TU- STATISTICS.pptx staticsts for bba students

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Statistic
Descriptive Statistic
BBA * 3rd
Semester *
Tabish University

Statistics
Statistics:
Statistics is the branch of
mathematics dealing with the
collection, analysis,
interpretation, presentation and
organization of data
OR

Contt
Statistics :
is the science of collecting, organizing, analyzing,
and interpreting data in order to make decisions

Collecting :
You are doing survey perform experiment.

Organizing :
After the collecting you are organizing the data making table
and graph

Analyzing :
what the data telling you find out the median mean and mode

Interpreting:
After the analyzing the data then we interpret the data.

Branches of Statistics
The study of statistics has two
major branches:
 Descriptive statistics
 Inferential statistics.

Contt
Descriptive Statistic : Consist of the collection,
organization, summarization , and
presentation of the data,
OR
Involves the organization, summarization,
and display of data
OR
It consist of methods for organizing displaying
and describing data by using tables and graph

Example of Descriptive Statistic
Descriptive statistic give information
that describes the data in some
manner . For Example . A supper store
Sells Egg, Bread , milk if 100 items are
sold and 30 out 100 is were milk.
Total Item is 100 and 30 item is milk it
is descriptive statistic and if there is
not given the figure and we are saying
30% is milk so it is inferential statistic

Contt
Second Example of Descriptive Statistic
The Same super store may conduct a
study on the number of bread sold each
day for the one month and determine
that an average of 20 bread each day .
This average is an example of
descriptive statistic

Inferential Statistic
 The process of drawing inference about a
population on the basis of information
contain in a sample taken from the
population is called statistical Inference
or Inferential statistic
 That is statistical inference is the art of
drawing conclusion about the population
from limited information contain in a
sample .

Contt
 Inferential Statistic based on estimation
when we are taking the sample it comes in
the inferential statistic .
 In inferential Statistic the answer is never
100% accurate because the calculation use a
sample taken from the population. This
Sample does not include every
measurement from the population

Contt
 Statistical Inference:
The process of drawing inferences about the
population on the basis of sample information is
called Statistical Inference.

Example
 If we saying that 30% Student of the BBA department daily
expenses is 8o AFN . It is the inferential statistic because we
are estimating and its is not 100% correct as you have studied
that inferential statistic is based on estimation. This inference
we have drawn from the population through sample . And
there is not exact number.

 Another Example : If we are saying that 112 people is
died from the corona disease in year 2020 so it is
descriptive statistic and the Exact figure is known.
Because the data is collected organize, analyze and
display which is meaningful

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CHARACTERISTICS OF STATISTICS
 Statistic deals with behavior of aggregate or large groups of data. It
has nothing to do with what is happening to a particular individual.
 ‫لري‬ ‫کار‬ ‫سره‬ ‫ارقامو‬ ‫او‬ ‫ګرپونو‬ ‫لویو‬ ‫د‬ ‫احصاییه‬
 Statistics are numerically express ‫شو‬ ‫ښودالی‬ ‫بڼه‬ ‫عددي‬ ‫په‬ ‫احصاییه‬
 Statistics are collected in systematic manner‫په‬ ‫ارقام‬ ‫او‬ ‫اعداد‬ ‫احصائیه‬
‫غونډوي‬ ‫توګه‬ ‫سیستماتیکه‬
 It must be enumerated or estimated accurately‫درست‬ ‫او‬ ‫سم‬ ‫په‬ ‫احصاییه‬
‫کړي‬ ‫ترسره‬ ‫تخمین‬ ‫توګه‬
 it should be collected for a predetermined purpo‫پالن‬ ‫نه‬ ‫مخکي‬ ‫د‬ ‫احصاییه‬
‫غونډوي‬ ‫معلومات‬ ‫موخه‬ ‫په‬ ‫هدف‬ ‫شوي‬
 It should be capable of being placed in relation to each other‫ترالسه‬
‫وي‬ ‫کې‬ ‫اړیکه‬ ‫په‬ ‫سره‬ ‫بل‬ ‫له‬ ‫یو‬ ‫باید‬ ‫معلومات‬ ‫شوي‬

Importance Of Statistics
Statistics is perhaps a subject that is used by everybody. The following functions and uses of statistics
in most diverse fields serve to indicate its importance.
 Statistics assists in summarizing the larger sets of data in a form that is easily understandable.
 Statistics assists in drawing general conclusions and in making predictions of how much thing will
happen under given conditions
 A businessman, an industrialist and research worker all employ statistical methods in their work.
Banks, insurance companies and governments all have their statistical departments.
 A modern administrator whether in public or private sector leans on statistical data to provide a
factual basis for decision making.
 A politician also uses statistical data to lent support and credence to his arguments.
 Statistical techniques are also used in biological and physical sciences, Physics, and Geology etc
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4
Tabish University 14

In the Next Chapter we will Study,
Population , Population parameter ,
Sample, Sample plan , Sampling
techniques , Types of data , Sources
of Data , Sampling error and Non
Sampling error with Example .

Lecture No 2
In this lecture we will cover
 Sample , population , Sampling
 sample techniques , Sampling error
 Non Sampling error
 Parameter and Statistic
 Types of Data
 Primary data , Secondary Data
 Sources of Data

Sample :
A sample is the subset of the population selected with the object that it will
represent the characteristic of the population
A population is the collection of all outcomes, responses,
measurement, or counts that are of interest.
CONTT
A sample is a subset of a population.
Sample is a part of the population

Populations And Samples
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Example:
In a recent survey, 250 college students at Kabul
university were asked if they smoked cigarettes
regularly. 35 of the students said yes. Identify
the population and the sample.
Responses of all students
at Kabul university
(population)
Responses of
students in
survey (sample)
Example:

Sampling : Sampling is a statistical
techniques which is used in almost in
every field in order to collect
information at on the basis of this
information inference about the
characteristic of population is drawn .
A techniques of drawing sample which
serves as a representative of population
is known as sampling

Error involved in sampling
 Sampling Error : A sample being
only a part of population can not
perfectly represent the population no
matter have carefully the sample is
selected . The difference between
sample mean and the population mean
is called Sample error ..

Contt
As the sample size increases the sampling error is reduced and at in
complete enumeration there is no sampling error as X become equal to
Memu
The difference between the result found from a sample and the result
found from the population from which the sample was selected

Contt
 Non Sampling Error :
Error which occur at the stage of gathering and processing
data are called non sampling error . Non sampling error
includes all kind of human errors due to faulty sampling fraud
biased method of selection of units
 These error can be avoided through the
 proper selection of questioner ,
 proper training of investigator ,
 Correct manipulating (calculation to apply methods )
of the collected information

Parameters & Statistics
A parameter is a numerical description of a
population characteristic.
A statistic is a numerical description of a sample
characteristic.
Parameter Population
Statistic Sample

Parameters & Statistics
Example:
Decide whether the numerical value describes a
population parameter or a sample statistic.
a.) A recent survey of a sample of 450 college students
reported that the average weekly income for students
is $325.
Because the average of $325 is based on a sample, this is
a sample statistic.
b.) The average weekly income for all students is $405
Because the average of $405 is based on a population,
this is a population parameter.

Types of Data
Primary Data
Secondary Data
Qualitative Data
Quantitative Data

Contt
Primary Data :
The Data which is collected for the first time is called
primary data. Primary data is available in the raw
form .
OR
The data By collected by investigator for his own
purpose for the first time from beginning to end is
called primary data . It is the first hand information.
It is collected from the source of origin that’s why
primary data is original.
The primary data collection sources includes survey,
observation , experiment , questionnaire, personal
interview etc .

Contt
The primary data collection requires a large amount of
resources like time , cost and main power .
 Secondary Data:
Secondary data is the already existing data collected by the
investigator agencies and organization earlier.
Secondary data collection source are government publication ,
websites, books, journal , articles , internal record .
Secondary data collection process is rapid and easy .
Secondary data is obtained when the statistical method is applied
on the primary data

Primary Data and Secondary data
Basis Primary data Secondary Data
Meaning Data originally collected in
the process of investigator
is called primary data
Data collected by the
other person is called
secondary data
Objective Primary data are always
related to a specific
objective of the investigator
Secondary data need to
be adjusted to suit the
objective of study in hand
Original It is original because these
are collected by the
investigator from the source
of their origin
Secondary data are
already in existence and
there for are not original
Basis of Collection Primary data is first hand
information
Second y data is second
hand information

Quantitative data and Qualitative Data
Quantitative Data
The Data which is expressed in numerical value is called quantitative Data
Consists of numerical measurements or counts.
Quantitative Data
As the name suggests is one which deals with quantity or numbers. It
refers to the data which computes the values and counts and can be
expressed in numerical terms is called quantitative data. In statistics, most
of the analysis are conducted using this data
Quantitative data may be used in computation and statistical test. It is
concerned with measurements like height, weight, volume, length, size,
humidity, speed, age etc.

Contt
 Qualitative Data
refers to the data that provides insights and understanding about a
particular problem. It can be approximated but cannot be computed.
Hence, the researcher should possess complete knowledge about
the type of characteristic, prior to the collection of data.
It is concerned with the data that is observable in terms of smell,
appearance, taste, feel,, gender, nationality and so on
OR
Simply Qualitative Data are those data which is not in the numerical
form or not expressed in the numerical form
Qualitative Data
Consists of attributes, labels, or nonnumeric entries.

Contt
Example:
The grade point averages of five students are listed in the table.
Which data are qualitative data and which are quantitative data?
Student GPA
Ahmad 3.22
Zahid 3.98
Shahab 2.75
Sina 2.24
Fawad 3.84
Qualitative data Quantitative data

Lecture No 3
Example of With replacement

Example Sample with replacement
Assume that a uniform population consists of 4 values 0,1,2
and 3.
a) Find the mean µ and the standard deviation σ.
b) Draw random samples of size 2 with replacement and
calculate the mean X of each sample.
c) Find the sampling distribution of X
d) Find the mean and the standard deviation of the sampling
distribution of X
a) Verify that and

 
x n
x

 

Properties of Sampling Distribution of X
The sampling distribution of Mean has the following properties.
1)
2) In case of sampling with replacement
In case of sampling without replacement

 
x
n
x

 
1



N
n
N
n
x



Contt
a) µ= = = = 1.5
Standard Deviation
σ = -
= -
= √1.25 = 1.1180
0+1+2+3
4
6
4
∑X
N
X X
0 0
1 1
2 4
3 9
6 14
2 ∑X
N
2
N
∑X
2
14
4
6
4
2

Example Page 211 Cont
b) Total number of possible samples
N= 4 =16
2
n
S.No Sample Mean (X) S.No Sample Mean( X)
1 (0,0) 0 9 (2,0) 1
2 (0,1) 0.5 10 (2,1) 1.5
3 (0,2) 1 11 (2,2) 2
4 (0,3) 1.5 12 (2,3) 2.5
5 (1,0) 0.5 13 (3,0) 1.5
6 (1,1) 1 14 (3,1) 2
7 (1,2) 1.5 15 (3,2) 2.5
8 (1,3) 2 16 (3,3) 3

Cont
c) Sampling Distribution of X
X f( X)
0 1/16
0.5 2/16
1 3/16
1.5 4/16
2 3/16
2.5 2/16
3 1/16

Cont
d) Mean and the standard deviation of the sampling
distribution of X
X f(X) X f(X) (X) f(X)
0 1/16 0 0
0.5 2/16 1/16 0.5/16
1 3/16 3/16 3/16
1.5 4/16 6/16 9/16
2 3/16 6/16 12/16
2.5 2/16 5/16 12.5/16
3 1/16 3/16 9/16
24/16 46/16
2

Cont
· Mean of X
= 24/16 = 1.5
Standard Error of X
= 46/16 – (24/16)
·
· = 2.875 – 2.25 = √ 0.625 = 0.79
2
 
x
f
x
x 


   
 2
2
x
f
x
x
f
x
x 





Cont
(e) Verification
As
and µ =1.5
Hence
Also
5
.
1

x


 
x
n
Hence
o
and
n
x
x











79
.
79
.
0
2
1180
.
1
1180
.
1

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Different types of variables
 According to relationship:
(a) Dependent variables
(b) Independent variables
According to values
(c) Quantitative variable
- Discrete variable
- Continues variable
(a) Qualitative variable

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Continue….
Independent Variables: are those who take values
by their own, and are not banded to they value of any
other object or variable.
Dependent Variables: variables whose values are
varies due the variation of independent variable, like the
effect of motivation factor on employee’s performance.
The motivation is independent variable and the employee's
performance is a dependent one.

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Continue…..
 Qualitative Variable:
When the characteristic being studied is nonnumeric, it is
called a qualitative variable or an attribute. Examples of
qualitative variables are gender, religious affiliation, type of
automobile owned, state of birth, and eye color. When the
data are qualitative, we are usually interested in how many
or what proportion fall in each category. For example, what
percent of the population has blue eyes?

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Continue…..
 quantitative variable:
When the variable studied can be reported numerically, the
variable is called a quantitative variable. Examples of
quantitative variables are the balance in your checking
account, the ages of company presidents, the life of an
automobile battery (such as 42 months), and the number of
children in a family.
 Quantitative variables are either discrete or continuous.

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Continue…..

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Presentation Of Data

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( ‫بندي‬ ‫صنف‬
Classification
)
‫په‬ ‫شیان‬ ‫یا‬ ‫مشاهدات‬ ‫سیټ‬ ‫یو‬ ‫چې‬ ‫کې‬ ‫کوم‬ ‫په‬ ‫ده‬ ‫عملیه‬ ‫هغه‬
:‫چې‬ ‫ډول‬ ‫داسې‬ ‫په‬ :‫کېږي‬ ‫ویشل‬ ‫باندې‬ ‫ګروپونو‬ ‫یا‬ ‫صنفونو‬ ‫مناسبو‬
.1
‫یو‬ ‫یې‬ ‫ځانګړنې‬ ‫کېږي‬ ‫تړل‬ ‫پورې‬ ‫صنف‬ ‫یو‬ ‫چې‬ ‫مشاهدات‬ ‫هغه‬
.‫وي‬ ‫شان‬
.2
‫مشاهداتو‬ ‫د‬ ‫صنف‬ ‫بل‬ ‫د‬ ‫باید‬ ‫مشاهدات‬ ‫ګروپ‬ ‫یا‬ ‫صنف‬ ‫هر‬ ‫د‬
.‫ولري‬ ‫توپیر‬ ‫سره‬

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‫موخې‬ ‫بندۍ‬ ‫صنف‬

‫رالنډول‬ ‫او‬ ‫خالصه‬ ‫ډیټا‬ ‫د‬

‫ښودل‬ ‫توپیرونو‬ ‫او‬ ‫مشابهتونو‬ ‫د‬ ‫ترمنځ‬ ‫مشاهداتو‬ ‫د‬

‫مخنیوی‬ ‫فشار‬ ‫ذهني‬ ‫د‬ ‫او‬ ‫مخنیوي‬ ‫تفصیل‬ ‫ضروري‬ ‫غیر‬ ‫د‬

‫مساعدول‬ ‫زمینې‬ ‫د‬ ‫لپاره‬ ‫استنباط‬ ‫او‬ ‫مقایسې‬ ‫د‬ ‫معلوماتو‬ ‫د‬

11/18/2024
‫ځانګړتیاوې‬ ‫بندۍ‬ ‫صنف‬

.‫شي‬ ‫کېدای‬ ‫ځای‬ ‫کې‬ ‫صنف‬ ‫یو‬ ‫په‬ ‫یوازې‬ ‫مشاهده‬ ‫یوه‬

.‫وي‬ ‫شوي‬ ‫ځای‬ ‫کې‬ ‫صنفونو‬ ‫ټولو‬ ‫په‬ ‫باید‬ ‫مشاهدات‬ ‫ټول‬

.‫واوسي‬ ‫پیچلې‬ ‫هم‬ ‫نه‬ ‫او‬ ‫ساده‬ ‫نه‬ ‫باید‬ ‫بندي‬ ‫صنف‬
:‫قاعدې‬ ‫بندۍ‬ ‫صنف‬ ‫د‬

،)‫ښځینه‬ ،‫(نارینه‬ ،‫جنس‬ ‫لکه‬ ،‫کېږي‬ ‫ترسره‬ ‫مطابق‬ ‫خاصیت‬ ‫او‬ ‫کیفیت‬ ‫د‬ ‫بندي‬ ‫صنف‬ ‫دلته‬ :‫بندي‬ ‫صنف‬ ‫کیفي‬
.‫نور‬ ‫او‬ ‫مذهب‬ ،‫کچه‬ ‫تعلیم‬

‫رقمي‬ ‫او‬ ‫عددي‬ ‫د‬ ‫مشاهدات‬ ‫او‬ ‫کېږي‬ ‫مطابق‬ ‫مقدار‬ ‫او‬ ‫کمیت‬ ‫د‬ ‫بندي‬ ‫صنف‬ ‫دلته‬ :‫بندي‬ ‫صنف‬ ‫کمي‬
‫نور‬ ‫داسې‬ ‫او‬ ‫اندازه‬ ‫ګټې‬ ‫د‬ ‫خرڅالو‬ ،‫عاید‬ ،‫وزن‬ ،‫لوړوالی‬ ‫لکه‬ ،‫کېږي‬ ‫بندي‬ ‫صنف‬ ‫مخې‬ ‫له‬ ‫مقیاسونو‬

‫بندي‬ ‫صنف‬ ‫اساس‬ ‫په‬ ‫سیمو‬ ‫جغرافیوي‬ ‫د‬ ‫ارقام‬ ‫کې‬ ‫ډول‬ ‫دې‬ ‫په‬ ‫بندۍ‬ ‫صنف‬ ‫د‬ :‫بندي‬ ‫صنف‬ ‫جغرافیوي‬
.‫نور‬ ‫داسې‬ ‫او‬ ‫زون‬ ،‫کلی‬ ،‫ولسوالي‬ ،‫والیت‬ ‫لکه‬ ،‫کېږي‬

.‫کېږي‬ ‫ترسره‬ ‫مطابق‬ ‫وخت‬ ‫د‬ ‫کې‬ ‫سلسله‬ ‫زماني‬ ‫یوه‬ ‫په‬ ‫بندي‬ ‫صنف‬ ‫دلته‬ :‫بندي‬ ‫صنف‬ ‫اساس‬ ‫په‬ ‫وخت‬ ‫د‬
‫کې‬ ‫موده‬ ‫په‬ ‫کلونو‬ ‫مختلفو‬ ‫د‬ ‫صادرات‬ ‫هیواد‬ ‫یوه‬ ‫د‬ ً‫مثال‬

11/18/2024 52
( ‫ویش‬ ‫دفعاتو‬ ‫د‬
Frequency distribution
)

‫هر‬ ‫او‬ ‫شي‬ ‫ځای‬ ‫په‬ ‫ځای‬ ‫کې‬ ‫جدول‬ ‫په‬ ‫ارقام‬ ‫شوي‬ ‫بندي‬ ‫صنف‬ ‫بیا‬ ،‫شي‬ ‫بندي‬ ‫صنف‬ ‫ارقام‬ ‫چې‬ ‫کله‬ ‫هر‬
.‫کيږې‬ ‫ویل‬ ‫ویش‬ ‫دفعاتو‬ ‫د‬ ‫ته‬ ‫شمېر‬ ‫دفعاتو‬ ‫دې‬ ‫د‬ ‫شي‬ ‫ولیکل‬ ‫دفعات‬ ‫صنف‬ ‫مربوطه‬ ‫ته‬ ‫صنف‬
‫د‬ ‫کالم‬ ‫صنفونو‬ ‫د‬
x
‫د‬ ‫کالم‬ ‫دفعاتو‬ ‫د‬ ‫او‬ ‫توري‬ ‫په‬
Y
‫د‬ ‫هم‬ ‫یا‬
Fi
.‫کېږي‬ ‫ښودل‬ ‫توري‬ ‫یا‬ ‫سمبول‬ ‫په‬
:‫مثال‬
Tabish University
‫شمېر‬ ‫حشراتو‬ ‫د‬
x ‫شمېر‬ ‫بوټو‬ ‫د‬
y
0
1
2
3
4
5
6
7
8
25
50
75
100
125
100
75
50
25
150
150
325
(0-2)
(3-5)
(6-8)

11/18/2024 53
‫جوړښت‬ ‫ویش‬ ‫د‬ ‫دفعاتو‬ ‫د‬

:)‫(صنف‬ ‫کالس‬
‫کله‬
‫کالس‬ ‫ورته‬ ‫کې‬ ‫احصائیه‬ ‫په‬ ‫نو‬ ‫شي‬ ‫وټاکل‬ ‫واټن‬ ‫یا‬ ‫فاصله‬ ‫ترمنځ‬ ‫عددونو‬ ‫دوو‬ ‫د‬ ‫چې‬
.‫کېږي‬ ‫ویل‬

:‫حدود‬ ‫صنفي‬
‫او‬ ‫پورتنی‬ ‫صنف‬ ‫هر‬ .‫کېږي‬ ‫بلل‬ ‫حدود‬ ‫کالس‬ ‫د‬ ‫نیسي‬ ‫کې‬ ‫بر‬ ‫په‬ ‫صنف‬ ‫یو‬ ‫چې‬ ‫عددونه‬ ‫دوه‬
.‫کېږي‬ ‫ویل‬ ‫حد‬ ‫لوړ‬ ‫کالس‬ ‫د‬ ‫ته‬ ‫عدد‬ ‫لوړ‬ ‫او‬ ‫حد‬ ‫ټیټ‬ ‫کالس‬ ‫د‬ ‫یې‬ ‫ته‬ ‫کوچني‬ .‫لري‬ ‫حد‬ ‫ښکتنی‬

:)‫(صنف‬ ‫کالس‬ ‫متمادي‬ ‫غیر‬ ‫او‬ ‫متمادي‬
‫کالس‬ ‫راتلونکي‬ ‫د‬ ‫حد‬ ‫پورتنی‬ ‫کالس‬ ‫وړ‬ ‫نظر‬ ‫د‬ ‫کې‬ ‫کالسونو‬ ‫متمادي‬ ‫په‬
.‫وي‬ ‫نه‬ ‫مساوي‬ ‫سره‬ ‫حدود‬ ‫کالسونو‬ ‫د‬ ‫کې‬ ‫متمادي‬ ‫غیر‬ ‫په‬ ‫مګر‬ ‫وي‬ ‫مساوي‬ ‫سره‬ ‫حد‬ ‫ښکتني‬ ‫د‬

:‫دفعات‬ ‫صنفي‬
.‫کېږي‬ ‫ویل‬ ‫دفعات‬ ‫صنفي‬ ‫ته‬ ‫شمېر‬ ‫حادثاتو‬ ‫او‬ ‫پېښو‬ ‫اړونده‬ ‫پورې‬ ‫کالس‬ ‫یوه‬ ‫د‬

:)‫(وسط‬ ‫نیمايي‬ ‫صنف‬ ‫د‬
‫د‬ ‫چې‬ ‫کله‬ ‫یعني‬ .‫ویشي‬ ‫برخو‬ ‫مساوي‬ ‫دوه‬ ‫پوره‬ ‫په‬ ‫اعداد‬ ‫صنف‬ ‫د‬ ‫چې‬ ‫دی‬ ‫عدد‬ ‫هغه‬
‫دوو‬ ‫په‬ ‫بیا‬ ‫او‬ ‫جمع‬ ‫سره‬ ‫حد‬ ‫ټیټ‬ ‫او‬ ‫لوړ‬ ‫صنف‬ ‫یوه‬
2
‫کالس‬ ‫د‬ ‫یې‬ ‫حاصل‬ ‫راغلې‬ ‫السته‬ ‫نو‬ ‫شي‬ ‫وویشل‬ ‫باندې‬
.‫کېږي‬ ‫بلل‬ ‫نیمايي‬ ‫یا‬ ‫وسط‬

:‫سرحدات‬ ‫صنف‬ ‫د‬
‫کله‬ .‫یادیږي‬ ‫نوم‬ ‫په‬ ‫سرحداتو‬ ‫صنفي‬ ‫د‬ ‫چې‬ ‫لري‬ ‫پولي‬ ‫ټاکلي‬ ‫خواته‬ ‫دوو‬ ‫خپلو‬ ‫صنف‬ ‫هر‬
‫د‬ ‫په‬ ‫بیا‬ ‫او‬ ‫جمع‬ ‫سره‬ ‫حد‬ ‫لوړ‬ ‫کالس‬ ‫مخکیني‬ ‫د‬ ‫حد‬ ‫ټیټ‬ ‫کالس‬ ‫وړ‬ ‫نظر‬ ‫د‬ ‫مونږ‬ ‫چې‬
2
‫ټیټ‬ ‫نو‬ ‫وویشو‬ ‫باندې‬
‫دوو‬ ‫په‬ ‫او‬ ‫جمع‬ ‫سره‬ ‫حد‬ ‫ټیټ‬ ‫له‬ ‫کالس‬ ‫وروستي‬ ‫د‬ ‫حد‬ ‫لوړ‬ ‫کالس‬ ‫وړ‬ ‫نظر‬ ‫د‬ ‫چې‬ ‫کله‬ ‫او‬ ‫راځي‬ ‫السته‬ ‫سرحد‬
.‫کېږي‬ ‫ترالسه‬ ‫سرحد‬ ‫لوړ‬ ‫کالس‬ ‫اړونده‬ ‫د‬ ‫نو‬ ‫شي‬ ‫وویشل‬ ‫باندې‬
Tabish University

11/18/2024 54
‫جوړښت‬ ‫ویش‬ ‫د‬ ‫دفعاتو‬ ‫د‬

:‫وسعت‬ ‫یا‬ ‫عرض‬ ‫صنفي‬
‫ګوته‬ ‫په‬ ‫ته‬ ‫مونږ‬ ‫پراخوالی‬ ‫یا‬ ‫وسعت‬ ‫صنف‬ ‫یوه‬ ‫د‬ ‫وسعت‬ ‫صنفي‬ ‫د‬
‫شوی‬ ‫ترالسه‬ ‫نو‬ ‫شي‬ ‫کړل‬ ‫منفي‬ ‫څخه‬ ‫سرحد‬ ‫لوړ‬ ‫د‬ ‫سرحد‬ ‫ټیټ‬ ‫کالس‬ ‫یوه‬ ‫د‬ ‫چې‬ ‫کله‬ .‫کوي‬
:‫ډول‬ ‫په‬ ‫مثال‬ .‫کېږي‬ ‫شمېرل‬ ‫وسعت‬ ‫کالس‬ ‫د‬ ‫حاصل‬
Class interval= upper boundary – lower boundary, CI= 5.5 – 2.5 = 3

:‫فاصله‬ ‫صنفي‬
‫ترمنځ‬ ‫مشاهدې‬ ‫کوچنۍ‬ ‫ټولو‬ ‫تر‬ ‫او‬ ‫مشاهدې‬ ‫لوې‬ ‫ترټولو‬ ‫کې‬ ‫مشاهداتو‬ ‫په‬
:‫کېږي‬ ‫ویل‬ ‫فاصله‬ ‫یا‬ ‫رنج‬ ‫ته‬ ‫توپیر‬
R= Xmax – Xmin

:‫معلومول‬ ‫شمېر‬ ‫صنفونو‬ ‫د‬
( ‫قاعدې‬ ‫له‬ ‫سټرجي‬
K= 1+ 3.3 log N
‫کوو‬ ‫پورته‬ ‫ګټه‬ ‫څخه‬ )

:‫فریکونسي‬ ‫نسبي‬
‫نو‬ ‫شي‬ ‫وښودل‬ ‫بڼه‬ ‫په‬ ‫فیصدۍ‬ ‫د‬ ‫فریکونسي‬ ‫کالس‬ ‫یوه‬ ‫د‬ ‫چې‬ ‫کله‬
‫ده‬ ‫فریکوینسي‬ ‫مجموعي‬ ،‫ډول‬ ‫په‬ ‫مثال‬ .‫کیږي‬ ‫بلل‬ ‫فریکوینسي‬ ‫نسبي‬
۵۰
‫کالس‬ ‫وړ‬ ‫نظر‬ ‫د‬ ‫او‬
‫ده‬ ‫فریکوینسي‬
۲۰
:‫له‬ ‫دی‬ ‫عبارت‬ ‫یې‬ ‫فریکونسي‬ ‫نسبي‬ ‫نو‬
Tabish University

Difference between Relative and
cumulative frequency
Relative frequency represents the ratio of the number
of times a value of the data occurs in a dataset. While
cumulative frequency represents the sum of the
relative frequency. Relative frequency can be written
in percentage & cumulative frequency is the collection
of all previous frequencies.

Relative Frequency
Relative Frequency Distribution
Class Marks Frequency
R.F=
(
R.F %=
RF 100
C.F
30-34 2 %
35-39 5
40-44 10
45-49 14
50-54 8
55-59 7
60-64 4
Total 50 1 = 100%
• Add two more columns to the
previous frequency table.
• In the third column calculate
relative frequency by dividing
the frequency of the class on
total class.
• In the fourth column multiply
relative frequency to 100 to get
R.F in percentage.
• Repeat the process for each
class.
• Total Relative frequency must
equal to 1. if not so then there's
something wrong with
calculations.
• Total Relative frequency in
percentage must equal to 100%.
if not so then there's something
wrong with calculations.

Cumulative Frequency
Cumulative Frequency Distribution
Class Marks Freq
uenc
y (f)
R.F
C.F
Less Than
Cumulative F
More than
Cumulative F
30-34 2 2 50
35-39 5 7 48
40-44 10 17 43
45-49 14 31 33
50-54 8 39 19
55-59 7 46 11
60-64 4 50 4
Total 50 1
• As relative frequency is the
percentage frequency of each class
in respect to the total frequency.
• Cumulative frequency is the
running to total of the values of
previous class to the specific class.
• It can be a less than C.F where
adding start from the top or it can
be more than C.F where adding
start from below.
• (I) we add the previous cumulated
frequency to the new class
individual frequency to get CF for
that class.
next class: .2+.5=.7

Measures of Central
Tendency / location
MEAN, MEDIAN,
MODE

Measure of central tendency
 It is descriptive statistical measure which shows the
midpoint of the data that describe the whole data.

‫په‬‫وسط‬‫یا‬‫نقطه‬‫منځنې‬‫سیټ‬‫ډیټا‬‫د‬‫ته‬‫مونږ‬‫چې‬‫کوم‬،‫کېږي‬‫شمیرل‬‫څخه‬‫وسایلو‬‫د‬‫احصائیې‬‫تشریحي‬‫د‬
.‫شو‬‫کوالی‬‫ترسره‬‫تشریح‬‫سیټ‬‫ډیټا‬‫ټول‬‫د‬‫مونږ‬‫مرسته‬‫په‬‫وسط‬‫همدې‬‫د‬.‫کوي‬‫ګوته‬
A measure of central tendency (also referred to as measures of centre or central location) is a summary
measure that attempts to describe a whole set of data with a single value that represents the middle or centre of
its distribution.
‫سیټ‬‫ډیټا‬‫ټول‬‫د‬‫ته‬‫مونږ‬‫چې‬‫کوم‬‫ده‬‫خالصه‬‫ګیریو‬‫اندازه‬‫د‬ ‫ي‬‫ګیر‬‫اندازه‬‫تمایالتو‬ ‫ي‬‫مرکز‬‫د‬
.‫ده‬‫نقطه‬‫منځنۍ‬‫مشاهداتو‬‫د‬‫هم‬‫یا‬.‫کوي‬‫ترسره‬‫وسیله‬‫په‬‫ښت‬‫ز‬‫ار‬‫واحد‬‫یوه‬‫د‬‫ښودنه‬

Measures of Central Location /Mean
1. The arithmetic mean:
also known as average, shortened to mean, is the most
popular & useful measure of central location.
 It is computed by simply adding up all the observations
and dividing by the total number of observations:
4.60
Sum of the observations
Number of observations
Mean = 𝑿=
∑ƒ 𝒙
∑ƒ

Notation…
 When referring to the number of observations in a population, we use
uppercase letter N
 When referring to the number of observations in a sample, we use
lower case letter n
 The arithmetic mean for a population is denoted with Greek letter
“mu”:
 The arithmetic mean for a sample is denoted with an “x-bar”:
4.61

Statistics is a pattern language…
4.62
Population Sample
Size N n
Mean

Population
Mean
 Where:
 µ = represents the population mean.
It is the
Greek letter “mu.”
 N = is the number of items in the
population.
 X = is any particular value
 Σ = indicates the operation of adding
all
the values. It is the Greek letter
“sigma.”
 ΣX = is the sum of the X values.
Sample Mean
 Xbar = represents the
sample mean.
 = is the number of items in
sample or sample size.
 = is any particular value
 Σ = indicates the operation
of adding all the values.
It is the Greek letter “sigma.”
 = is the sum of the X values.

Calculate Sample and population mean
 Example: A sample of five executives received the following bonus
last year
14.0, 15.0, 17.0, 16.0, 15.0
= = 14+14+17+16+15/5 = 77/5 = 15.4
 Example: A population of executives received the following bonus
last year
10, 12, 8, 13, 14, 7
= 10+12+8+13+14+7/6 = 64/6 = 10.66

Calculating Mean for Frequency Distribution
When group data is given with frequency we use the following formula.
Sample mean for frequency distribution:
Where,
= sample mean
= frequency of each class
= class mid point or class marks
= product of (F) and (X) values.
is the sum of all the fx values.
= sum of all the frequencies.

The following series relates to the marks secured by students
in an examination.
Requirement: Calculate mean for the
following data.
Marks
No. of
students
0-10 11
11-20 18
21-30 25
31-40 28
41-50 30
51-60 33
Example:

Cont’d
Illustration:
 Midpoint (x): x value in a grouped frequency distribution is taken
by dividing the sum of a class upper and lower value by 2. for
example for the first class of our example 0+102= 5
 (f): Frequency for each class.
 Fx: Column (F) multiply by column (X) is equal to FX. We then
sum all the values of each column and put the sum values in the
formula.

Solution: Marks (f) Midpoint (x) fx
0-10 11 5 55
10-20 18 15 270
20-30 25 25 625
30-40 25 35 980
40-50 30 45 1350
50-60 33 55 1815
∑ 145 5095

Median & Mode
For grouped and ungrouped frequency distributions

Median
 Is known as midpoint of all data set or when we divide data
into two portions 50% by 50%. The midpoint or border line is
known as median.
Difference between Mean and Median
The main difference b/w median and mean is data distribution or
shape.
 When data distribution is normal we use mean and when the
data shape is not normal we use median. Which means that in
start the values of the data is small and at the end the data
value or data point is really high.

Example:
 The middle number; found by ordering all data points and
picking out the one in the middle (or if there are two middle
numbers, taking the mean of those two numbers).
 Example: The median of 4, 1, and 7 is 4 because when the
numbers are put in order (1 , 4, 7) , the number 4 is in the
middle

Median
 The Median is that value of the data which divides the
group into two equal parts, one part comprising all values
greater, and the other, all values less than median.
 When odd number of values is given we can find median
by following formula:
 When even number of values is given then Median is the
average of the & items.
 to identify the first boundary and to identify the 2nd
boundary.

Median
 Example when the numbers of values are odd.
Find median for 25, 18, 27, 10, 8, 30, 42, 20, 53
 Solution: Arranging the data in the increasing order 8, 10, 18, 20, 25,
27, 30, 42, 53
 The middle value is the 5th
item i.e., 25 is the median
 Using formula

Median
 Example when the number of values are even.
Find median for 5, 8, 12, 30, 18, 10, 2, 22
 Solution: Arranging the data in the increasing order 2, 5, 8, 10, 12, 18,
22, 30
 Here median is the mean of the middle two items (i.e.) mean of (10,
12) i.e.
 = 4th
 Item
 So median is between the 4th
and the 5th
item (i.e.) mean of (10, 12)

Median (grouped frequency distributions)
 The steps given below are followed for the calculation of median in continuous series.
 Step1: Find cumulative frequencies.
 Step2: Find (N / 2)th item (Sum of total frequency /2)
 Step3: find the cumulative frequency value which is near to the sum of total frequency. Then the
corresponding class interval is called the Median class. Then apply the formula
 Where;
 l lower boundary of the median class
 h is the class interval
 f is the frequency of the median class
 N is The sum of frequency
 Cf is cumulative frequency of the preceding class

 Example: Find median for the distribution of examination marks given below.
 Solution:
Marks 30 - 39 40 – 49 50 - 59 60 - 69 70 - 79 80 - 89 90 - 99
No of
students
(f)
8 87 190 304 211 85 20
Class
Boundaries
Midpoints
(x)
Frequency
(f )
Cumulative frequency
(Cf )
29.5 – 39.5 34.5 8 8
39.5 – 49.5 44.5 87 95
49.5 – 59.5 54.5 190 285
59.5 – 69.5 64.5 304 589
69.5 – 79.5 74.5 211 800
79.5 – 89.5 84.5 85 885
89.5 – 99.5 94.5 20 905

→ Median = marks obtained by students
→ Median = marks obtained by
→ Median = marks obtained by
→ Which corresponds to the class 59.5 – 69.5
→ Using the following formula:
→ Putting values in the above formula
→
→
→
→

Mode
 Most re-occurring value in data is known as mode.
For example: If we ask 10 students about their marks and
the response is 85, 79, 85, 90, 86, 90, 90, 83, 88, 90
The mode is 90 because it the most re occurring value in
the data.
 We use Mode mostly is market analysis. Such as if we
collect data that which number of shoes are sold. We
will produce the one which have market value.

Mode
 The MODE refers to that value in a distribution, which occur most
frequently. It is an actual value, which has the highest concentration of
items in and around it.
For ungrouped data:
For ungrouped data or a series of individual observations, mode is often
found by mere inspection.
Example: Find out Mode for the following data?
2, 7, 10, 15, 10, 17, 8, 10, 2
Ascending order ( 2, 2, 7, 8, 10, 10, 10, 15, 17)
Mode = M0 =10
 In some cases the mode may be absent while in some cases there may
be more than one mode. If 2 also comes 3 times it is know as bio-model
data and if 7 also come 3 time then it is know as tri-model data.

Mode (for frequency distribution)
Mode for grouped frequency distribution:
 See the highest frequency then the corresponding value of class interval
is called the modal class. Then apply the formula
Where;
 l = lower class boundary of the Model Class,
 f m = frequency of the Model Class, (higher frequency in the data is
FM)
 f1 = frequency of the class preceding the model class,
 f2 = frequency of the class following the model class, and
 h = width of the class interval

Example: Calculate mode for the following data:
C-I F
0-50 5
50-100 14
100-150 40
150-200 91
200-250 150
250-300 87
300-350 60
350-400 38
400-450 15

Solution:
 The highest frequency is 150 and corresponding class
interval is 200 –250, which is the modal class.
While,
l = 200
f m = 150,
f1 = 91,
f2 = 87
h = 50

Cont`d
Putting values in the formula

11/18/2024

Correlation
Two or more variables are said to be
correlated if the change in one
variable results in a corresponding
change in the other variable.
According to Simpson and Kafka,
“Correlation analysis deals with the
association between two or more
variables”.

 Lun chou defines, “ Correlation analysis
attempts to determine the degree of
relationship between variables”.
 Boddington states that “Whenever some
definite connection exists between two or
more groups or classes of series of data,
there is said to be correlation.”

 Correlation can be classified in
different ways. The following are the
most important classifications
 Positive and Negative correlation
 Simple, partial and multiple
correlation
 Linear and Non-linear correlation

Classification of Correlation

 When the variables are varying in the same
direction, it is called positive correlation. In
other words, if an increase in the value of
one variable is accompanied by an
increase in the value of other variable or if
a decrease in the value of one variable is
accompanied by a decree se in the value of
other variable, it is called positive
correlation.
Positive and Negative Correlation
Positive Correlation

 When the variables are moving in
opposite direction, it is called negative
correlation. In other words, if an increase
in the value of one variable is
accompanied by a decrease in the value of
other variable or if a decrease in the value
of one variable is accompanied by an
increase in the value of other variable, it is
called negative correlation
Negative Correlation

 Simple Correlation
 In a correlation analysis, if only two variables are studied it is
called simple correlation. Eg. the study of the relationship
between price & demand, of a product or price and supply of a
product is a problem of simple correlation
Simple, Partial and Multiple correlation

 Multiple correlation
 In a correlation analysis, if three or more variables are studied
simultaneously, it is called multiple correlation. For example,
when we study the relationship between the yield of rice with
both rainfall and fertilizer together, it is a problem of multiple
correlation.

 Correlation exists in various degrees
 Perfect positive correlation
 If an increase in the value of one variable is
followed by the same proportion of increase in
other related variable or if a decrease in the
value of one variable is followed by the same
proportion of decrease in other related variable,
it is perfect positive correlation. eg: if 10% rise in
price of a commodity results in 10% rise in its
supply, the correlation is perfectly positive.
Similarly, if 5% full in price results in 5% fall in
supply, the correlation is perfectly positive.
Degrees of correlation:

 If an increase in the value of one variable is
followed by the same proportion of decrease in
other related variable or if a decrease in the value
of one variable is followed by the same
proportion of increase in other related variably it
is Perfect Negative Correlation. For example if
10% rise in price results in 10% fall in its demand
the correlation is perfectly negative. Similarly if
5% fall in price results in 5% increase in demand,
the correlation is perfectly negative
Perfect Negative correlation

 If there is no correlation between
variables it is called zero correlation. In
other words, if the values of one variable
cannot be associated with the values of
the other variable, it is zero correlation.
Zero Correlation (Zero Degree correlation)

Correlation analysis is of
immense use in practical life
because of the following
reasons:
1 :Correlation analysis helps us to find a single
figure to measure the degree of relationship exists
between the variables
SIGNIFICANCE OF CORRELATION ANALYSIS

2: Correlation analysis helps to
understand the economic behavior
3: Correlation analysis enables the
business executives to estimate cost,
price and other variables.

• 4 : Correlation analysis can be used as a basis for
the study of regression. Once we know that two
variables are closely related, we can estimate the
value of one variable if the value of other is
known.
5: Correlation analysis helps to reduce the range of
uncertainty associated with decision making. The
prediction based on correlation analysis is always
near to reality

6:It helps to know whether the
correlation is significant or not. This is
possible by comparing the correlation
co-efficient with 6PE. It ‘r’ is more than
6 PE, the correlation is significant.

 Karl Pearson’s Coefficient of Correlation is the most
popular method among the algebraic methods for
measuring correlation. This method was developed
by Prof. Karl Pearson in 1896. It is also called
product moment correlation coefficient.
Karl Pearson’s Co-efficient of Correlation

 Pearson’s Co-efficient of correlation
always lies between +1 and -1. The
following general rules will help to
interpret the Co-efficient of correlation:
Interpretation of Co-efficient of Correlation

 When r - +1, It means there is
perfect positive relationship
between variables.
 When r = -1, it means there is
perfect negative relationship
between variables.
 When r = 0, it means there is no
relationship between the variables.

 When ‘r’ is closer to +1, it means there is
high degree of positive correlation
between variables.
 When ‘r’ is closer to – 1, it means there is
high degree of negative correlation
between variables.
 When ‘r’ is closer to ‘O’, it means there is
less relationship between variables.

 If there is correlation between variables,
the Co-efficient of correlation lies
between +1 and -1.
 If there is no correlation, the coefficient
of correlation is denoted by zero (ie r=0)
 It measures the degree and direction of
change
 If simply measures the correlation and
does not help to predict cansation.
Properties of Pearson’s Co-efficient of
Correlation

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