Business statistics

BUSINESS STATISTICS FOR BBA (HONS) 4TH SEMESTER
Presentation by Sajjad Hussain (Sajjad Chitrali)
INTRODUCTION
There are different definitions of “Statistics” and each researcher has defined it in their own
terms. For example;
 Statistics is a science of sampling and estimation.
 Statistics is a science of probability.
 Statistics is a science of collecting information/data.
 Statistics is a science of presentation of data either in qualitative or
quantitative form.
 Statistics is a science of analyzing the data.
 Statistics is a science of collection, presentation, analysis and
interpretation of numerical data.
IMPORTANCE OF STATISTICS
 Statistical methods are used for summarization of a large set of data.
 Statistical methods are used for analyzing the data related to filed and lab
experiments.
 Statistical methods are used for conducting sampling surveys, also the data coming
from surveys can be analyzed by using statistical methods to find solutions of the
problems under study.
 Statistical methods are helpful in effective planning in any field of inquiry.
 Statistical methods are used in each and every filed of scientific discipline like
agriculture, business, medical, biological, genetics, physical and social sciences etc.
 Banks, insurance companies, government and semi-government organizations, are
using statistical techniques as a tool for data analysis.
 Statistics helps in drawing general conclusions about the characteristics of a
population/aggregate on the basis of sample data;
 Statistical methods are also helpful in making prediction (forecasting).
What Is Business Statistics?

Briefly defined, business statistics can be described as the collection, summarization,
analysis, and reporting of numerical findings relevant to a business decision or situation.
Naturally, given the great diversity of business itself, it’s not surprising that statistics can be
applied to many kinds of business settings. We will be examining a wide spectrum of such
applications and settings. Regardless of your eventual career destination, whether it be
accounting or marketing, finance or politics, information science or human resource
management, you’ll find the statistical techniques explained here are supported by examples
and problems relevant to your own field.
Types of Statistics
As we have seen, statistics can refer to a set of individual numbers or numerical facts, or to
general or specific statistical techniques. A further breakdown of the subject is possible,
depending on whether the emphasis is on (1) simply describing the characteristics of a set of
data or (2) proceeding from data characteristics to making generalizations, estimates,
forecasts, or other judgments based on the data. The former is referred to as descriptive
statistics, while the latter is called inferential statistics. As you might expect, both
approaches are vital in today’s business world.
Descriptive Statistics
Descriptive statistics deals with concepts and methods related with the summarization and
description of the important aspects of numerical data. It consists of condensation of data,
STATISTICS
DescriptiveStatistics
Presentation of Data
(Graphs and Diagrams)
Tabulation and
Classification
Measures of Central
Tendency and
Dispersion
Inferential Statistics
Estimation of
Parameters
Testing of Hypothesis

their graphical displays and computation of numerical quantities that can provide information
about the Centre and spreadness of observations of a data set.
In descriptive statistics, we simply summarize and describe the data we’vecollected. For
example, upon looking around your class, you may find that 35%of your fellow students are
wearing Casio watches. If so, the figure “35%” is adescriptive statistic. You are not
attempting to suggest that 35% of all collegestudents in the United States, or even at your
school, wear Casio watches. You’remerely describing the data that you’ve recorded. For now,
however, just remember that descriptivestatistics are used only to summarize or describe.
Inferential Statistics
Inferential statistics deals with methods and procedures used for drawing inferences about the
true but unknown characteristics of a population based on the sample data derived from the
same population. Inferential statistics can be further classified into estimation of parameters
and testing of hypothesis.
In inferential statistics, sometimes referred to as inductive statistics, we go beyondmere
description of the data and arrive at inferences regarding the phenomenonor phenomena for
which sample data were obtained. For example, based partially on an examination of the
viewing behaviour of several thousand television households, the ABC television network
may decide to cancel a prime-time television program. In so doing, the network is assuming
that millions of other viewers across the nation are also watching competing programs.
Key Terms for Inferential Statistics
In surveying the political choices of a small number of eligible voters, political pollsters are
using a sample of voters selected from the population of all eligible voters. Based on the
results observed in the sample, the researchers then proceed to make inferences on the
political choices likely to exist in this larger population of eligible voters. A sample result
(e.g., 46% of the sample favour Charles Grady for president) is referred to as a sample
statistic and is used in an attempt to estimate the corresponding population parameter (e.g.,
the actual, but unknown, national percentage of voters who favour Mr. Grady). These and
other important terms from inferential statistics may be defined as follows:
• Population Sometimes referred to as the universe, this is the entire set of people or objects
of interest. It could be all adult citizens in the United States, all commercial pilots employed
by domestic airlines, or every roller bearing ever produced by the Timken Company.

An aggregate or totality having some common characteristics on interest is called
population. It is also called Universe. For example, total number of students enrolled in
IBMS-Peshawar, total number of markets in Peshawar city, total number of banks in
Hayatabad, number of industries in the province, monthly/yearly sales of the stores in
Peshawar district, etc.
There are different types of population e.g. finite and infinite population, homogeneous and
heterogeneous population etc.
Sample This is a smaller number (a subset) of the people or objects that exist within the
larger population. The retailer in the preceding definition may decide to select her sample by
choosing every 10th person entering the store between 9 a.m. and 5 p.m. next Wednesday.
A small representative part of population is called sample. For example, a small portion/part
of students of IBMS-Peshawar will constitute a sample of students. Similarly, a
randomly/purposively selected number of markets from a bulk of markets are called a sample
of markets.
A sample is said to be representative if its members tend to have the same characteristics
(e.g., voting preference, shopping behaviour, age, income, educational level) as the
population from which they were selected. For example, if 45% of the population consists of
female shoppers, we would like our sample to also include 45% females. When a sample is so
large as to include all members of the population, it is referred to as a complete census.
Parameter:Any numerical quantity like mean, standard deviation etc. computed/obtained
from population data is known as parameter. For example, average monthly/yearly sale of all
the stores located in district Peshawar etc. Parameters are generally used to specify the
distribution of data.
This is a numerical characteristic of the population. If we were to take a complete census of
the population, the parameter could actually be measured. As discussed earlier, however, this
is grossly impractical for most business research. The purpose of the sample statistic is to
estimate the value of the corresponding population parameter (e.g., the sample mean is used
to estimate the population mean).
Statistic: Any numerical quantity like mean, standard deviation etc. computed from sample
data is called statistic. For example, the average GPA of the 50 students that are selected from

a population of 300 students. Similarly, the average sale of 100 stores instead of 1000 stores
etc. is the examples of statistic(s).
VARIABLE AND ITS TYPES
Variable
Any characteristic of interest which takes on different values is called variable. For example,
price of a commodity at different places in Peshawar city, profit of a business firm at
different months of a year, production, cost, temperature, sale of a market, consumption etc.
Variable is broadly divided into qualitative and quantitative variables.
Variables express how much of an attribute is possessed. Discrete quantitative variables can
take on only certain values along an interval, with the possible values having gaps between
them, while continuous quantitative variables can take on a value at any point along an
interval. When a variable is measured, a numerical value is assigned to it, and the result will
be in one of four levels, or scales, of measurement—nominal, ordinal, interval, or ratio. The
scale to which the measurements belong will be important in determining appropriate
methods for data description and analysis. By helping to reduce the uncertainty posed by
largely uncontrollable factors, such as competitors, government, technology, the social and
economic environment, and often unpredictable consumers and voters, statistics plays a vital
role in business decision making. Although statistics is a valuable tool in business, its
techniques can be abused or misused for personal or corporate gain. This makes it especially
important for businesspersons to be informed consumers of statistical claims and findings.
Qualitative and Quantitative Variables
A variable is defined to be qualitative which is not capable of numerical measurement but
one can feel the presence or absence of a particular phenomenon. For example, honesty,
beauty, race, like and dislike, pass or fail, gender classification etc.
A variable is defined to be quantitative which is capable of numerical measurement. For
example, cost of production, price of a commodity, monthly consumption of households etc.
Discrete and Continuous Variables
A variable is said to be discrete if it takes isolated integral values or a variable which take the
values on jumps is called a discrete variable. For example, number of rooms in a house,

number of students in the class, number of Banks in different cities, size of a household,
number of shops in a market etc.
A type of variable which takes all possible values with in a given interval/range (a, b). For
example, consumption, production, temperature, monthly sale of a market, height, weight and
age etc.
Dependent and Independent Variables
A type of variable which is influenced by other variable/variables is called dependent
variable. It is also called random or stochastic variable. OR
A variable which depends on one or more other variables is called dependent variable. OR
A variable of primary interest that lends itself for investigation as a function of other cause
variables is known as dependent variable.
For example, in economics, consumption of a commodity (say apple) depends upon the
income, household size, and price etc. of the commodity. In this example, consumption of
apple is a dependent variable which will vary from one family to other family; while the other
variables like income, household size and price are independent variables.
A variable which influence a dependent variable in either direction (positive or negative) is
called independent variable.
Meaning and Purpose of Data
Data means observations or evidences. OR, the raw facts and figures/collection of meaningful
information is called data. Data are both qualitative and quantitative in nature.
The data are needed in a research work to serve the following purposes:
1. Quality of data determines the quality of research.
2. It provides a direction and answer to a research inquiry. Data are very essential for
conducting a research.
3. The main purpose of data collection is to verify the hypotheses.
4. Data are necessary to provide the solution of the problem.

5. Data are also employed to ascertain the effectiveness of new device for its practical utility.
6. Statistical data are used in two basic problems of any investigation:
(a). Estimation of population parameters, which helps in drawing generalization about the
population characteristics.
(b).The hypotheses of any investigation are tested with the help of data.
Types of Data
Primary Data: The data which is collected for the first time from its source, is called primary
data.
Secondary Data: When the primary data is passed through any sort of statistical or
mathematical treatment, the data is known as secondary data.
OR, the data that are collected and compiled by an outside source or by someone in the
organization who may later provide access to the data to other users.
Collection of Data
Collection of Primary Data
1. Primary data can be collected through:
2. Direct personal investigation
3. Indirect investigation or personal interviews
4. Collection through questionnaire
5. Collection through enumerators
6. Collection through local sources
Collection of Secondary Data
1. Secondary data can be collected from:
2. Collection from official records
3. Collection from semi-official records
Collection of Primary Data
i. Direct Personal Investigation

According to this method, the researcher/investigator collect information in person from the
selected respondents. In this method an investigator has a degree of freedom and open
choices of asking a variety of questions (open ended and closed ended or mixed). The data
collected through this method is complete; however, personal bias can be present due to
personal involvement of the investigator. Also this method is very costly and time
consuming.
ii.Indirect Investigation
In some cases, it is not possible to take direct information from the respondents due to certain
limitations. So, in such a circumstances indirect investigation is carried out by involving a
third party for collecting the required information. This method is useful in conducting the
inquiries or the information are the information required are complex.
iii. Collection through questionnaire
In this method, a list of questions (called questionnaire) is prepared by the
researcher/investigator covering all aspects of the study being required. A list of questions is
send to the respondents through mail or email with a request to send back after answering all
the listed questions. In this method it is possible that the respondents keep some of the
questions blank due to no understanding or don’t want to give information about those
questions. In addition, some of the respondents are not willing at all to given any of the
information that are contained in the questionnaire.
iv. Collection through enumerators
According to this method, trained peoples are send to the area under study for collecting
information on a pre-specified Performa. Information collected through this method will be
more useful as compared to the questionnaire method. In this method, the enumerators can
take information from the respondents directly or may be his/her closed relatives (if not
available on the spot).
V. Collection through local sources
As the name indicates that by using this method, data are collected through local sources.
Local sources means that information are not directly collected from the respondents but the
desired information are collected from the people belong the area about which information
are required.

Time Series Data
The data collected at different interval of time regarding a commodity or group of
commodities (or organization/firm) is called time series data. For example,
Time Series data of a company showing profit, production and sale.
Year Profit Production Sale
1990 12 120 110
1991 13 140 132
1992 14 150 145
1993 13.5 140 123
1994 10 103 90
1995 11 115 100
1996 12.5 123 122
1997 13.8 140 135
1998 15 160 145
2000 11.6 120 115
2001 15 162 150
2002 16 165 145
Cross Sectional Data
Widely dispersed data (such as) relating to one period, or data related to households, data
collected from the field survey i.e. monthly profit of the selected stores or monthly profit of
different companies related to only one period etc.
Cross sectional data of 12 different households showing profit, production and sale.

Household Profit Production Sale
1 12 120 110
2 13 140 132
3 14 150 145
4 13.5 140 123
5 10 103 90
6 11 115 100
7 12.5 123 122
8 13.8 140 135
9 15 160 145
10 11.6 120 115
11 15 162 150
12 16 165 145
Frequency
Repetition of an observation in a data set is called frequency of that particular
observation/data point/individual. OR
Total number of observations in a class is called the frequency of that class. For example,
consider the following data showing the monthly salaries of 50 employees of a certain
University. In this example, 20 is the frequency of the class (employees) having salary Rs.
40, 000 per month, and 3 is the frequency of the employees drawing Rs. 90,000 per month
salary.
Salary (000) 40 50 60 70 80 90

Number of employees 20 10 8 5 4 3
Class Boundaries: In a grouped frequency distribution, if upper limit of a class is repeated as
a lower limit of the next class, such classes are called class boundaries. For example, consider
the following data set:
Salary (000) 5-10 10-15 15-20 20-25 25-30 30-35
Class Limits: In a grouped frequency distribution, if upper limit of a class is not repeated as
a lower limit of the next class, such classes are called class limits. For example, consider the
following data set:
Salary (000) 5-9 10-14 15-19 20-24 25-29 30-34
How to convert class limits in to class boundaries:
In the given data, classes shows class limits. To convert class limits in to class boundaries,
calculate the mid-way-value as: = (10-9)/2 = 0.5
Now subtract 0.5 from each of the lower limit of the class, and add 0.5 to each of the upper
class limits. See the example for further understanding.
Classes Class Boundaries
5-9 4.5-9.5
10-14 9.5-14.5

15-19 14.5-19.5
20-24 19.5-24.5
25-29 24.5-29.5
30-34 29.5-34.5
Frequency Distribution
Arrangement of data in to different classes or group in such a way that each class/group has
their own frequency is called frequency distribution. For example, the following data shows
the frequency distribution of the salary of 50 employees of a firm. This frequency distribution
is called discrete frequency distribution.
Salary (000) 40 50 60 70 80 90
Whereas, the data below indicate the grouped/continuous frequency distribution of the
amount of salary of 50 employees
Salary (000) 5-9 10-14 15-19 20-24 25-29 30-34
PRESENTATION OF DATA: DIAGRAMS
City Name No. of Industries No. of Banks
Peshawar 50 35
Islamabad 40 45
Karachi 120 90

Lahore 70 55
Faisalabad 90 30
Quetta 15 8
In this section, we will examine several other methods for the graphical representation of
data, then discuss some of the ways in which graphs and charts can be used (by either the
unwary or the unscrupulous) to deceive the reader or viewer. We will also provide several
Computer Solutions to guide you in using Excel and Minitab to generate some of the more
common graphical presentations. These are just some of the more popular approaches. There
are many other possibilities.
The Bar Chart
Like the histogram, the bar chart represents frequencies according to the relative lengths of a
set of rectangles, but it differs in two respects from the histogram: (1) the histogram is used in
representing quantitative data, while the bar chart represents qualitative data; and (2) adjacent
rectangles in the histogram share a common side, while those in the bar chart have a gap
between them.
Figure: Summary of the number of industries in different cities of Pakistan
0
20
40
60
80
100
120
140
Peshawar Islamabad Karachi Lahore Faisalabad Quetta
City
NumberofIndustries

Multiple Bar Diagram:
Figure: Summary of the number of industries and Banks in different cities of Pakistan
Component bar Diagram:
Figure: Summary of the number of industries and Banks in different cities of Pakistan
The Line Graph
The line graph is capable of simultaneously showing values of two quantitative variables (y,
or verticalaxis, and x, or horizontal axis); it consists of linear segments connecting points
observed or measuredfor each variable. When x represents time, the result is a time series
view of the y variable.
0
20
40
60
80
100
120
140
City Name
No. of Industries No. of banks
50 40
120
70
90
15
35 45
90
55
30
8
0
50
100
150
200
250
City Name
Number
No. of Industries No. of banks

The Pie Chart
The pie chart is a circular display divided into sections based on either the number of
observations within or the relative values of the segments. If the pie chart is not computer
generated, it can be constructed by using the principle that a circle contains 360 degrees. The
angle used for each piece of the pie can be calculated as follows: Number of degrees for the
category 5 Relative value of the category 3 360.
For example, if 25% of the observations fall into a group, they would be represented by a
section of the circle that includes (0.25 3 360), or 90 degrees. Computer Solutions 2.6 shows
the Excel and Minitab procedures for generating a pie chart to show the relative importance
of four major business segments in contributing to Home Depot Corporation’s overall profit.
0
5
10
15
20
25
30
35
40
45
2.7 2.9 3.1 3.3 3.5 3.7 3.9 4.1
GPA
NumberofStudents
Peshawar
13%
Islamabad
10%
Karachi
32%
Lahore
18%
Faisalabad
23%
Quetta
4%

Figure: Summary of the number of industries in different cities of Pakistan
Statistical Tools and Methods
 Measures of Central Tendency
 Measures of Dispersion
 Reliability Analysis
 Inferential Statistics
 Mean Comparison (statistical tests)
 Analysis of Variance (ANOVA)
 Tests of Association
 Regression and Correlation Analysis
 Non-parametric Tests
MEASURES OF CENTRAL TENDENCY
We saw how raw data are converted into frequency distributions, histograms, and visual
displays. We will now examine statistical methods for describing typical values in the data as
well as the extent to which the data are spread out. Introduced in Chapter 1, these descriptors
are known as measures of central tendency and measures of dispersion:
A data set can be summarized into a single value, usually lies somewhere in the center and
represent the whole data set. Such a single value that represents the central part of a data set
is called central value.The tendency of samples of a given measurement to cluster around
some central value. Tendency of observations that cluster in the central part of the data set is
called central tendency. Most commonly used measures of central tendency are given in the
following diagram:
Arithmetic Mean
Defined as the sum of the data values divided by the number of observations, the arithmetic
mean is one of the most common measures of central tendency. Also referred to as the
arithmetic average or simply the mean, it can be expressed as _ (the population mean,

pronounced “myew”) or} ( x ) (the sample mean, “x bar”). The population mean µ applies
when our data represent all of the items with in the population. The sample mean ( x ) is
applicable whenever data representa sample taken from the population.
Simply it is called mean or average and mostly used measure of central tendency in every
field of research. “Arithmetic mean is a value obtained by dividing the sum of all
observations in a data set by the number of observations”.
Mathematical Description of Arithmetic Mean
Mathematically, Arithmetic mean is expressed as
Example: The following data shows the consumption (in thousand of Rs.) of 9 MBA students
per semester in a certain University, compute arithmetic mean and interpret the result. The
data is: 39, 36, 48, 36, 41, 37, 32, 46 and 45.
It indicates that on the average, each MBA student is consuming Rs. 40,000 per semester.
Arithmetic Meanfor Grouped Data
Sum of all observations
Mean =
Total number of observations
1 2 1
1 2 1
= = [Population data]
= = [Sample data]
N
i
N i
n
i
n i
x
x x x
N N
x
x x x
X
n n
 

  
  


1 39 36 48 36 41 37 32 46 45 360
= 40
9 9
n
i
i
x
X
n
        
  

1 1 2 2
1 2
1
1 2
1 2
=
=
, where (total number of observations)
where, , , , are corresponding frequencies
of , , ,
n n
n
n
i i
i
i
i
n
n
f x f x f x
X
f f f
f x
f
fx
f n
f
f f f
x x x

  
  
 


 


Example: Using the following data showing the profits (in thousand of Rs.) of 60 different
industries, calculate the mean profit (average profit) of the industries.
Profit (000) 65-
84
85-
104
105-
124
125-
144
145-
164
165-
184
185-
204
Total
Number of
industries
9 10 17 10 5 4 5 60
To compute the mean, first we convert the class intervals into mid points (X)
Profit (000) 65-
84
85-
104
105-
124
125-
144
145-
164
165-
184
185-
204
Total
Number of
industries (f)
9 10 17 10 5 4 5 60
Mid point (X) 74.5 94.5 114.5 134.5 154.5 174.5 194.5
fx 670.5 945 1946.5 1345 772.5 698 972.5 7350
Frequency distributions and histograms
To more concisely communicate the information contained, raw data can be visually
represented and expressed in terms of statistical summary measures. When data are
quantitative, they can be transformed to a frequency distribution or a histogram describing the
number of observations occurring in each category.
Visual Description of Data
1 7350
= 122.5
60
n
i i
i
i
f x
X
f

 



The set of classes in the frequency distribution must include all possible values and should be
selected so that any given value falls into just one category. Selecting the number of classes
to use is a subjective process. In general, between 5 and 15 classes are employed. A
frequency distribution may be converted to show either relative or cumulative frequencies for
the data.
• Stem-and-leaf displays, dot plots, and other graphical methods
The stem-and-leaf display, a variant of the frequency distribution, uses a subset of the
original digits in the raw data as class descriptors (stems) and class members (leaves). In the
dot plot, values for a variable are shown as dots appearing along a single dimension.
Frequency polygons, ogives, bar charts, line graphs, pie charts, pictograms, and sketches are
among the more popular methods of visually summarizing data. As with many statistical
methods, the possibility exists for the purposeful distortion of graphical information.
• The scatter diagram
The scatter diagram, or scatterplot, is a diagram in which each point represents a pair of
known or observed values of two variables. These are typically referred to as the dependent
variable (y) and the independent variable (x). This type of analysis is carried out to fit an
equation to the data to estimate or predict the value of y for a given value of x.
• Tabulation and contingency tables
When some of the variables represent categories, simple tabulation and cross tabulation
are used to count how many people or items are in each category or combination of
categories, respectively. These tabular methods can be extended to include the mean or other
measures of a selected quantitative variable for persons or items within a category or
combination of categories.
MODE
Mode is a value which has maximum frequency as compared to other items of a data set. OR,
the most frequent value of a data set is called mode.
A distribution/data set having only one mode is called uni-modal distribution. Similarly, a
distribution is defined to be bi-modal if it has two modes. Generally, a distribution having
more than one modes is called multi-modal distribution. For example:

a). 2, 4, 6, 4, 8, 10 (mode = 4)
b). 2, 4, 6, 4, 8, 10, 8 (mode = 4 and 8)
c). 2, 4, 6, 4, 8, 10, 8, 10 (mode = 4, 8 and 10 )
If all the observations of a data set have the same frequencies (repeated the same number of
times), the data set will have no mode. For example: 2, 4, 6, 4, 8, 10, 8, 10, 6: this data set
has no mode because each and every observation is repeated the
same number of times.
Mode is the appropriate average for qualitative/nominal data.
MODE FOR CONTINEOUS SERIES
MEDIAN
Median is a value which divide and arranged data set into two equal parts i.e. half (50%) of
the observations will lies below and half (50%) will come above that value.
For example: what will be the median of the following data showing weekly profit (000) of
seven stores as: 10, 20, 15, 13, 14, 9 and 12.
Arranged data (increasing order): 9, 10, 12, 13, 14, 15, 16
Median = 13
Consumption f
4.5-9.5 10
9.5-14.5 8
14.5-19.5 20
19.5-24.5 5
24.4-29.5 4
29.5-34.5 3
Total 50
1
1 2
1
2
( )
Mode = +
(2 )
= lower limit of the modal group
frequency of the modal group
frequency preceeding the modal group
frequency exceeding the modal group
= width of class
m
m
m
f f h
l
f f f
l
f
f
f
h
 
 



Mode lies in the group (14.5-19.5) as it has maximum frequency, so
(20 8) 5 60
Mode = 14.5 + 14.5
(2 20 8 5) 27
14.5 2.22 16.72
 
 
  
  

Similarly, for the data set having the size (even number) divisible by 2, median will be the
average of two middle values, for example:
9, 10, 12, 13, 14, 15, 16, 20 (here n = 8) so
Numerical Examples-Continuous Frequency Distribution
The following data shows the frequency distribution of the salary of 50 employees of a firm.
Calculate the following
1. Arithmetic mean
2. Median
Salary
(000)
5-9 10-14 15-19 20-24 25-29 30-34
Numberof
employees
20 10 8 5 4 3
To calculate the required quantities, we take the following steps:
Salary (000) Class boundary f cf X fX
5-9 4.5-9.5 20 20 7 140
10-14 9.5-14.5 10 30 12 120
15-19 14.5-19.5 8 38 17 136
20-24 19.5-24.5 5 43 22 110
25-29 24.4-29.5 4 47 27 108
30-34 29.5-34.5 3 50 32 96
50 710
Median = (13+14)/2 = 13.5

QUANTILES: Quartiles, Deciles and Percentiles are collectively called quintiles. Generally,
quantiles are also called measures of position.
We have already seen how the median divides data into two equal-size groups: one with
values above the median, the other with values below the median. Quantilesalso separate the
data into equal-size groups in order of numerical value. There are several kinds of quantiles,
of which quartiles will be the primary topic for our discussion.
PERCENTILES divide the values into
Quartiles:After values are arranged from smallest to largest, quartiles are calculated similarly to
the median. It may be necessary to interpolate (calculate a position between) two values to
identify the data position corresponding to the quartile.
The three points which divide an arranged (ascending order) data set into four equal parts are
called quartiles. Quartiles are denoted by Q1, Q2 and Q3.
Q1 = lower quartile or first quartile
Q2 = second quartile = median
Q3 = upper quartile or third quartile
Deciles:The nine points which divide an arranged (ascending order) data set into 10 equal parts
are called deciles. Deciles are denoted by D1, D2 -----D9.
D1 = first decile, D2 = second decile, …., D9 = 9thdecile
710
AM = 14.2
50
For median, n/2 = 50/2 = 25. It implies that median lies in the
group (9.5-14.5), so
5
Median = ( /2 ) = 9.5 + (50/2 20) 12
10
fX
f
h
l n c
f
 
   


10% 10% 10% 10% 10% 10% 10% 10% 10% 10%
D
1
D
2
D
3
D
4
D
5
Q2
Median
D
6
D
7
D
8
D
9

D1 is a value from which 10% observations lies below and 90% lies above;
D2 is a value from which 20% observations lies below and 80% lies above;
------
------
D9 = 90% observations below and 10% lies above
Percentiles: Percentiles divide an arranged data set into 100 equal parts. These are 99 points
to do so. Percentiles are denoted by Pi( i = 1,2 , 3, …., 99).
P1 is a value from which 1% observations lies below and 99% lies above;
P2 is a value from which 2% observations lies below and 98% lies above;
------
------
P99 = 99% observations below and 1% lies above
COMPUTATIONAL FORMULAE
1% 1%
P
50
Q2
Median
D
5
----P
75
---
Q
3
Q = + ( )
4
= lower limit of the quartile group
= width of class
= frequency of the quartile group
= cumulative frequency preceding the quartile group
j
h jn
l c
f
l j th
h
f j th
c j th




D = + ( )
10
= lower limit of the decile group
= width of class
= frequency of the decile group
= cumulative frequency preceding the decile group
j
h jn
l c
f
l j th
h
f j th
c j th





COMPUTATIONAL FORMULA FOR PERCENTILES
MEASURES OF DISPERSION
Although the mean and other measures are useful in identifying the central tendency of
values in a population or a set of sample data, it’s also valuable to describe their dispersion,
or scatter.
Measures of central tendency (mean, median, mode, GM and HM) do not provide all
information about the observations contained in a data set that how the individual
observations are scattered around the central value. It is possible with the help of measures of
dispersion.
A single value which measure that how the individual observations of a data set are
scattered/dispersed around the central value, is called measure of dispersion. Measures of
dispersion are classified as “Absolute measures” and “Relative measures” of dispersion.
TYPES OF MEASURES OF DISPERSION
A type of dispersion which can be expressed in the same unit of measurement in which the
original series/data set/ distribution is given is called “Absolute measure” of dispersion. For
example, Range, Quartile deviation, Mean deviation, Standard deviation etc. Similarly,a type
of dispersion which is independent of unit of measurement is called “Relative measure” of
dispersion. For example, coefficient of range, coefficient of quartile deviation, coefficient of
mean deviation, coefficient of variation etc.
Absolute Measures of Dispersions
1. Range
P = + ( )
100
= lower limit of the percentile group
= width of class
= frequency of the percentile group
= cumulative frequency preceding the
percentile group
j
h jn
l c
f
l j th
h
f j th
c
j th





2. Inter quartile range
3. Semi inter quartile range or Quartile Deviation (QD)
4. Mean deviation
5. Variance and
6. Standard deviation
Relative Measures of Dispersions
1. Coefficient of Range
2. Coefficient of Inter quartile range
3. Coefficient Semi inter quartile range
4. Coefficient Mean deviation
5. Coefficient of Variation
RANGE AND ITS COEFFICIENT
The simplest measure of dispersion, the range is the difference between the highest and
lowest values. OR
Range is an absolute measure of dispersion. “It is the difference between maximum (Xm) and
minimum (X0) values of a data set”. Mathematically, range is defined as:
Range = Xm – X0
Its relative measure is called coefficient of range and can be defined as:
For Example: The following data indicate the amount of fill (in ml) of 5 different bottle by a
soft drink company. The data is 12.5, 12.3, 12, 13, 12.8
So, Range = 13-12 = 1 ml.
Coefficient of Range = (13-12)/(13+12) = 1/25.
Inter Quartile Range and Quartile Deviation
Inter Quartile Range (IQR) is an absolute measure of dispersion. “It is the difference
between upper quartile (Q3) and lower quartile (Q1) of a data set”. Mathematically, IQR is
defined as:
IQR = Q3 – Q1
Quartile Deviation (Semi inter quartile range): It is half of the inter quartile range. It is also
called quartile deviation (QD) and is expressed as:
SIQR = QD = (Q3 – Q1)/2
m 0
m 0
X X
Coefficient of Range =
X + X


A relative measure of IQR and SIQR is called coefficient of IQR and coefficient of SIQR
(coefficient of QD), respectively and can be expressed as:
Coefficient IQR = (Q3 – Q1)/ (Q3 + Q1)
Coefficient of SIQR =Coefficient of QD = (Q3 – Q1)/ (Q3 + Q1)
MEAN DEVIATION AND ITS COEFFICIENT
Mean deviation is an arithmetic mean of absolute deviations taken from any central value
(mean, median, and mode) of a data set. It is an absolute measure of dispersion.
With this descriptor, sometimes called the average deviation or the average
absolutedeviation, we now consider the extent to which the data values tend to differ fromthe
mean. In particular, the mean absolute deviation (MAD) is the average of theabsolute values
of differences from the mean and may be expressed as follows:
In the preceding formula, we find the sum of the absolute values of the differences between
the individual values and the mean, and then divide by the number of individual values. The
two vertical lines (“z z”) tell us that we are concerned with absolute values, for which the
algebraic sign is ignored.
COEFFICIENT OF MEAN DEVIATION
Its relative measure is called coefficient of mead deviation and can be expressed as:
|X-X|
MD = ( individual series)
|X-X|
MD = ( discrete/contineous series)
for
n
f
for
f



Mean Deviation
Coefficient of MD =
Arithmatic Mean
MD
Coefficient of MD =
X

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