WHAT IS STATISTICS?:CHARACTERISTICS OF THE SCIENCE OF STATISTICS

<< RATIO AND PROPORTION MERCHANDISING:Punch recipe, PROPORTION

WHAT IS STATISTICS?:COMPONENT BAR CHAR, MULTIPLE BAR CHART >>

MTH001 Elementary Mathematics

LECTURE # 22

WHAT IS STATISTICS?

That science which enables us to draw conclusions about various phenomena on

the basis of real data collected on sample-basis

A tool for data-based research

Also known as Quantitative Analysis

A lot of application in a wide variety of disciplines ... Agriculture, Anthropology,

Astronomy, Biology, Economic, Engineering, Environment, Geology, Genetics,

Medicine, Physics, Psychology, Sociology, Zoology .... Virtually every single

subject from Anthropology to Zoology .... A to Z!

Any scientific enquiry in which you would like to base your conclusions and

decisions on real-life data, you need to employ statistical techniques!

Now a day, in the developed countries of the world, there is an active movement

for of Statistical Literacy.

THE NATURE OF

THIS DISCIPLINE:

DESCRIPTIVE statistics

Probability

INFERENTIAL STATISTICS

MEANINGs OF `STATISTICS':

Page

143

MTH001 Elementary Mathematics

The word "Statistics" which comes from the Latin words status, meaning a political

state, originally meant information useful to the state, for example, information about the

sizes of population sand armed forces. But this word has now acquired different meanings.

In the first place, the word statistics refers to "numerical facts systematically

arranged". In this sense, the word statistics is always used in plural. We have, for

instance, statistics of prices, statistics of road accidents, statistics of crimes, statistics

of births, statistics of educational institutions, etc. In all these examples, the word

statistics denotes a set of numerical data in the respective fields. This is the meaning

the man in the street gives to the word Statistics and most people usually use the

word data instead.

In the second place, the word statistics is defined as a discipline that includes

procedures and techniques used to collect, process and analyze numerical data to

make inferences and to research decisions in the face of uncertainty. It should of

course be borne in mind that uncertainty does not imply ignorance but it refers to the

incompleteness and the instability of data available. In this sense, the word statistics

is used in the singular. As it embodies more of less all stages of the general process

of learning, sometimes called scientific method, statistics is characterized as a

science. Thus the word statistics used in the plural refers to a set of numerical

information and in the singular, denotes the science of basing decision on numerical

data. It should be noted that statistics as a subject is mathematical in character.

Thirdly, the word statistics are numerical quantities calculated from sample

observations; a single quantity that has been so collected is called a statistic. The

mean of a sample for instance is a statistic. The word statistics is plural when used in

this sense.

CHARACTERISTICS OF THE

SCIENCE OF STATISTICS:

Statistics is a discipline in its own right. It would therefore be desirable to know the

characteristic features of statistics in order to appreciate and understand its general nature.

Some of its important characteristics are given below:

Statistics deals with the behaviour of aggregates or large groups of data. It

has nothing to do with what is happening to a particular individual or object of

the aggregate.

ii)

Statistics deals with aggregates of observations of the same kind rather than

isolated figures.

iii)

Statistics deals with variability that obscures underlying patterns. No two

objects in this universe are exactly alike. If they were, there would have been

no statistical problem.

iv)

Statistics deals with uncertainties as every process of getting observations

whether controlled or uncontrolled, involves deficiencies or chance variation.

That is why we have to talk in terms of probability.

Statistics deals with those characteristics or aspects of things which can be

described numerically either by counts or by measurements.

Page

144

MTH001 Elementary Mathematics

vi)

Statistics deals with those aggregates which are subject to a number of

random causes, e.g. the heights of persons are subject to a number of

causes such as race, ancestry, age, diet, habits, climate and so forth.

vii)

Statistical laws are valid on the average or in the long run. There is n

guarantee that a certain law will hold in all cases. Statistical inference is

therefore made in the face of uncertainty.

viii)

Statistical results might be misleading the incorrect if sufficient care in

collecting, processing and interpreting the data is not exercised or if the

statistical data are handled by a person who is not well versed in the subject

mater of statistics.

THE WAY IN WHICH STATISTICS WORKS:

As it is such an important area of knowledge, it is definitely useful to have a fairly good idea

about the way in which it works, and this is exactly the purpose of this introductory course.

The following points indicate some of the main functions of this science:

Statistics assists in summarizing the larger set of data in a form that is easily

understandable.

Statistics assists in the efficient design of laboratory and field experiments as well as

surveys.

Statistics assists in a sound and effective planning in any field of inquiry.

Statistics assists in drawing general conclusions and in making predictions of how much

of a thing will happen under given conditions.

IMPORTANCE OF STATISTICS

IN VARIOUS FIELDS

As stated earlier, Statistics is a discipline that has finds application in the most diverse fields

of activity. It is perhaps a subject that should be used by everybody. Statistical techniques

being powerful tools for analyzing numerical data are used in almost every branch of

learning. In all areas, statistical techniques are being increasingly used, and are developing

very rapidly.

A modern administrator whether in public or private sector leans on statistical data to

provide a factual basis for decision.

A politician uses statistics advantageously to lend support and credence to his

arguments while elucidating the problems he handles.

A businessman, an industrial and a research worker all employ statistical methods in

their work. Banks, Insurance companies and Government all have their statistics

departments.

A social scientist uses statistical methods in various areas of socio-economic life of a

nation. It is sometimes said that "a social scientist without an adequate

understanding of statistics, is often like the blind man groping in a dark room for a

black cat that is not there".

The Meaning of Data:

The word "data" appears in many contexts and frequently is used in ordinary

conversation. Although the word carries something of an aura of scientific mystique, its

Page

145

MTH001 Elementary Mathematics

meaning is quite simple and mundane. It is Latin for "those that are given" (the singular form

is "datum"). Data may therefore be thought of as the results of observation.

EXAMPLES OF DATA

Data are collected in many aspects of everyday life.

Statements given to a police officer or physician or psychologist during an interview

are data.

So are the correct and incorrect answers given by a student on a final examination.

Almost any athletic event produces data.

The time required by a runner to complete a marathon,

The number of errors committed by a baseball team in nine innings of play.

And, of course, data are obtained in the course of scientific inquiry:

the positions of artifacts and fossils in an archaeological site,

The number of interactions between two members of an animal colony during a

period of observation,

The spectral composition of light emitted by a star.

OBSERVATIONS AND VARIABLES:

Observation:

In statistics, an observation often means any sort of numerical recording of information,

whether it is a physical measurement such as height or weight; a classification such as

heads or tails, or an answer to a question such as yes or no.

Variables:

A characteristic that varies with an individual or an object, is called a variable. For example,

age is a variable as it varies from person to person. A variable can assume a number of

values. The given set of all possible values from which the variable takes on a value is

called its Domain. If for a given problem, the domain of a variable contains only one value,

then the variable is referred to as a constant.

QUANTITATIVE AND

QUALITATIVE VARIABLES:

Variables may be classified into quantitative and qualitative according to the form of

the characteristic of interest.

A variable is called a quantitative variable when a characteristic can be expressed

numerically such as age, weight, income or number of children. On the other hand, if the

characteristic is non-numerical such as education, sex, eye-colour, quality, intelligence,

poverty, satisfaction, etc. the variable is referred to as a qualitative variable. A qualitative

characteristic is also called an attribute. An individual or an object with such a characteristic

can be counted or enumerated after having been assigned to one of the several mutually

exclusive classes or categories.

Discrete and Continuous Variables:

A quantitative variable may be classified as discrete or continuous. A discrete

variable is one that can take only a discrete set of integers or whole numbers, which is the

values are taken by jumps or breaks. A discrete variable represents count data such as the

Page

146

MTH001 Elementary Mathematics

number of persons in a family, the number of rooms in a house, the number of deaths in an

accident, the income of an individual, etc.

A variable is called a continuous variable if it can take on any value-fractional or

integralwithin a given interval, i.e. its domain is an interval with all possible values without

gaps. A continuous variable represents measurement data such as the age of a person, the

height of a plant, the weight of a commodity, the temperature at a place, etc.

A variable whether countable or measurable, is generally denoted by some symbol

such as X or Y and Xi or Xj represents the ith or jth value of the variable. The subscript i or j

is replaced by a number such as 1,2,3, ... when referred to a particular value.

Measurement Scales:

By measurement, we usually mean the assigning of number to observations or

objects and scaling is a process of measuring. The four scales of measurements are briefly

mentioned below:

NOMINAL SCALE:

The classification or grouping of the observations into mutually exclusive qualitative

categories or classes is said to constitute a nominal scale. For example, students are

classified as male and female. Number 1 and 2 may also be used to identify these two

categories. Similarly, rainfall may be classified as heavy moderate and light. We may use

number 1, 2 and 3 to denote the three classes of rainfall. The numbers when they are used

only to identify the categories of the given scale, carry no numerical significance and there is

no particular order for the grouping.

ORDINAL OR RANKING SCALE:

It includes the characteristic of a nominal scale and in addition has the property of

ordering or ranking of measurements. For example, the performance of students (or players)

is rated as excellent, good fair or poor, etc. Number 1, 2, 3, 4 etc. are also used to indicate

ranks. The only relation that holds between any pair of categories is that of "greater than" (or

more preferred).

INTERVAL SCALE:

A measurement scale possessing a constant interval size (distance) but not a true

zero point, is called an interval scale. Temperature measured on either the Celsius or the

Fahrenheit scale is an outstanding example of interval scale because the same difference

exists between 20o C (68o F) and 30o C (86o F) as between 5o C (41o F) and 15o C (59o F). It

cannot be said that a temperature of 40 degrees is twice as hot as a temperature of 20

degree, i.e. the ratio 40/20 has no meaning. The arithmetic operation of addition,

subtraction, etc. is meaningful.

RATIO SCALE:

It is a special kind of an interval scale where the sale of measurement has a true

zero point as its origin. The ratio scale is used to measure weight, volume, distance, money,

etc. The, key to differentiating interval and ratio scale is that the zero point is meaningful for

ratio scale.

ERRORS OF MEASUREMENT:

Experience has shown that a continuous variable can never be measured with

perfect fineness because of certain habits and practices, methods of measurements,

instruments used, etc. the measurements are thus always recorded correct to the nearest

units and hence are of limited accuracy. The actual or true values are, however, assumed to

exist. For example, if a student's weight is recorded as 60 kg (correct to the nearest

kilogram), his true weight in fact lies between 59.5 kg and 60.5 kg, whereas a weight

recorded as 60.00 kg means the true weight is known to lie between 59.995 and 60.005 kg.

Thus there is a difference, however small it may be between the measured value and the

true value. This sort of departure from the true value is technically known as the error of

measurement. In other words, if the observed value and the true value of a variable are

denoted by x and x + ε respectively, then the difference (x + ε) x, i.e. ε is the error. This

Page

147

MTH001 Elementary Mathematics

error involves the unit of measurement of x and is therefore called an absolute error. An

absolute error divided by the true value is called the relative error. Thus the relative error

, which when multiplied by 100, is percentage error. These errors are independent

x+ε

of the units of measurement of x. It ought to be noted that an error has both magnitude and

direction and that the word error in statistics does not mean mistake which is a chance

inaccuracy.

BIASED AND RANDOM ERRORS:

An error is said to be biased when the observed value is consistently and constantly higher

or lower than the true value. Biased errors arise from the personal limitations of the

observer, the imperfection in the instruments used or some other conditions which control

the measurements. These errors are not revealed by repeating the measurements. They are

cumulative in nature, that is, the greater the number of measurements, the greater would be

the magnitude of error. They are thus more troublesome. These errors are also called

cumulative or systematic errors.

An error, on the other hand, is said to be unbiased when the deviations, i.e. the excesses

and defects, from the true value tend to occur equally often. Unbiased errors and revealed

when measurements are repeated and they tend to cancel out in the long run. These errors

are therefore compensating and are also known as random errors or accidental errors.

Page

148

Table of Contents: