# BAST 1206 HCT Statistics Managerial Statistics Exam Practice

HIGHER COLLEGE OF TECHNOLOGYDEPARTMENT: BUSINESS STUDIES

Final Examination: Assignment Based Assessment

Semester: II

A. Y.: 2019 / 2020

Start Date: Tuesday, 19 May

Time: 09.00 AM

Due Date: Thursday, 21 May

Time: 09.00 AM

Student Name

Student ID

Specialization

Section

01 to 11

Level

DIPLOMA FIRST YEAR

Course Name

MANAGERIAL STATISTICS FOR BUSINESS

Course Code

BAST1206

For official Use Only

Question No.

Max.

Marks

5

1

5

2

5

2

5

3

5

3

5

4

5

4

5

5

5

5

5

6

5

6

5

7

5

7

5

8

5

8

5

9

5

9

5

10

5

10

5

Grand Total Marks

50

Question No.

Max. Marks

1

Obtained Marks

Obtained

Marks

50

First Marker:

Second Marker:

Date:

Date:

Guidelines for Students to Submit the Assignment:

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1)

The final assessment for semester 2, 2019-20 will be done through comprehensive assignment

for a maximum of 50 marks. The schedule of the final assessment is available in the college website.

https://www.hct.edu.om/about/the-college/announcements/final-assessment-timetable-041620

2)

All the students are expected to have only one assignment at one time. In case, if the students

have more than one assignment on the same day, please report to the exam committee through the

following mail id. exam.bus@hct.edu.om as soon as possible.

3)

All students are given 48 hours to complete and submit each assignment from the day, date and

time the assignment is uploaded. Students are advised not to wait till the last moment of the deadline

to submit the assignment.

4)

The students can check the assignment anytime and any number of times from the opening of the

assignment. The answer to the assignment need to be uploaded in e-learning within 48 hours.

5)

The answer to the assignment can be uploaded only one time. No requests for resubmission of

the assignment will be entertained.

6)

The students may contact the following mail Ids if they face any difficulties while related to final

assignment.

For Academic related support :

Business Courses

anand.kalimani@hct.edu.om

karri.krishna@hct.edu.om

For Technical Writing 1

ramil-ecot@hct.edu.om

For Technical Writing 2

khulood.aiadi@hct.edu.om

For Technical Communication

jocelyn.balili@hct.edu.om

For issues related to e-mail

office365support@hct.edu.om

accounts and Microsoft Teams

Any issues related to E-Learning

support.elearning@hct.edu.om

Moodle

Any other IT Troubleshooting

helpdesk2@hct.edu.om

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7)

Students may contact their respective lecturer through college email (within the 48-hour period

given) if they have any doubts and clarifications on the assignments.

8)

Students should be aware that this assignment is an independent assessment. Students are not

allowed to get help from any other person during the assessment period.

9)

Students assignment will be checked for plagiarism through Turnitin software. This assignment

will be assessed as per the College Assessment Policy. Student will be investigated in case of

plagiarism as per the College policy and procedures. The maximum acceptable similarity index is

25%.

10) In case the students face any technical issues regarding the submission of assignment, the answer

to the assignment can be mailed to the concerned lecturer within the 48-hour period using college

email.

11) Any assignment submitted after the 48-hour period will not be considered for evaluation.

12) The assignment should be submitted only with the file in MS Word document. No other format is

acceptable at all (e.g. pictures, JPEG, PDF, etc).

13) The students need to answer the assignment in the prescribed number of words as mentioned in

the assignment.

14) The students need to follow the following format while preparing the assignment :

Font Style: Times New Roman

Font Size: 12 point for body and 14 point for Headings

Line Spacing: 1.5

Margin: 2.54cm (One inch) on all the sides

Page Number : At the bottom right hand corner of each page

Colour: All words should be in black colour

15) Students who will fail to submit their assignment as per the deadline given are required to make

an online appeal along with the valid excuses as the guidelines which will be announced through the

college website or e-learning portal within three days from the date of submission deadline.

ANSWER ALL THE QUESTIONS, SHOW ALL CALCULATIONS, WORKING NOTES AND EXPLANATION

(EACH QUESTION WILL CARRY 05 MARKS)

Q 1: Mr. Irfan, Plant Manager of Al Khuwair Furniture LLC has recently installed two plants A and B

for their production of 2 Seater Polyster Sofa.

The productivity of the plant A for the past 10 days is 9, 14, 10, 8, 12, 16, 9, 12, 8 and 14 sofas

The productivity of the plant B for the past 10 days is 10, 14, 7, 9, 10, 11, 8, 13, 10 and 9 sofas

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a) Find out which plant is more consistent in productivity based on Standard Deviation (SD) and

give reason for your answer.

b) Which method will give you precise results, Coefficient of Variation (CV) or Standard

deviation? Discuss analytically

(3+2=5 Marks)

Q 2: During COVID-19 lockdown, Mr. Umair, a Sales Manager of Oasis Retail LLC is worried about

the sales and revenue of the store. He decided to find out the average sales revenue for the last 60 days.

He collected the revenue details of last 60 days from the accounts.

Revenue

(’00 RO)

No. of days

00-10

10-20

20-30

30-40

40-50

50-60

60-70

70-80

9

5

10

10

9

8

3

6

a) Find out the average revenue of the Oasis Retail LLC for the last 60 days

b) Find out the value which has occurred most frequently in the given data set.

c) Critically compare measures of central tendency mean and mode.

(2+2+1=5 Marks)

Q 3: The Times of Oman, leading newspaper had organized reading competition for the students and

the competition was attended by 40 Male students and 20 Female students. The exam result shows that

18 Male students and 15 female students obtained A Grade and remaining students got either B or C

Grade.

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a) If a student is selected randomly, what is the probability that the student is going to be

Female student?

b) If a student is selected randomly, what is the probability that the student is male student?

c) If a student is selected randomly, what is probability that we get a male student or A grade

student?

d) If a student is selected randomly, what is probability that we get a male student or female

student?

e) What will happen if probability is more than 1? Discuss critically

(5Q*1M=5 Marks)

Q 4: Mr. Ahmed who is a manager at PVR Multiplex Muscat has conducted a survey to investigate

the number of people visited the theatre in the last 30 days.

The number of people visited the theatre in the last 30 days are: 30, 36, 39, 37, 36, 54, 38, 33, 37, 33,

33, 36, 32, 48, 42, 38, 36, 35, 30, 33, 39, 37, 33, 36, 30, 39, 44, 32, 44 and 50.

a) Find out the grouped frequency distribution using 5 classes.

b) Which graph will be suitable to present the grouped frequency distribution? Give reasons to

your answer

c) Cumulative frequency and relative frequency are same. Discuss critically

(3+1+1=5 Marks)

Q 5: Al Matra LLC has recently hired Mr. Musab, a professional market researcher. Mr. Musab is

interested to find out relationship between cost incurred and revenue earned by the company. He

collected the data of last 7 years from the accounts department, but he is confused about the method

which can find out the relationship.

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Cost Incurred

Year

(OMR in

Millions)

Revenue

Earned

(OMR in

Millions)

2013

8

20

2014

10

22

2015

9

22

2016

7

18

2017

11

20

2018

12

24

2019

10

25

a) Suggest Mr. Musab a suitable method to find out the relationship between cost and revenue.

b) Find out the correlation value and give interpretation of the result

c) If correlation value is 1.25, discuss the result analytically.

(1+3+1=5 Marks)

Q 6 : (Using the Q. 5 data) If the cost incurred is OMR 20 Million, what will be the volume of revenue?

(05 Marks)

Q.7: Mr. Hamood, a research scholar of Business Department in HCT, would like to find out the

association between the mid-term marks, quiz marks, assignment marks and end semester marks with

the Total Marks/GP obtained by the students in the final exam. He has decided to collect the data from

the college students studying in the Sultanate of Oman. He came to know that there are more than

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138,000 students studying in all the different colleges in Oman and realized that it is not possible to

collect the data from all the students considering the time to complete the study.

(5Q*1M=5 Marks)

a) How will you decide the sample size? Discuss with reasons

b) Find out the dependent variable and independent variables

c) Who are the respondents?

d) How will you collect the data? Discuss the instrument and process

e) Inferential statistics is applicable. Discuss analytically

Q.8: Following are the details of survey conducted by a survey firm Mini LLC. They would like to

find out the trend of urban population since 2013.

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Population

Urban

Population

2013

2,843,415

2,086,983

2014

3,041,434

2,285,997

2015

4,267,348

3,416,565

2016

4,479,219

3,650,429

2017

4,665,928

3,874,042

2018

4,829,473

4,083,206

2019

4,974,986

4,273,762

2020

5,106,626

4,442,970

Year

a) Moving average or semi average which method will be suitable to find out the trend.

b) Find out the 3 yearly trend values using moving average method and comment on the trend

c) Critically compare moving average and semi average method

(1+3+1=5 Marks)

Q 9: In Bank Muscat Al Khuwair branch, there are 32 employees and average weight of the branch is

48 kg and average height is 155cm excluding branch manager.

Find out the following:

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a) If 08 employees having weight of 44 kg, 46 kg, 48 kg, 47 kg, 40 kg, 42 kg, 41 kg and 40 kg

have been terminated, what will be the average weight of the branch excluding branch

manager?

(02 Marks)

b) If 08 new employees having height 140 cm, 135 cm, 137 cm, 135 cm, 145 cm, 142 cm, 143

cm and 140 cm joined, what will be the mean height of the branch excluding branch manager?

(02 Marks)

c) If the weight of the branch manager (76 kg) is included, what will be the mean weight of the

branch?

(01 Mark)

Q 10: Mr. Adheel, a Manager with Construction Company working in Ibra and his friend Mr. Anees

who is also working as a Manager with Construction Company in Muscat. The average weekly milk

products expenditure for both in the last 07 years is as per below:

Year

Expenditure in Expenditure

Muscat (RO)

in Ibra (RO)

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2013

30

12

2014

33

15

2015

30

30

2016

28

27

2017

34

18

2018

24

23

2019

27

24

a) If cost of living is positively correlated with milk expenditure, which city will have consistent

cost of living? Find out using coefficient of variation

b) Why do you think coefficient of variation is more precise than Standard deviation? Discuss

critically

(3+2=5 Marks)

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INTRODUCTION TO

STATISTICS

Statistics

❑ The science of collecting, organizing, presenting,

analyzing, and interpreting data to assist in making

more effective decisions

❑ Statistical analysis – used to manipulate summarize,

and investigate data, to provide useful information

for decision-making.

Why study statistics?

1. Data are used everywhere.

2. Statistical techniques are used to make many decisions that affect

our day to day lives

3. Irrespective of your career, you will make professional decisions

that involve data. An understanding of statistical methods will help

you make these decisions effectively.

Types of statistics

• Descriptive statistics – Methods of organizing, summarizing, and

presenting data in an informative way

• Inferential statistics – The methods used to make conclusion about a

population on the basis of a sample

• Population –The entire set of individuals or objects of interest or the

measurements obtained from all individuals or objects of interest

In general population means number of people but in statistics meaning

of population is different.

Sample – A portion, or part, of the population of

interest

Descriptive Statistics

• Collect data

• e.g., Survey

• Present data

• e.g., Tables and graphs

• Summarize data

X

• e.g., Sample mean =

n

i

Examples of Descriptive Statistics Tools

• Mean

• Weighted mean

• Median

• Quartiles

• Mode

• Variance

• Range

• Mid range

Inferential Statistics

• Inference is the process of drawing conclusions or making

decisions based on sample results about a population

• Estimation.For example, Estimate the population average

height using the sample average height

• Hypothesis testing . For example, Test the claim that the

population average height is 161cm

• Hypothesis means assumptions or presumptions or claim

Sampling

A sample should have the same characteristics of the population from which sample is drawn.

Sampling methods can be:

• Random sampling :each member of the population has an equal chance of being selected. A

simple random sample is an unbiased

surveying technique.

• Non-random sampling is a sampling technique where the samples are gathered in a process that

doesnot give all the individuals in the population

equal chances of being selected.(biased)

The actual process of sampling causes sampling errors. For example, the sample may not be large

enough or representative of the population.

Factors not related to the sampling process cause non-sampling errors. A defective counting device

can cause a non-sampling error.

Statistical data

The collection of data that are relevant to the problem being

studied is commonly the most difficult, expensive, and timeconsuming part of the entire research project.

Statistical data are usually obtained by counting or measuring

items.

Primary data are collected for first time

specifically for the analysis desired. Methods are:

➢ Questionnaire

➢ Interview

➢ Observation

➢ Projective technique

• Secondary data have already been compiled

and are available for further statistical analysis

Questionnaire

Definition: Questionnaire is a

set of questions for obtaining

statistically useful or personal

information from individuals.

(www.merriam-webster.com)

Source: Batehet.al (2015),Using Statistics for Better Business Decisions, Business Expert Press, ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/momp/detail.action?docID=4201910. Created from momp on 2019-01-08 20:56:25.

Types of Data

Data means information in raw or unorganized form (such as alphabets, numbers, or

symbols) that refer to, or represent, conditions, ideas, or objects. Data is limitless and

present everywhere in the universe.

(Read more:http://www.businessdictionary.com/definition/data.html)

Statistical data are usually obtained by counting or measuring items. Most data can be put

into the following categories:

Qualitative – Qualitative data are generally described by words or letters. (Gender, blood

groups, hair colour etc). Many numerical techniques do not apply to the qualitative

data. For example, it does not make sense to find an average hair color or blood type.

• Quantitative – data are observations that are measured on

a numerical scale (distance traveled to college, number of

children in a family, etc.)

Quantitative data can be separated into two subgroups:

• discrete (if it is the result of counting (the number of students of a given ethnic group in

a class, the number of books on a shelf, …)

• continuous (if it is the result of measuring (distance traveled, weight of luggage, …)

Variables and Distributions

We finally need a flexible term to denote what is being measured

through the sample survey. That term is called as variable.

A variable is an item of interest that can take on many different

numerical values.

• Variable is the term used to record a particular characteristic of the

population we are studying. Example: Marks, Age, Gender, etc

• For example, if our population consists of pictures taken from Mars,

we might use the following variables to capture various

characteristics of our population:

• Quality of a picture • Title of a picture • Latitude and longitude of the

center of a picture • Date the picture was taken

Variables and Distributions

• It is useful to put variables into different categories, as different statistical procedures

apply to different types of variables. Variables can be categorized into two broad

categories, numerical and categorical:

• Categorical variables are variables that have a limited number of distinct values or

categories. They are sometimes called discrete variables.

Categorical variables again split up into two groups, ordinal and nominal variables.

•

Ordinal variables represent categories with some intrinsic order (e.g., low,

medium, high; or strongly agree, agree, disagree, strongly disagree). Ordinal

variables could consist of numeric values that represent distinct categories (e.g., 1 =

low, 2 = medium, 3 = high). These numbers are merely codes.

•

Nominal variables represent categories with no intrinsic order (e.g., job category,

company division, and race). Nominal variables could also consist of numeric values

that represent distinct categories (e.g., 1 = male, 2 = female).

• Numeric variables refer to characteristics that have a numeric value. They are usually

continuous variables, that is, all values in an interval are possible.

Variables and Distributions

• Example:

An experiment is conducted to test whether a particular drug will

successfully lower the blood pressure of people. The data collected

consists of the sex of each patient, the blood pressure measured, and

the date the measurement took place. The blood pressure is measured

three times, once before the patient was treated, then one hour after

administrating the drug, and again two days after administrating the

drug. What variables comprise this experiment?

Variables and Distributions

• The distribution of a variable refers to the set of all possible

values of a variable and the associated frequencies or

probabilities with which these values occur.

• Sometimes variables are distributed so that all outcomes are

equally, or nearly equally likely. Other variables show results that

“cluster” around one (or more) particular value.

• A heterogeneous distribution is a distribution of values of a

variable where all outcomes are nearly equally likely.

• A homogeneous distribution is a distribution of values of a

variable that cluster around one or more values, while other

values are occurring with very low frequencies or probabilities.

Statistics and Microsoft Excel

we recommend spending some time using the resources available on

Microsoft’s website at https://support.office.com. Search for “Basic

Tasks in Excel .”

Frequency Distributions and

Graphical Presentation of Data

Frequency distribution

Frequency distribution is the organization of raw data in a table form,

using classes and frequencies.

Types of frequency distribution

• Categorical frequency distribution

• Ungrouped frequency distribution

• Grouped frequency distribution

Categorical frequency distribution

• Categorical Frequency Distributions are used for data that can be

placed in specific categories, such as nominal or ordinal level data.

Example: Educational Qualifications of 40 individuals.

Example

Following are the grades scored by the students in Business

Mathematics. You are required to construct a frequency distribution.

A

C

D

D

C

D

A

B

B

C

C

B

C

B

A

A

C

F

A

C

A

C

A

D

F

A

D

C

D

D

A

D

D

A

D

F

D

A

A

D

A

F

B

F

B

A

F

B

B

D

F

B

C

F

B

D

A

F

A

D

F

A

A

Answer

Grade

A

B

C

D

F

Tally Bars

Frequency

Ungrouped Frequency Distribution

• The frequency is the number of times a particular data point occurs in

the set of data.

Example:

Given below are the marks obtained by 20 students in Accounting out

of 25.

21, 23, 19, 17, 12, 15, 15, 17, 17, 19, 23, 23, 21, 23, 25, 25, 21, 19, 19,

19

Marks

Tally Bars

Frequency

12

I

1

15

II

2

17

III

3

19

IIIII

5

21

III

3

23

IIII

4

25

II

2

Example

• From the below information construct an ungrouped frequency

distribution table.

Answer

Variable

5

10

15

20

Tally Bar

Frequency

Grouped Frequency Distribution

• In grouped frequency distribution, we need to find classes and frequencies.

Important terms in Grouped Frequency Distribution:

1.Class interval/width

2.Class limit

3.Inclusive method

4.Exclusive method

Steps :

1.

Find Highest and Lowest value

2.

Find Range

3.

Decide No. Of classes required

4.

Find Class width

5.

Find Upper and Lower class limit

6.

Talley the data

7.

Find frequencies

Example

Weekly wages for the 15 workers in a company are listed below. Construct a frequency

distribution with 5 classes.

21

10

16

32

23

23

26

20

29

34

18

22

19

27

20

Solution:

Step 1: Determine the classes.

➢Find the highest and lowest value. H = 34 and L = 10

➢Find the range: R = Highest value – Lowest Value

➢R = 34 – 10 = 24

➢Select the number of classes desired (usually between 5 and 20). In this case 5 classes

➢Find the class width by dividing the range by the number of classes

➢Width

=

(Range )/(Number of classes) = 24/5 = 4.8 (round off to 5)

Example-continued

• Step 2: tally the data

• Step 3: Find the numerical frequencies from the tallies

Class Limit

Tally

Frequencies

10 – 15

1

1

15 – 20

111

3

20 – 25

11111

5

25 – 30

1111

4

30 – 35

11

2

Cumulative Frequencies and Relative Frequencies

Class Limit

10 – 15

15 – 20

20 – 25

25 – 30

30 – 35

Total

Cumulative

Relative

Frequencies Frequencies Frequencies (%)

1

1

1÷15= 6.67%

3

1+3=4

3 ÷15=20%

5

1+3+5=9

5÷15=33.33%

4

1+3+5+4=13

4÷15=26.67%

2

1+3+5+4+2=15 2÷15=13.33%

15

100%

Exercise 1

• Example :- Following are the marks obtained by 30 students in an

examination. Prepare a grouped frequency distribution with 6 classes.

Also find out cumulative frequencies and relative frequencies.

25

15

26

65

41

55

35

54

78

55

65

22

32

62

22

45

13

15

16

46

62

19

42

33

24

Exercise 2

102

104

140

136

152

132

158

193

128

141

130

133

147

148

141

129

133

137

179

147

152

114

124

138

129

164

135

128

139

154

168

148

152

116

107

136

167

143

139

152

Construct a grouped frequency distribution with 7

classes. Also find out cumulative frequencies and

relative frequencies.

Graphical Presentation

BAR DIAGRAM/CHART

• A bar chart presents categorical data/ungrouped frequency distributions.

• A bar diagram makes it easy to compare sets of data between different

groups at a glance.

• The graph represents categories on one axis and frequencies in the other.

• Examples:-

Example

• The table shown here displays the number of students joined in

different specialization in a college during 2018. Construct a bar chart

for the data.

Specializations

Frequency

Accounting

Human

Resource

200

Marketing

120

E-Business

90

260

Pie Diagram/Chart

• Pie charts represent categorical data. Not suitable for numerical data.

• Pie charts are useful when the categories are not numerous( eight

categories or less)

• Pie charts are generally used to show percentage or proportional data.

• Each percentage is represented by a slice of pie.

Example

• The favorite flavors of ice-cream for the children in a locality are given

follow. Draw a pie chart to represent the given information.

Flavors

Vanilla Strawberry Chocolate Kesar-pista Mango zap

Number of

50

children

30

20

60

40

Answer

Flavors

Vanilla

Strawberry

Chocolate

Kesar-Pista

Mango Zap

Percentage of Children

(50/200)*100=25%

15%

10%

30%

20%

Pie chart

Vanilla

25%

Mango Zap

20%

Strawberry

15%

Kesar-Pista

30%

Chocolate

10%

Note: Pie chart is drawn based on percentage of frequencies

Histogram

Histograms are used to represent the grouped frequency distribution.

Steps for drawing histogram:

• Step 1: Draw and label the x and y axes. The x axis is always the horizontal

axis, and the y axis is always the vertical axis.

• Step 2: Represent the frequency on the y axis and the class limits on the x

axis.

• Step 3: Using the frequencies as the heights, draw vertical bars for each

class.

Example

Class

Limit Frequencies

10 – 15

1

15 – 20

3

20 – 25

5

25 – 30

4

30 – 35

2

Draw a Histogram from the above data.

CHAPTER 3

DESCRIPTIVE STATISTICS

MEASURES OF CENTRAL TENDENCY

ARITHMETIC MEAN/AVERAGE

Definition: The mean represents the average of all observations.

Mean = (sum of all measurements)/(number of measurements).

The Greek letter m (mu) is used to denote the mean of the entire

population, or population mean.

The symbol x̅ (read as “x bar”) is used to denote the mean of a

sample, or sample mean

Mean from Raw data.

• Formula:

• x̅ =

σ𝐗

𝐍

• Example:1. Find Mean from the following score by a student in 10 different

tests.

25, 65, 32, 46, 28, 34, 52, 64, 68, 36

2. Find the average profit of 6 small trading companies in OMR

250, 630, -330, 450, 350, 510

Mean from Ungrouped data

f(X)

• Formula : Mean =

N

Example:- Find the mean wages of 100 workers

Wages in OMR(X)

No of workers(f)

05

15

08

18

10

30

12

15

15

10

18

7

20

5

Example No. 2 – Find average cash flows of a shop

for 49 days.

Daily income in RO

No. Shops

100

5

-150

8

230

12

350

10

400

7

425

4

450

3

Mean from grouped data

Example:- Find average marks of 33 students in a class.

Marks

No of Students

0-10

2

10-20

6

20-30

8

30-40

10

40-50

4

50-60

2

60-70

1

Mean =

f(Xm)

N

Xm=Mid points of X values

Example No. 2

• Find the average monthly income of 52 families in Muscat

Income in

No. Families

OMR

00- 50

50- 100

100- 150

150- 200

200- 250

250- 300

300- 350

5

10

12

15

6

3

1

What are the Advantages and Disadvantages of

Mean?

Advantages:

The mean is easier to compute than the median since it does not

require sorted observations.

• The mean is based on each value of a variable that makes it more

useful than the median.

Disadvantage:

• The mean is influenced by extreme values.

Median

• Median is a value that divide a series of numbers into two equal parts.

• Definition: The median is that number from a population or sample

chosen so that half of all numbers are larger and half of the numbers are

smaller than that number.

Median from Raw data

• Rule:1. ARRANGE DATA IN ASCENDING ORDER

2. If N is odd number then, (N+1)/2th value

3. If N is an even number then, find (N+1)/2th value and take the

average of two middle values

Median from ungrouped data

• Step 1. Find cumulative frequency

• Step 2 Use the rule (N+1)/2th value.

• Example: Find Median from the following marks

Marks

Number of

Students

10

2

20

5

30

10

40

12

50

4

60

3

70

1

Cumulative

frequency

Median from Grouped data

• Step 1. Find cumulative frequency

• Step 2 Use the rule N/2th value.

• Step 3 Use the following formula to find the exact value of median

Median from Grouped data-Example

• Find the median income of 52 families in Muscat

Income in

No. Families

OMR

00- 50

50- 100

100- 150

150- 200

200- 250

250- 300

300- 350

5

10

12

15

6

3

1

Mode

• The mode is the value that occurs most often (with the highest

frequency) in the data set.

• A data set can have more than one mode or no mode at all.

• A group of data set may be

➢Uni-modal : Only one mode

➢Bi-modal : Two modes

➢Multi-modal: More than two modes

Mode from Raw data

• Value that has highest frequency is the mode.

Examples:

• 1,2,3,4,5,6,7,8,9,10 – No Mode

• 1,2,2,3,4,5,6,7,8,9 – Mode is 2 (Uni-modal)

• 1,2,2,3,4,5,5,6,7,8 – Modes are 2 and 5( Bi-Modal)

• 1,1,2,3,3,4,5,5,6,7,7– Modes are 1,3,5 and 7 ( Multi modal)

• 1,4,3,3,2,4,3,5,4,2,6,4,7,4– Mode is 4 ( Uni modal)

Mode from Ungrouped data

• Value that has highest frequency is the mode.

• Find Mode from the following marks:

Size of Shoe

Frequency

38

34

39

45

40

60

41

23

42

22

Mode from Grouped data

Step1: Find the modal class with highest frequency.

Step 2: Use the following formula to find the exact value of mode.

Where L = Lower limit of the modal class

f1 = frequency of the modal class

f0 = frequency of the class preceding the modal class

f2 = frequency of the class succeeding the modal class

c= class width

Mode from Grouped data-Example

Find Mode from the following marks:

MEASURES OF VARIABILITY

Meaning of Variability

• Variability measures how much values in a set of data differ from

each other.

• Variability is also called as dispersion or spread.

• Data sets with similar values are said to have little variability, while

data sets that have values that are spread out have high variability.

Methods of measuring variability

• Range

• Quartile deviation

• Variance

• Standard deviation

• Co-efficient of variation

RANGE

• The range is the difference between the largest and the smallest

value of the data set.

Example: Suppose two machines produce nails which are on average

10 inches long. A sample of 11 nails is selected from each machine.

Machine A: 6, 8, 8, 10, 10, 10, 10, 10, 12, 12, 14

Machine B: 6, 4, 6, 8, 8, 10, 12, 12, 14, 14, 14

• For Machine A data, the range is 14-6= 8

• For Mechanic B data, the range is 14-4= 10

Conclusion: Performance of machine A is better than machine B as nails

produced by machine A show less variation compared to machine B.

Quartile Deviation from raw data

• Quartile deviation= (Quartile3-Quartile 1)/2

Quartile Deviation = (29.50-16.25)/2= 6.625

Quartile deviation from ungrouped data

• Q1 = (n + 1) / 4 th value (using cf)

• Q3 =3 (n + 1) / 4 th value( using cf)

• Quartile deviation= (Quartile3-Quartile 1)/2

• Example: Find quartile deviation from below information:

Quartile deviation from grouped data

Formula :Modify median formula accordingly.

Q1– N/4 instead of N/2

Q3 — 3N/4 instead of N/2

Quartile deviation= (Quartile3-Quartile 1)/2

Standard Deviation and Variance

• The Standard deviation measures the spread of the data around the

mean.

• Standard deviation is the absolute measure of dispersion which

expresses variation in the same units as the original data.

• Variance is the square of standard deviation.

• Standard deviation is denoted by “σ” or “s”

• Variance is denoted by “σ²”or “s²”

Standard Deviation and Variance from raw data

•

Formula :

Example: Suppose two machines produce nails which are on average 10 inches long. A

sample of 11 nails is selected from each machine.

Machine A: 6, 10, 8, 10, 12, 10, 14

Machine B: 6, 4, 8, 10, 12, 14, 14

Find the value of standard deviation and variance of size of nails for each machine.

Standard Deviation and Variance from raw data

Answer

Machine A

Machine B

X

ഥ

X-𝑿

ഥ )2

(X-𝑿

8

-2

4

4

-6

36

8

-2

4

10

0

0

12

2

4

14

4

16

14

4

16

ഥ )2=80

∑(X-𝑿

Mean = 70/7=10

Standard deviation= 2.39

Variance=(2.39)2

= 5.71

Mean = 70/7=10

Standard deviation= 2.64

Variance=(2.64)2

= 6.96

Conclusion: Based on standard

deviation, nails produced by

machine A is better than nails

produced by machine B as

variability is comparatively less

for nails produced by machine A.

Standard Deviation and Variance from Ungrouped Data

• Formula for standard deviation

• Example:

Marks

Number of

students

10

20

30

40

50

2

5

10

8

3

Find standard deviation and variance

Standard Deviation and Variance from Ungrouped Data

Answer

X

f

10

2

20

5

30

10

40

8

50

3

F(x)

x2

f (x2)

Standard Deviation and Variance from Grouped Data

Formula:

Marks Students (f) Mid-point( x) F(x)

0- 10

2

10- 20 5

20- 30 10

30- 40 8

40- 50 3

x2

f (x2)

Co-efficient of Variation(CV)

• The Coefficient of Variation(CV) expresses the standard deviation as a

percentage of mean.

• Researchers can easily compare variability of more than one variable

using CV.

• Formula

Co-efficient of Variation(CV)-Example 1

• Following are the mean and standard deviation of marks scored by

students in two different sections of Managerial Statistics.

• Section 1– Mean = 64.43, SD = 12.02,

• Section 2– Mean = 46.68, SD = 14.76,

1. Calculate coefficient of variation.

2. Which section shows higher performance of students? Give reason.

Co-efficient of Variation(CV)-Example 2

Following are the average weekly wages in Riyals and standard deviations in Riyals in two

factories located in Gala Industrial Area:

Factory

X

Y

Average

Wages

24.5

Wages(

38.5

Standard Deviation

4

6

No. of workers

512

624

1. Which factory X or Y pays out a larger amount as weekly wages?

2. Which factory X or Y has greater variability in individual wages?

Co-efficient of Variation(CV)-Example 3

Prices of a particular commodity in six years in Muscat and Salalah are given below:

Price in Muscat

22

20

19

23

16

18

Which city has more stable prices?

Price in Salalah

10

20

18

12

15

16

Chapter 4 &5

PROBABILITY,NORMAL DISTRIBUTION, ESTIMATION AND HYPOTHESIS TESTING

What is a probability?

What does it mean to say that a probability of a fair coin is one half,

or that the chances I pass this class are 80 percent,

First, think of some event where the outcome is uncertain.

Examples of such outcomes would be the roll of a die,

➢ the amount of rain that we get tomorrow,

➢ the state of the economy in one month.

In each case, we don’t know for sure what will happen. For example, we don’t know

exactly how much rain we will get tomorrow.

http://www-math.bgsu.edu/~albert/m115/probability/interp.htm

• A probability is a numerical measure of the likelihood of the event. It is a

number that we attach to an event, say the event that we’ll get over an inch of

rain tomorrow, which reflects the likelihood that we will get this much rain.

• A probability is a number from 0 to 1.

• If we assign a probability of 0 to an event, this indicates that this

event never will occur.

• A probability of 1 attached to a particular event indicates that this

event always will occur.

• What if we assign a probability of .5? This means that it is just as likely for

the event to occur as for the event to not occur.

THE PROBABILITY SCALE

+—————————-+—————————-+

0

event never

will occur

.5

1

event and “not event”

event are likely

always

will occur

to occur

http://www-math.bgsu.edu/~albert/m115/probability/interp.htm

Experiment

• An experiment is defined as the process that generates or

provides an outcome (result).

• The sample point is an individual outcome of an experiment.

• The set of all possible sample points in an experiment is

called a sample space.

• An event of a random Experiment is defined as a subset of

the sample space of the random Experiment.

• When you toss one coin the possible outcomes or the

sample space will include a head or tail.

Classical probability and Empirical probability.

• The classical probability uses the sample space to determine the

numerical probability that an event will occur.

• It assumes that all events have the same probability of occurring

(equally likely). For example, if you toss a coin the probability that you

1

get a head is .

2

• The empirical probability relies on actual experience to determine

the likelihood of outcomes. For example, suppose you have data on

number of people with different blood types and we get the

following:

• A: 22

B: 5

AB: 2

O: 21

• The probability that a person has type O is

21

50

which is 0.42.

Some sample spaces for various probability experiments are shown here

Experiment

Sample space

Tossing a coin

S = {H, T}

Tossing a coin twice

S = {HH},( HT), (TH),(TT)

Throwing a die

S = (1,2,3,4,5,6)

Throwing two die

S = (1,1), (1,2),(1,3), (1,4),(1,5),(1,6)

(2,1), (2,2),(2,3), (2,4),(2,5),(2,6)

(3,1), (3,2),(3,3), (3,4),(3,5),(3,6)

(4,1), (4,2),(4,3), (4,4),(4,5),(4,6)

(5,1), (5,2),(5,3), (5,4),(5,5),(5,6)

(6,1), (6,2),(6,3), (6,4),(6,5),(6,6)

EVENT OF AN PROBABILITY EXPERIMENT

• An event of a random Experiment is defined as a subset of the sample

space of the random Experiment.

• Example: 1

• Consider the random Experiment of the following a die and getting an

even number. Here sample space

• S = (1,2,3,4,5,6) and the event getting an even number is E= ( 2,4,6).

• Example 2

• There are 2 children in a family. Find the event that:

• Both children are boys

• Only one of the children is a girl

• There is at least one girl

• Here S = (BB,BG,GB,GG)

• Answer?

Probability of an Event

• Probability of an event E is given by

Total number of outcomes in E

P(E)= ________________________________________________

Total number of outcomes in the sample space

• The probability is denoted by

• P (E) =

𝒏(𝑬)

𝒏(𝒔)

Probability Rules

Rule 1: The probability of any event E is a number between and

including 0 and 1.

• (probability cannot be negative or greater than 1 )

Rule 2: If an event E cannot occur, its probability is 0.

• Example: When a single die is rolled , find the probability of getting 9

• Sample space S= ( 1,2,3,4,5,6,)

• There is no point in 9.

• .. P (getting 9) =

0

6

Probability Rules

Rule 3: If an event E is certain, then the probability of E is 1.

• Example: When a single die is rolled , What is the probability of

getting a number less than 7

• Sample space S= ( 1,2,3,4,5,6,). The event of getting a number less

than 7 is certain.

• P (getting a number less than 7) = 6 =

6

6

=1

Rule 4: The sum of the probabilities of the outcomes in the sample

space is 1.

Example: Tossing a single coin- Two out comes( Head or Tail)

Probability of getting head is

Total probability is

𝟏 𝟏

+

𝟐 𝟐

=1

1

2

and probability of getting tail is

1

.

2

Addition Rule for probability

• Addition Rule 1: When two events, A and B, are mutually exclusive,

the probability that A or B will occur is the sum of the probability of

each event.

• P(A or B) = P(A) + P(B)

• Experiment 1 :A day of the week is selected at random. Find the

probability that it is a weekend day.

• P ( Friday or Saturday) = P (friday) + P (saturday ) =

1

7

1

7

+ =

2

7

• Experiment 2: A single 6-sided die is rolled. What is the probability of

rolling a 2 or a 5?

P(2 or 5) =

P(2)

=

1

6

+

+

1

6

P(5)

=

2

6

=

1

3

• Rule 2: If A and B are not mutually exclusive, then

P (A or B) = P (A) + P( B) – P (A and B ).

• Example

• In a hospital unit there are 8 nurses and 5 physicians; 7 nurses and 3

physicians are females. If a staff person is selected, find the probability that

the subject is a nurse or a male.

• Solution

• The sample space is shown here. Answer:

Staff

Females

Males

Total

Nurses

7

1

8

Physicians

3

2

5

Total

10

3

13

The probability is

P(nurse or male) = P(nurse) + P(male) – P(male and nurse)

𝟖

𝟑

𝟏

𝟏𝟎

=

+

=

𝟏𝟑

𝟏𝟑

𝟏𝟑

𝟏𝟑

NORMAL DISTRIBUTION

• NORMAL DISTRIBUTION is a continuous probability distribution in

which the relative frequencies of a continuous variable are distributed

according to the normal probability law.

• In other words ,it is a symmetrical distribution in which the frequencies

are distributed evenly about the mean of the distribution.

• The normal distribution of a variable, when represented graphically,

takes the shape of a symmetrical curve, known as the Normal Curve.

• It helps us to find the proportion of

measurements that falls within a

certain range above, or below or

between selected values.

Symmetrical curve or Normal Curve.

Summary of the properties of the Theoretical Normal Distribution

The normal distribution curve is bell – shaped

The mean, median and mode are equal and located at the center of the distribution

The normal distribution curve is uni modal (i.e., it has only

one mode)

ESTIMATION AND HYPOTHESIS TESTING

Point Estimate

•

A Point estimate is a specific numerical value estimate of a

parameter. The best point estimate of the population mean (µ) is the

sample mean(x̄) .

•

Suppose a College Dean wishes to estimate the average age of

students attending classes this semester. The College Dean could select

a random sample of 100 students and find the average age of these

students (x̄), say, 22.3 years. For the sample mean, the College Dean

could infer that the average age of all the students (µ) is 22.3years. This

type of estimate is called a point estimate.

•

Sample measures are used to estimate population measures.

These statistics are called the estimators.

•

A good estimator should satisfy the three properties described

below.

Three properties of Good Estimator

• The estimator should be an unbiased estimator. That is the

expected value of the mean of the estimate obtained from samples of

a given size is equal to the parameter being estimated.

• The estimator should be consistent. For a consistent estimator, as

sample size increase, the value of the estimator approaches the value

of the parameter estimated.

• The estimator should be relatively efficient estimator. That is, of all

the statistics that can be used to estimate a parameter, the relatively

efficient estimator has the smallest variance.

Interval estimate

• An interval estimate of a parameter is an interval or a range of values

used to estimate the parameter. This estimate may or may not

contain the value of the parameter being estimated.

• In an interval estimate, the parameter is specified as being between

two values. For example, an interval estimate for the average age of

all students might be 26.9 < µ < 27.7
Confidence Level and Confidence Interval
• The Confidence Level of an interval estimate of a parameter is the
probability that the interval estimate will contain the parameter.
• A Confidence Interval is a specific interval estimate of a parameter
determined by using data obtained from a sample and by using the
specific confidence level of the estimate.
•
HYPOTHESIS TESTING
• Hypothesis testing is a decision – making process for evaluating claims
about a population.
• In hypothesis testing, the researcher must define the population under
study, state the particular hypotheses that will be investigated, give the
significance level, select a sample from the population, perform the
calculations required for the statistical test, and reach a conclusion.
• A statistical hypothesis is a conjecture about a population parameter.
• This conjecture may or may not be true.
• There are two types of statistical hypotheses for each situation: the null
hypothesis and the alternative hypothesis.
Null Hypothesis
• The null hypothesis symbolized by H0, is a statistical hypothesis that
states that there is no difference between a parameter and a specific
value, or there is no difference between two parameters.
Alternative Hypothesis
• The alternative hypothesis symbolized by H1, is a statistical
hypothesis that states the existence of a difference between a
parameter and a specific value, or states that there is a difference
between two parameters.
Chapter 6
Correlation and Regression
Correlation
Correlation is a statistical technique used to determine the degree to
which two variables are related
For example
❑whether the volume of sales for a given month is related to the amount of
advertising .
❑whether the number of hours a student studies is related to the student’s
score on a particular exam
❑Whether person’s age and his or her blood pressure are related?
These are only a few of the many questions that can be answered by using
the techniques of correlation and regression analysis
Types of relationship between variables
There are two types of relationships
• Simple relationship: Relationship between only two
variables, X and Y.
➢For example, Relationship between Oil price and economic
development
•
Simple relationship can also be Positive or Negative.
Types of relationship between variables
• Positive relationship or positive correlation :
• If the value of two variables, X and Y, move in the same direction such
as:
❖an increase in the value of one variable results, on an average, in a
corresponding increase in the values of the other variable.
❖a decrease in the value of one variable results, on an average, in a
corresponding decrease in the other variable .
• Example: The height and weight of a growing child.
Types of relationship between variables
• Negative relationship or Negative correlation: the correlation is said
to be negative or inverse if the two variables X and Y deviate in the
opposite direction.
• i.e. if the increase (or decrease) in the values of one variable results,
on an average, in a corresponding decrease (or increase) in the
values of other variable
Example: the price and demand of a commodity .
Types of relationship between variables
• Multiple relationships or multiple correlation: Examines the relationship
between more than two variables.
• For example an educator may wish to investigate the relationship
between a student’s success in college and factors such as the number of
hours devoted to studying, the student’s previous GPA and the student’s
high school background.
Types of variables in correlation analysis
• Independent variable: is the variable in regression that can be
controlled or manipulated. In this case, “number of hours of study” is
the independent variable and is designated as the x variable.
• Dependent variable: is the variable in regression that cannot be
controlled or manipulated. The grade the student received on the
exam is the dependent variable, designated as the y variable.
How to measure correlation?
Correlation can be measured using:
• Scatter diagram( graphical method)
• Correlation coefficient( Numerical method)
Scatter Diagram
• The scatter diagram is drawn with two variables, usually the first
variable is independent and the second variable is dependent on the
first variable
Source:https://www.statisticshomeworktutors.com/Scatter-DiagramAssignment-Homework-Help.php
Scatter Diagram-Example
Draw a scatter diagram from the following information
Hours of study x
Grade y (%)
Chart Title
100
90
6
3
92
63
80
70
60
50
40
1
47
30
5
88
10
20
0
0
2
58
4
75
1
2
3
4
5
6
7
Exercise : Scatter Diagram
Draw scatter diagrams from the following data sets
X
Y
X
Y
X
Y
50
10
10
50
50
20
40
8
20
40
30
50
30
6
30
30
20
40
20
4
40
20
40
10
10
2
50
10
10
30
Interpretation of scatter diagrams
Computation of Correlation Co-efficient
• Correlation coefficient is a numerical measure of correlation.
• Correlation Co-efficient measures the strength of directions of a linear
relationship between two variables.
Types of Correlation Co-efficient
➢Karl Pearson's Correlation Co-efficient
➢Spearman's Correlation Co-efficient
Karl Pearson's Correlation Co-efficient
• The symbol for the Karl Pearson's correlation coefficient is r.
• The range of the correlation coefficient is from -1 to +1.
• If there is a strong positive relationship between the variables, the
value of r will be close to +1.
• If there is a strong negative relationship between the variables, the
value of r will be close to -1.
• When there is no relationship between variables or only a weak
relationship, the value of r will be close to 0 (zero).
Formula-Karl Pearson's Correlation Co-efficient
https://study.com/academy/lesson/pearson-correlation-coefficient-formula-examplesignificance.html
Example 1
Compute the value of the Karl Pearson’s Correlation Coefficient for the data
obtained in the study of age and blood pressure
Student
Age (x)
Pressure y
A
43
12
B
48
12
C
56
13
D
61
14
E
67
14
F
70
15
Answer
Student
Age
Pressure y
XY
X2
Y2
∑xy=
∑X2=
∑Y2=
x
A
43
12
B
48
12
C
56
13
D
61
14
E
67
14
F
70
15
Total
∑x=
∑y=
Interpretation of correlation coefficient
https://www.chegg.com/homework-help/definitions/pearson-correlation-coefficient-pcc-31
Exercise : Compute the value of Karl Pearson’s Correlation Coefficient from
the following data.
X
Y
12
5
14
10
16
6
18
10
20
12
22
9
24
10
XY
X2
Y2
Spearman's Rank Correlation Coefficient
• The symbol for the Spearman’s correlation coefficient is rs
• The range of the correlation coefficient is from -1 to +1.
• If there is a strong positive relationship between the variables, the
value of r will be close to +1.
• If there is a strong negative relationship between the variables, the
value of r will be close to -1.
• When there is no relationship between variables or only a weak
relationship, the value of r will be close to 0 (zero).
Spearman's Rank Correlation Coefficient : Steps and
formula
• Rank the two data sets( Rx and Ry). Ranking is achieved by giving the
ranking '1' to the biggest number in a column, '2' to the second biggest
value and so on. The smallest value in the column will get the lowest
ranking. This should be done for both sets of measurements.
• Tied scores are given the mean (average) rank.
• Find Rx-Ry =d
• Square the differences (d²) to remove negative values and then sum
them (∑d²).
Formula
n= number of paired values in the given
data set
Example 1
Compute the value of the Spearman’s Correlation Coefficient for the data obtained in
the study of age and blood pressure
Student
Age (x)
Pressure y
A
43
12
B
48
12
C
56
13
D
61
14
E
67
14
F
70
15
Answer
Students
Age
Pressure y
Rank X
Rank Y
Rx-Ry =d
d²
x
A
43
12
6
5.5
B
48
12
5
5.5
C
56
13
4
4
D
61
14
3
2.5
E
67
14
2
2.5
F
70
15
1
1
∑d²=
Exercise : Compute the value of the Spearman’s Correlation Coefficient
for the following data.
X
Y
10
6
12
12
10
10
9
5
11
9
10
5
Rank X
Rank Y
Rx-Ry=d
d²
Regression
Regression Analysis is a basic and commonly used
type of predictive analysis
Regression Analysis is used to predict dependent
variable(Y) when any one of the independent
variable (X) is varied.
Difference Between Correlation and Regression
• Correlation is described as the analysis which lets us
know the association or the absence of the relationship
between two variables ‘x’ and ‘y’.
• Regression analysis, predicts the value of the
dependent variable based on the known value of the
independent variable, assuming that average
mathematical relationship between two or more
variables.
• https://keydifferences.com/difference-between-correlation-and-regression.html
Difference Between Correlation and Regression
Correlation describes the strength of a linear relationship
between two variables. Linear means “straight line”
Regression tells us how to draw the straight line described by
the correlation. Also known as “best-fit” line .
• Using scatter diagram, one must be able to draw the line of best fit
• Best fit means that the sum of the squares of the vertical distances
from each point to the line is at a minimum
Formula to calculate Regression line
Formulas for the regression line y!= a + b x
𝑎=
( 𝑦)( 𝑥 2 ) − ( 𝑥)( 𝑥𝑦)
𝑛
𝑏=
𝑥 2 − ( 𝑥)
𝑛( 𝑥𝑦) −
𝑛
𝑥
𝑥 2 − ( 𝑥)
Where a is the y! intercept
And b is the slope of the line
2
𝑦
2
Example
Using formula find the equation of the regression line from the following data
Students
Age (x)
Pressure y
A
43
128
B
48
120
C
56
135
D
61
143
E
67
141
F
70
152
Solution:
Step 1:
Find the values of xy, and x2 using following table.
Students
Age (x)
Pressure y
A
43
128
B
48
120
C
56
135
D
61
143
E
67
141
F
70
152
Total
∑x=345
∑y=819
XY
X2
∑xy=
∑X2=
Solution
Step 4:
• Step 2 :
• 𝑎=
𝑥2
𝑦
𝑛
−
𝑥2 −
𝑥
𝑥
The equation of the regression line is
𝑥𝑦
2
y=a+bx
y = 81.048 + 0.964 x
• 𝑎=
819 20399 − 345 47634
6 20399 − 345 2
= 81.048
Question:
If the value of Age(X)= 80, find the value of
Pressure(Y).
• Step 3:
𝑏=
• 𝑏=
𝑛
𝑥𝑦 −
𝑛
𝑥2
𝑥
−
6 47634 − 345 819
6 20399 − 345 2
𝑦
𝑥
2
= 0.964
Answer:
Exercise:
Find the equation of the regression line from the following data. Also
find the value of sales when the advertisement expenditure is OMR
10 millions.
Advertisement Expenditure
Sales
(OMR in Millions)
(OMR in Millions)
3
12
8
16
6
13
1
10
7
14
7
15
Chapter 7: Time series
Definition
• According to Merriam Webster Dictionary, time series is a set of
data collected sequentially and usually at fixed intervals of time.
• Example : The number of packets of milk sold in a small shop
Day
No. of packets of milk sold
Monday
90
Tuesday
88
Wednesday
85
Thursday
75
Friday
72
Saturday
90
Sunday
102
Meaning
Time series data is a sequence of observations
• collected from a process
• with equally spaced periods of time.
Example: Oil production in Oman in Barrels/Day
Example: Population in Oman ( in millions)
Importance of Time Series Analysis in Business
• Profit Planning
• Sales Forecasting
• Stock Market Analysis
• Process and Quality Control
• Economic Forecasting
• Risk Analysis & Evaluation of changes
COMPONENTS OF TIME SERIES
• Any time series can contain some or all of the following components:
1.Secular trend
• The increase or decrease in the movements of a time series is called
Secular trend.
• Secular trend is a long term movement in a time series.
• A time series data may show upward trend or downward trend for a
period of years and this may be due to factors like:
• Increase in population
• Change in technological progress,
• Large scale shift in consumers’ demands
COMPONENTS OF TIME SERIES
2. Seasonal variation
• Seasonal variation are short-term fluctuation in a time series which occur
periodically in a year.
• This continues to repeat year after year.
• The major factors that are weather conditions and customs of people.
• More woolen clothes are sold in winter than the season of summer.
• Each year more ice creams are sold in summer and very little in winter
season
• The sales in the departmental stores are more during festive seasons that
in the normal days.
COMPONENTS OF TIME SERIES
3. Cyclical variation:
• Cyclical variations are recurrent upward or downward
movements in a time series but the period of cycle is greater
than a year.
• Also these variations are not regular as Seasonal variation.
COMPONENTS OF TIME SERIES
• Irregular variation:
➢Irregular variation are fluctuations in time series that are short in
duration, erratic in nature and follow no regularity in the
occurrence pattern.
➢Irregular fluctuations results due to the occurrence of unforeseen
factors.
➢This component is unpredictable.
• Floods
• Earthquakes
• Wars
• Famines
MEASUREMENT OF SECULAR TREND
• Free hand curve method or eye inspection method
• Semi average method
• Method of moving average
Free hand curve method or eye inspection method
In this method the data is denoted on graph paper.
Show “Time” on “X” axis and “ Data” on the “Y” axis.
On graph there will be a point for every point of time.
Draw a smooth hand curve with the help of this plotted points.
Example : Draw a free hand curve on the basis of the following data:
Year
1989
(Profit 148
in
‘000)
1990
1991
1992
1993
1994
1995
1996
149
149.5 150.5 152.2 153.7 153.7 153
Solution: Free hand curve
Semi-Average Method
• In this method the given data are divided in to two parts, preferably
with the equal number of years.
• An average is obtained for each part.
• Each such average is shown against the mid-point of the half period.
• Obtain two points on a graph paper based on the averages of each
part.
• By joining these points, a straight line trend is obtained.
Example:
• Find the trend line from the following data by semi-Average method:
Year
1989 1990
Production 150
152
1991
1992
1993
1994
1995
1996
153
151
154
153
156
158
Solution: Semi-Average Method
• There are total 8 years .
• Divide it into equal parts of 4 years.
• Calculate average for each part.
• First Part =
• Second part =
150+152+153+151
4
154+153+156+158
4
= 151.50
= 155.25
• Obtain two points on a graph paper based
on the averages of each part. Join these
points to get a straight line trend.
Moving Average Method
• This method is based on a series of arithmetic means as shown in the below example.
• Calculate 3-yearly moving average for the following data.
Year
Production
3-Yearly moving average (trend values)
2000- 2001
40
2001- 2002
45
(𝟒𝟎+𝟒𝟓+𝟒𝟎)
𝟑
= 41.67
2002- 2003
40
(𝟒𝟓+𝟒𝟎+𝟒𝟐)
𝟑
= 42.33
2003- 2004
42
(𝟒𝟎+𝟒𝟐+𝟒𝟔)
=
𝟑
2004- 2005
46
2005- 2006
52
2006- 2007
56
2007- 2008
61
(𝟒𝟐+𝟒𝟔+𝟓𝟐)
=
𝟑
(𝟒𝟔+𝟓𝟐+𝟓𝟔)
=
𝟑
(𝟓𝟐+𝟓𝟔+𝟔𝟏)
𝟑
=
42.67
46.67
51.33
56.33

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