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WHAT IS STATISTICS?:CHARACTERISTICS OF THE SCIENCE OF STATISTICS

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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':
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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:
i)
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.
v)
Statistics deals with those characteristics or aspects of things which can be
described numerically either by counts or by measurements.
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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
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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
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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
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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
ε
=
, which when multiplied by 100, is percentage error. These errors are independent
x
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.
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Table of Contents:
  1. Recommended Books:Set of Integers, SYMBOLIC REPRESENTATION
  2. Truth Tables for:DE MORGAN’S LAWS, TAUTOLOGY
  3. APPLYING LAWS OF LOGIC:TRANSLATING ENGLISH SENTENCES TO SYMBOLS
  4. BICONDITIONAL:LOGICAL EQUIVALENCE INVOLVING BICONDITIONAL
  5. BICONDITIONAL:ARGUMENT, VALID AND INVALID ARGUMENT
  6. BICONDITIONAL:TABULAR FORM, SUBSET, EQUAL SETS
  7. BICONDITIONAL:UNION, VENN DIAGRAM FOR UNION
  8. ORDERED PAIR:BINARY RELATION, BINARY RELATION
  9. REFLEXIVE RELATION:SYMMETRIC RELATION, TRANSITIVE RELATION
  10. REFLEXIVE RELATION:IRREFLEXIVE RELATION, ANTISYMMETRIC RELATION
  11. RELATIONS AND FUNCTIONS:FUNCTIONS AND NONFUNCTIONS
  12. INJECTIVE FUNCTION or ONE-TO-ONE FUNCTION:FUNCTION NOT ONTO
  13. SEQUENCE:ARITHMETIC SEQUENCE, GEOMETRIC SEQUENCE:
  14. SERIES:SUMMATION NOTATION, COMPUTING SUMMATIONS:
  15. Applications of Basic Mathematics Part 1:BASIC ARITHMETIC OPERATIONS
  16. Applications of Basic Mathematics Part 4:PERCENTAGE CHANGE
  17. Applications of Basic Mathematics Part 5:DECREASE IN RATE
  18. Applications of Basic Mathematics:NOTATIONS, ACCUMULATED VALUE
  19. Matrix and its dimension Types of matrix:TYPICAL APPLICATIONS
  20. MATRICES:Matrix Representation, ADDITION AND SUBTRACTION OF MATRICES
  21. RATIO AND PROPORTION MERCHANDISING:Punch recipe, PROPORTION
  22. WHAT IS STATISTICS?:CHARACTERISTICS OF THE SCIENCE OF STATISTICS
  23. WHAT IS STATISTICS?:COMPONENT BAR CHAR, MULTIPLE BAR CHART
  24. WHAT IS STATISTICS?:DESIRABLE PROPERTIES OF THE MODE, THE ARITHMETIC MEAN
  25. Median in Case of a Frequency Distribution of a Continuous Variable
  26. GEOMETRIC MEAN:HARMONIC MEAN, MID-QUARTILE RANGE
  27. GEOMETRIC MEAN:Number of Pupils, QUARTILE DEVIATION:
  28. GEOMETRIC MEAN:MEAN DEVIATION FOR GROUPED DATA
  29. COUNTING RULES:RULE OF PERMUTATION, RULE OF COMBINATION
  30. Definitions of Probability:MUTUALLY EXCLUSIVE EVENTS, Venn Diagram
  31. THE RELATIVE FREQUENCY DEFINITION OF PROBABILITY:ADDITION LAW
  32. THE RELATIVE FREQUENCY DEFINITION OF PROBABILITY:INDEPENDENT EVENTS