Introduction to Statistics

INTRODUCTION
to
STATISTICS
Mr. Khalil Ahmad
Faculty of Management Studies,
The University of Faisalabad

Introduction
“Statistical thinking will one
day be as necessary for
efficient citizenship as the
ability to read and write.”
[H.G. Wells]
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Introduction
The late W. Edwards Deming, a noted statistician
and quality-control exert insisted that statistics
education should begin before high school. He
liked to tell the story of an 11-year-old who
devised a quality-control chart to track the on-
time performance of his school bus.
Deming commented,
“He has got a good start in life.”
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Why study Statistics?
Statistics is required for many programs. Why this
is so? Why is statistics required in so many
majors?
The first reason is that numerical information is
everywhere. Look in the newspapers, news
magazines, business magazines, or general
interest magazines, or sports magazines, and
you will be bombarded with numerical
information.
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Why study Statistics?A second reason for taking a statistics course is that
statistical techniques are used to make decisions that
affect our daily lives. That is, they affect our personal
welfare. Here are a few examples:
Insurance companies use statistical analysis to set rates for home,
automobile, life, and health insurance. Tables are available that summarize
the probability that a 25-year-old woman will survive the next year. On the
basis of these probabilities, life insurance premiums can be established.
The Environmental Protection Agency is interested in the water quality at a
certain city. They periodically take water samples to establish the level of
contamination and maintain the level of quality.
Medical researchers study the cure rates for diseases using different drugs
and different forms of treatment. For example, what is the effect of treating a
certain type of knee injury surgically or with physical therapy? If you take
an aspirin each day, does that reduce your risk of a heart attack?
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A third reason for taking a statistics
course is that the knowledge of
statistical methods will help you
understand how decisions are
made and give you a better
understanding of how they affect
you.
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Why study Statistics?No matter what line of work you select, you will find yourself
faced with decisions where an understanding of data
analysis is helpful. In order to make an informed decision,
you will need to be able to:
Determine whether the existing information is adequate or
additional information is required.
Gather additional information, if it is needed, in such a
way that it does not provide misleading results.
Summarize the information in a useful and informative
manner.
Analyze the available information.
Draw conclusions and make inferences while assessing the
risk of an incorrect conclusion.

In summary, there are at least three
reasons for studying statistics:
Data are everywhere
Statistical techniques are used to make many
decisions that affect our lives
No matter what your career, you will make
professional decisions that involve data. An
understanding of statistical methods will help
you make these decisions more effectively.
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Logical Reasoning
Deduction
Drawing conclusions from general to
particular (specific)
Produce Exact results
Induction
Drawing conclusions from particular
(specific) to general
Always chances of error
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Population
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A population is the totality of the observations made
on all the objects (under investigation) possessing
some common specific characteristics, which are
of particular interest to researchers. It is the entire
group whose characteristics are to be estimated.
For example, the heights of all the students
enrolled at UAF in a given year, the wages of all
employees of a mill in a given year, etc. A
population may be finite or infinite. The number
of observations in a finite population is called the
size of the population and is denoted by the letter
N.

Parameter
A parameter is a numerical
characteristic of a population,
such as its mean or standard
deviation, etc. Parameters are
fixed constants that
characterize a population. They
are denoted by Greek letters.
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Sample
A sample is a representative part of the population
which is selected to obtain information concerning
the characteristics of the population. The number
of observations in a sample is called the size of the
sample which is denoted by n.
Statistic
A statistics is a numerical characteristic of a sample
such as its mean or standard deviation, etc. The
statistics are used to draw valid inferences about
the population. They are denoted by Latin letters.
Statistics are variables.

Why take a Sample?
Why take a sample instead of studying every
member of the population?
A sample of registered voters is necessary because of the
prohibitive cost of contacting millions of voters before an
election.
Testing wheat for moisture content destroys the wheat, thus
making a sample imperative.
If the soft drink tasters tested all the soft drink, none would be
available for sale.
It would be physically impossible for a few marine biologists to
capture and tag all the seals in the ocean.
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Why take a Sample?
Taking a sample to learn something about a population is done
extensively in business, agriculture, politics, and government, as
cited in the following examples:
Television networks constantly monitor the popularity of their programs
by hiring organizations to sample the preferences of TV viewers.
These program ratings are used to set advertising rates or to cancel
programs.
A public accounting firm selects a random sample of 100 invoices and
checks each invoice for accuracy. There were at least one error on
five of the invoices; hence the accounting firm estimates that 5
percent of the population of invoices contains at least one error.
A random sample of 1,260 accounting graduates from four-year
institutes showed their mean starting salary was $42,694. We
therefore estimate the mean starting salary for all accounting
graduates of four-year institutions to be $42,694.
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Statistics
At a most basic level,
statistics is concerned with
the transformation of raw
data into knowledge
[Wegman, 1988].
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The word statistics is generally used to express the
following three different meanings:
Statistics is a branch of science that makes use of
scientific methods to statistical observations for
the purpose of drawing valid inferences about
the population parameter with an associated
degree of their reliability for making reasonable
decisions. Scientific methods comprise of
collecting, condensing, describing, analyzing,
and interpreting the statistical data. In this sense
the word statistics is used in a singular form.
Statistics
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Statistics
Statistics are the sequence of
numerical facts about some
characteristic of the objects involved
in the field of study. This sequence
of observations is also called
statistical data (plural of a Latin
word datum). In this sense the word
statistics is use in a plural form.
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Statistics
Statistic is a descriptive measure
obtained from the sample
observations to estimate a
population parameter. For example,
the mean, variance, etc., of a sample
are statistics. In this sense the word
statistics may be used as a plural of
the word statistics.
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Types of
Statistics
Descriptive
Statistics
Inferential
Statistics
Statistics
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Descriptive Statistics
Descriptive Statistics is that branch of
Statistics that summarizes, presents
and analyzes the great bodies of
statistical data for describing their
salient features. Descriptive statistic
includes methods of organizing,
summarizing, analyzing, and
presenting data in an informative
way.

21
Inferential Statistics
Another facet of statistics is inferential
statistics-also called statistical
inference and inductive statistics.
Statistical inference is that branch of
Statistics that deals with drawing
valid inferences about the population
parameters on the basis of sample
data along with an associated degree
of their reliability.

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Variable
Any characteristic or property that may
vary either quality or quantity from on
individual or object to another is
called a variable. Examples of
variables are: height of an individual,
weight of a person, family size,
education level, etc. The variables are
usually represented by last Latin
uppercase letters as X, Y, Z, etc.

Types of Variables
Types of
Variables
Qualitative Quantitative
Discrete Continuous
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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, eye
colour, etc. When the data are qualitative, we are
usually interested in how many or what proportion
fall in each cat-egory. For example, what percent of
the population has blue eyes? How many Muslims
and Non-Muslims are there in Pakistan? What
percent of the total number of cars sold last year
was Honda? Qualitative data are often sum-marized
in charts and bar graphs.

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 employees, the
life of an automobile battery (such as
42 months), and the number of children
in a family, etc.
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Discrete Variable
Discrete variables can assume only certain values, and
there are usually “gaps” between the values.
Exam-ples of discrete variables are the number of
bedrooms in a house (1, 2, 3, 4, etc.), and the number
of students in each section of a statistics course (25 in
section A, 42 in section B, and 18 in section C), etc.
We count, for example, the number of bedrooms in a
house, and we count the number of statistics students
in each section. Notice that a home can have 3 or 4
bedrooms, but it cannot have 3.56 bedrooms. Thus,
there is a “gap” between possible values. Typically,
discrete vari-ables result from counting.
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Continuous Variable
Continuous variable can assume any value
within a specific range, i.e., its domain is an
interval with all possible values without gaps.
The continuous variable flows without a break
from one value to the next with no limit to the
number of distinct values. Examples of
continuous variables are the air pressure in a
tire and the weight of a shipment of tomatoes,
height of a student, etc. Typically, continuous
variables result from measuring.
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The information obtained by observing the values of a
variable is called Data.
Qualitative Data
Data obtained by observing the values of a qualitative variable is called
Qualitative Data.
Quantitative Data
Data obtained by observing the values of a quantitative variable is called
Quantitative Data.
Discrete Data
Data obtained by observing the values of a discrete variable is called
Discrete Data.
Continuous Data
Data obtained by observing the values of a continuous variable is called
Continuous Data.
Data
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Types of Data
Types of
Data
Qualitative Quantitative
Discrete Continuous
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Measurement
The process of assigning numbers
or labels to objects, persons,
states or, events in accordance
with specific logically accepted
rules for representing quantities
or qualities of attributes or
characteristics.
30

Measurement Scales
Data can be classified according to
levels of measurement. The level
of measurement of the data often
dictates the calculations that can
be done to summarize and present
the data. It will also determine the
statistical tests that should be
performed.
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Measurement Scales
Types of
Measurement
Scales
Nominal Interval RatioOrdinal
Data may only
be classified
Data are
ranked
Meaningful
Difference
Between values
Meaningful Zero
point and Ratio
Between values
Eye colour,
Religion,
Sex, etc.
Cricket teams
standings in
ICC ranking
Students’ grades, etc
Temperature,
Shoe Size,
IQ Scores
Bank Balance,
Weight,
Height, etc.
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Measurement Scales
There are actually four levels of
measurement: nominal, ordinal,
interval, and ratio [Stevens 1951].
The lowest, or the most primitive,
measurement is the nominal level.
The highest, or the level that gives
the most information about the
observation, is the ratio level of
measurement.

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Nominal Scale
The nominal-level data have the
following properties:
Data categories are mutually
exclusive and exhaustive.
Data categories have no logical
order.
For example, eye colour, religion, sex,
etc.

Nominal Scale
Mutually Exclusive
A property of a set of categories such that an individual or
object is included in only one category.
Exhaustive
A property of a set of categories such that each individual
or object must appear in a category.
Mutually Exclusive and Exhaustive
In general, if categories are mutually exclusive and
exhaustive, then exactly one of them must occur.
35

Ordinal Scale
The ordinal-level data have the
Data categories are mutually exclusive and
exhaustive.
Data classifications are ranked or ordered
according to the particular trait they
possess.
For example, cricket teams standings in ICC
ranking, students’ grades, etc.
36

Interval Scale
The interval-level data have the
Data categories are mutually exclusive and
exhaustive.
Data classifications are ranked or ordered according
to the particular trait they possess.
Equal differences in the characteristic are
represented by equal differences in the
measurements.
For example, temperature, shoe size and IQ scores
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Ratio Scale
The ratio-level data have the
Data categories are mutually exclusive and exhaustive.
Data classifications are ranked or ordered according to
the particular trait they possess.
Equal differences in the characteristic are represented
by equal differences in the measurements.
The zero point is the absence of the characteristic.
For example, bank balance, weight, height, etc.
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Measurement ScalesIn the measurement hierarchy, ratio variables are
highest, interval variables are next, ordinal
variables are next, and nominal variables are
lowest. Statistical methods designed for variables
of one type can also be used with variables at
higher levels, but not at lower levels. For instance,
statistical methods for ordinal variables can also
be used with interval variables (by using only the
ordering of levels and not their distances); they
can’t be used with nominal variables’ since
categories of such variables have no meaningful
ordering. Normally, it is best to apply methods
appropriate for the actual scale.
39

Measurement Scales
Types of
Measurement
Scales
Nominal Interval RatioOrdinal
Data may only
be classified
Data are
ranked
Meaningful
Difference
Between values
Meaningful Zero
point and Ratio
Between values
Eye colour,
Religion,
Sex, etc.
Cricket teams
standings in
ICC ranking
Students’ grades, etc
Temperature,
Shoe Size,
IQ Scores
Bank Balance,
Weight,
Height, etc.
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Introduction to Statistics

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