two major areas of statistics are known as
descriptive statistics, which describes the

properties of sample and population data, and
inferential statistics, which uses those
properties to test hypotheses and draw
conclusions.

Sample survey methods are used to collect data
from observational studies, and experimental
design method are used to collect data from
experimental studies.

2

WHAT IS DATA?

Data is a collection of information
assembled by observation, research
measurements, or analysis. Data may be
classified as either quantitative or

qualitative.

Qualitative data represent some characteristics
or attribute. These descriptions that may be
observed but cannot be computed.

Quantitative data represent numerical value.
These can be numerically computed and
calculations can be performed.

3

TYPES OF DATA

PRIMARY DATA

• Data collected for the first time

SECONDARY DATA

• Data that is sourced by someone
other than the user

DISCRETE DATA

• Data that can take only specific
value

CONTINUOUS DATA

• Data that can take values from a
given range

4

WHAT IS
UNGROUPED

DATA?

Ungrouped data is the data that
obtained from direct observation that
gather from an experiment or study. The

data is called as raw data.

These data is not sorted into categories,
classified or otherwise grouped.

An ungrouped set of data is basically a
list of numbers.

5

WHAT IS
GROUPED

DATA?

Grouped data is the type of data
which is classified into groups after

the collection. These data is
categorized into various groups and a

table is created.

The purpose of the table is to show
the data points occurring in each
group.

6

CONCEPT MAP

STATISTICS

Grouped Data Ungrouped Data

Measures of central Measures of
tendency dispersion

Mean Mode Mean Standard
deviation deviation

Median Variance
7

SUBTOPIC 1.2

A table which the values for a sample are grouped into
the classes and the frequency of the observation into
each class are recorded.

Step 1: Calculate the number of classes by using
Sturge's Formula, k = 1+3.3 log n

Step 2: Find the size of class

Class size ≥ highest value - smallest value

number of classes

Step 3: Create a table with three column [class interval,
tally marks, frequency] and
number of row = number of classes.

Step 4: Make a tick mark in the appropriate row for each
data. Count the tally marks to determine the
frequency.

9

EXAMPLE 1

The weekly wages, in RM, of a group of
28 workers in a factory are as follow.
Construct a frequency distribution table
for this data.

10

SOLUTION

STEP 1: Calculate the number of classes by using
Sturge's Formula:

k = 1 + 3.3 log (28)
= 5.775 ~ 6

Round to the nearest integer

STEP 2: Find size of class:

• Highest value = 187
• Smallest value = 153

class size 187 153
6

= 5.667 ~ 6

Round up to the next integer

11

Create a table with 3 column and number of
STEP 3:

rows = number of classes

153 - 158
159 - 164
165 - 170
171 - 176
177 - 182
183 - 188

12

STEP 4: Make a tick mark in appropriate row for
each data and count the tally marks to
determine the frequency

153 - 158 / 1
159 - 164 //// 4
165 - 170 //// //// 9
171 - 176 //// /// 8
177 - 182 //// 4
183 - 188 // 2

13

EXAMPLE 2

The data below shows the time
taken (in seconds) for 40
students to complete a Quiz.
Construct a frequency distribution
table for this data.

40 49 43 45 42 43 46 36
42 36 37 44 39 41 31 45
38 48 44 51 38 53 35 32
30 43 41 52 46 43 50 40
39 41 48 47 32 52 47 42

14

SOLUTION

STEP 1: Calculate the number of classes by using
Sturge's Formula:

k = 1 + 3.3 log (40)
= 6.287 ~ 6

Round to the nearest integer

STEP 2: Find size of class:

• Highest value = 53
• Smallest value = 30

class size 53 30
6

= 3.833 ~ 4

Round up to the next integer

15

STEP 3: Create a table with 3 column and number of
rows = number of classes

30 - 33
34 - 37
38 - 41
42 - 45
46 - 49
50 - 53

16

STEP 4: Make a tick mark in appropriate row for
each data and count the tally marks to
determine the frequency

30 - 33 //// 4
34 - 37 //// 4
38 - 41 //// /// 9
42 - 45 //// //// / 11
46 - 49 //// // 7
50 - 53 //// 5

17

LET'S TRY

QUESTION 1

The data below shows the marks obtained by
50 students in a Mathematics Test. Construct a
frequency distribution table for this data with
number of classes is 7.

72 85 51 63 75 90 35 39 53 63
71 82 48 60 68 78 31 25 44 57
69 80 46 59 72 83 28 34 49 61
74 88 53 63 68 78 39 22 44 56
75 92 55 65 76 96 40 42 55 66

18

LET'S TRY

QUESTION 2

Construct a frequency distribution table
for the following data, starting with the
first class of 502 - 536.

592 534 663 631 541
622 708 541 528 546
631 599 637 502 592
618 546 641 603 650
641 622 547 586 567
644 578 483 578 619
695 645 578 644 531
536 644 689 557 612

19

LET'S TRY

QUESTION 3

The electricity bills for 50 houses are recorded
as in the table below. Construct a frequency
distribution table for the data with a class
interval of 10.

35 86 71 99 88 43 54 48 68 77
93 70 84 78 68 63 47 56 66 57
80 52 67 59 60 79 62 55 52 90
64 87 65 64 50 71 72 64 71 67
40 56 74 69 97 67 81 77 77 57

20

SUBTOPIC 1.3

WHAT IS
MEASURES OF

CENTRAL
TENDENCY?

A measures of central tendency is a summary
measure that attempts to describe a whole set of

data with a single value that represents the
middle or center of its distribution.

There are three main measures of central
tendency which is the mode, the median and the

mean. Each of these measures describes a
different indication of the typical or central value

in the distribution.

22

MEASURES OF CENTRAL TENDENCY

Is the most commonly
occurring value in a sequence

Is the middle value in a
sequence after arranged in

ascending order

Is the sum of the value in a
sequence divided by the
number of data

23

EXAMPLE 3

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

1, 3, 4, 9, 7, 2, 9

24

SOLUTION

1+3+4+9+7+2+9
ҧ = 7

=5

25

EXAMPLE 4

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

2, 5, 2, 3, 5, 8

26

SOLUTION

4+5+2+3+5+8
ҧ = 6

= 4.5

27

EXAMPLE 5

Given that the mean for series of
number m-2, m+4, 2m+5 and m+2
is 16. Find the value of m.

28

SOLUTION

σ
, ҧ =

16 = − 2 + + 4 + 2 + 5 + ( + 2)

4

64 = + + 2 + − 2 + 4 + 5 + 2

64 = 5 + 9

5 = 55

=

29

LET'S TRY

QUESTION 1

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

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

30

LET'S TRY

QUESTION 2

Mean for the series of number a-2, 2a+5
and a+1 is 8. Find the value of a.

QUESTION 3

The mean of a set positive integers; 3, 11,
9, 6, 15, 7, 2k and k+3 is 9. Find the
value of k.

31

MEASURES OF CENTRAL TENDENCY

σ
ഥ = σ

෥ = + ൗ −


ෝ = +
+

32

σ f = frequency
ഥ = σ x = midpoint

How to calculate midpoint?
Lower limit + Upper limit

Midpoint =
2

33

EXAMPLE 6

The table below show the waiting time
for 40 customers in a restaurant. Find
the mean of the data.

Time (minutes) Number of customers

1–4 3

5–8 8
9 – 12 13

13 – 16 9
17 – 20 4

21 – 24 2

25 – 28 1

34

SOLUTION

Step 2: Find fx

Step 1: Find the midpoint, x

Time Frequency Midpoint,
(minutes)
3 2.5 7.5
1–4 8 6.5 52
5–8 13 10.5 136.5
9 – 12 9 14.5 130.5
13 – 16 4 18.5 74
17 – 20 2 22.5 45
21 – 24 1 26.5 26.5
25 – 28

σ = 40 σ = 472

Step 3: σ 472
Substitute into ҧ = σ = 40

the formula = .

35

EXAMPLE 7

Calculate the mean from the
following data below.

Score Number of students

11 – 20 4
21 – 30 12
31 – 40 18
41 – 50 6

36

SOLUTION

Step 2: Find fx

Step 1: Find the midpoint, x

Score
4 62
11 – 20 12 15.5 306
21 – 30 18 25.5 639
31 – 40 6 35.5 273
41 – 50 45.5
σ = 40 σ = 1280

Step 3: Substitute into the formula
σ 1280

ҧ = σ = 40
=

37

LET'S TRY

QUESTION 1

The table shows the ages of the participants
of a badminton tournament. Calculate the
mean of the age of the participants.

Age (years) Number of participants

45 - 48 16
49 - 52 20
53 - 56 9
57 - 60 5
61 - 64 2

38

LET'S TRY

QUESTION 2

The table shows the lengths of fish in a pond.
Find the mean of the length of the fish.

Length (cm) Number of fish
1-5 5
6 - 10 16
11 - 15 12
7
16 - 20

39

෥ = + ൗ −


L = lower boundary median class
n = number of data
F = cumulative frequency before

median class
fm = frequency median class
c = size of class

40

How to calculate lower boundary?
The Lower boundary of each class is
calculated by subtracting half of the gap
value = 0.5 from the class lower limit.

How to calculate cumulative frequency?
The cumulative frequency is calculated by
adding each frequency from a frequency
distribution table

How to find median class?
The median class is the class interval
in which the median lies.

41

EXAMPLE 8

The table below shows the mass
of a group of students in a class.
Calculate the median of the data.

Mass (kg) Number of students
30 – 39 6
40 – 49 12
50 – 59 17
60 – 69 5
70 – 79 2

42

SOLUTION

Mass (kg) Number of students Cumulative
frequency
30 – 39 6
40 – 49 12 6
50 – 59 17 18
60 – 69 5 35
70 – 79 2 40
42

Therefore, ෥ = + ൗ −


෤ = 49.5 + 42ൗ2 − 18 (10)
17

= .

43