1. BA 301
ENGINEERING
MATHEMATICS 3
TOPIC 1 :
STATISTICS

2. Introduction
 Statistics
is a branch of mathematics which
involved the collecting , recording, arranging ,
studying and analyzing information consisting
of numerical data.
 In
our daily life , statistics is widely used to
provide important and useful information
about finance, insurance , tax, agricultural and
manufacturing output, medical records,
economy, import and export data,
employment , education and others.

4. Introduction
This topic explains basic terminologies
of statistics. Data presentation is
made in the form of graphs and
frequency distribution tables. Measure
of central tendency is determined
using formulaic and graphical
methods. This topic also discusses the
calculation of mean
deviation, variance and standard
deviation.

5. Learning Outcome
 Know
the basics of statistics
 Understand frequency distribution
 Compute measures of central
tendency
 Understand cumulative frequency
distribution
 Compute measures of dispersion for
sample

6. Basics of Statistics
Data Presentation
Frequency Distribution
Measures of Central
Tendency & Dispersion

7. Basics of Statistics
There are four components:

collecting data

summarizing data

analyzing data

presenting data

8. Basics of Statistics
Data
 Collection
i.
ii.
of information – facts or numbers
Quantitative data: consist of numbers representing
counts or measurements
Continuous data – length, weight, mass
Discrete data – Number of students
Qualitative data: can be separate into different categories
that are distinguished by some no numerical characteristic
– genders (male/female)

9. Basics of Statistics
Data
Collection of
information – facts or
numbers
a. Quantitative data:
consist of numbers
representing
counts or
measurements
b. Qualitative data:
can be separate
into different
categories that
are distinguished
by some no
numerical
characteristic –
genders
(male/female)

Population


Any entire
collection of
people, anim
als, plants or
things
Example:
Students of
PTSB
Sample


A group of
units
selected
from a larger
group
Example:
Students of
JKE, PTSB

10. Data Presentation
Line Graph Bar Chart
Pie Chart
Histogram
Ogive

11. Frequency Distribution
The following the data represent 25 plates survey, length in mm.
Arrange all data in a frequency table .
9.9
20.8
19.2
15.4
24.1
22.1
18.4
17.0
20.5
13.4
11.8
28.6
15.9
9.2
19.9
SOLUTION :
i. Num of data, N = 25
ii High value = 28.6
Low value = 5.4
iii. Difference between highest value and lowest value
iv. Range(Julat) = highest value – lowest value = 28.6 – 5.4 = 23.2
v.
Num of class,K
K = 1 + 3.3 log N
= 1 + 3.3 log 25
= 5.6 ~ 6
Size of class interval , C
C = Range / num of class
= 23.2 / 6
= 3.866 ~ 4
15.6
12.6
16.8
12.7
19.5
8.8
23.3
5.4
14.3
7.8

12. FREQUENCY TABLE
Class - 0.05
Class
interval
boundaries
+ 0.05
Frequency
f
9.4 – 5.4 = 4 ( size of class
interval )
Cumulative
Frequency
F
5.4 – 9.3
5.35 – 9.35
4
4
9.4 -13.3
9.35 – 13.35
4
8
13.4 – 17.3
13.35 – 17.35
7
16
17.4 – 21.3
17.35 – 21.35
6
21
21.4 – 25.3
21.35 -25.35
3
24
25.4 – 29.3
25.35 - 29.35
1
25
(9.4 – 9.3) / 2 = 0.05
25

13. EXAMPLE
Arrange all data in a frequency table
Ages of NASCAR Nextel Cup Drivers in Years (NASCAR.com) (Data is ranked--Collected Spring 2008)
21
21
21
23
23
23
24
25
25
26
26
26
26
27
27
28
28
28
28
29
29
29
29
30
30
30
30
31
31
31
31
31
32
34
35
35
35
36
36
37
37
38
38
39
41
42
42
42
43
43
43
44
44
44
44
45
45
46
47
48
48
48
49
49
49
50
50
51
51
65
72

14. Measure of Central Tendency
Measurement
Ungroup Data
Mean , x
x
x
=
x
N
= sum of all the
data
N = number of data
Group Data
x
=
fx
f
f = sum of the
value of
freq x
midpoint
f = sum of freq

15. Measurement
Ungroup Data
Group Data
Median, m
-
-
Lm
Value located at the
centre of data.
- Arranged in a ascending
order
can find from Ogive
chart
- Find out the located of
median at the centre of
data.
-
N
2
F
fm
c
Lm = lower boundary
of class in which the
median lies.
-
N = total of frequency
-
F = cumulative freq
before
the class in which the
median lies.
-
fm = frequency of the
class
in which the median
lies.
c = class size

16. Measurement
Ungroup Data
L mo
Mode, mo
-
Value of the highest
frequency
-
can find using
Histrogram chart
Group Data
Find the highest frequency
/ often number
d1
d1
d2
c
Lm = lower boundary of
class in which the
mode lies.
d1 = difference between
the frequency of
mode
class and class before
d2 = difference between
the frequency of
mode
class and class after
c = class size

17. Measure of Dispersion
Measurement
Ungroup Data
xi
Mean deviation , E
Group Data
x
x
xf
f
N
xi = data
x
= midpoint
x = mean of data
x
= mean of data
N = number of data
f = sum of freq

18. Measurement
Ungroup Data
Variance , s2
xi
x
2
Group Data
2
x x f
i.
f
N
xi = data
x
x = midpoint
= mean of data
x = mean of data
N = number of
f
= sum of freq
data
OR
fd 2
f
ii.
f
fd
2
f
= frequency
d = x (midpoint)

19. Measurement
Standard deviation, s
Ungroup Data
s2
s2 = variance
Group Data
s2
s2 = variance

20. EXAMPLE
Ungrouped Data
Given the data : 5, 3, 6, 2, 1, 8 , 2, 2, 3, 2, 1.
Find Mean, Median, Mode, Mean Deviation,
Variance and Standard Deviation.

21. •Mean,
x =
=
x =
x
N
use mean formula, x bar equal
total x over by number
35
11
3.182
•Median , m
arrange the data to number rise and cut left and
right, you have one number, there is median
answer
1 ,1, 2, 2, 2, 2, 3, 3, 5, 6, 8
m=2
•
Mode, m0
Recurring / often
number
1, 1 , 2 , 2 ,2 ,2,3,3, 5, 6 , 8
m0 = 2
. Mean Deviation, E =
xi
sign negative
should be neglect
x
n
E = (1-3.182)+(1-3.182)+(2-3.182)+(2-3.182)+ (2-3.182)+ (2-3.182)+(33.182)+ (3-3.182)+ (5-3.182)+ (6-3.182)+ (8-3.182)
11
= 1.719

22. . Variance,
s2
=
xi
x
2
n
= (1-3.182)2+(1-3.182)2+(2-3.182)2+ (2-3.182)2+ (2-3.182)2+ (2-3.182)2 +
(3-3.182)2+ (3-3.182)2+ (5-3.182)2+ (6-3.182)2+ (8-3.182)2
11
= 49.6367/11
= 4.5124
. Standard deviation, s =
s2
4.5124
= 2.12