NOTE STATISTIC BA301

BA 301 ENGINEERING MATHEMATICS 3

TOPIC 1 : STATISTICS

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.

Introduction

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.

Learning Outcome

Know the basics of statisticsUnderstand frequency distributionCompute measures of central

tendencyUnderstand cumulative frequency

distributionCompute measures of dispersion for

sample

Basics of Statistics

Data Presentation

Frequency Distribution

Measures of Central Tendency & Dispersion

collecting data

summarizing data

analyzing data


presenting data

There are four components:


DataCollection of information – facts or numbers

Quantitative data: consist of numbers representing counts or measurementsi. Continuous data – length, weight, massii. Discrete data – Number of students

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, animals, plants or things

Example: Students of PTSB

Sample A group of

units selected from a larger group

Example: Students of JKE, PTSB

DataCollection of

information – facts or numbersa. 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)


Line Graph Bar Chart Pie Chart Histogram Ogive

Data Presentation

Frequency DistributionThe following the data represent 25 plates survey, length in mm. Arrange all data in a frequency table .

9.9 15.4 18.4 13.4 15.9 15.6 12.7 23.3 14.3 20.8 24.117.0 11.8 9.2 12.6 19.5 5.4 7.8 19.2 22.1 20.5

28.6 19.9 16.8 8.8SOLUTION :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

Class interval

Class boundaries

Frequencyf

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

25

FREQUENCY TABLE- 0.05 + 0.05

9.4 – 5.4 = 4 ( size of class interval )

(9.4 – 9.3) / 2 = 0.05

EXAMPLE

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

Arrange all data in a frequency table

Measure of Central Tendency

Measurement Ungroup Data Group Data

Mean ,

=

= sum of all the

data

N = number of data

=

= sum of the value of freq x midpoint = sum of freq

xN

x

x

x f

fx

f

f

x


Median, m

- Value located at the centre of data.

- can find from Ogive chart

- Arranged in a ascending order - Find out the located of median at the centre of data.

- 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

cf

FN

Lm

m

2


Mode, mo

- Value of the highest frequency

- can find using Histrogram chart

Find the highest frequency / often number

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

cdd

dL mo

21

1

Measure of DispersionMeasurement Ungroup Data Group Data

Mean deviation , E

xi = data

= mean of data

N = number of data

x = midpoint

= mean of data

= sum of freq

N

xxi

x

f

fxx

x

f


Variance , s2

xi = data

= mean of data

N = number of

data

i.

x = midpoint

= mean of data

= sum of freq

OR

ii.

= frequency

d = x (midpoint)

N

xxi 2

x

f

fxx2

x

f

22

f

fd

f

fd

f


Standard deviation, s

s2 = variance

s2 = variance

2s2s

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.

x

x

N

x

n

xxi sign negative should be neglect

arrange the data to number rise and cut left and right, you have one number, there is median answer

Recurring / oftennumber

•Mean,

1 ,1, 2, 2, 2, 2, 3, 3, 5, 6, 8 m = 2

• Mode, m0

1, 1 , 2 , 2 ,2 ,2,3,3, 5, 6 , 8

m0 = 2

E = (1-3.182)+(1-3.182)+(2-3.182)+(2-3.182)+ (2-3.182)+ (2-3.182)+(3-3.182)+ (3-3.182)+ (5-3.182)+ (6-3.182)+ (8-3.182)

11

=

11

35=

use mean formula, x bar equal total x over by number

= 3.182

•Median , m

. Mean Deviation, E =

= 1.719

n

xxi2

. Variance, s2 =

. Standard deviation, s =

= (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

2s

5124.4

= 2.12

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NOTE STATISTIC BA301