Theory of attributes | Introduction

Most Welcome

I am Tajamul Islam prepared the paper for you.

Tajamul sir on youtube

Theory of Attributes-

Basic concept and their applications. (part 1)

Introduction.

Literally, an attribute means a quality or characteristic. Theory of attributes deals with qualitative characteristics which are not amenable to quantitative measurements.Examples of attributes are drinking, smoking, blindness, health, honesty, etc.

Data might be such that it may not be possible for an investigator to measure their magnitude. In such cases, the observer can only study the presence or absence of a particular quality in a group of individuals. Examples of such phenomena are blindness, insanity, deaf-mutism, sickness, honesty, extravagance etc. In such cases. an observer cannot measure the magnitude of the data ; for example, he cannot measure the extent of blindness or honesty in quantitative form. All he can do is to count the number of persons who are blind or who are honest. He has to take decision on the basis of some standard definition of the term in question. Such data in which the quantitative measurement of the magnitude is not possible and in which only the presence or absence of an attribute can be studied are called statistics of attributes.

CLASSIFICATION OF DATA

In the analysis of statistics relating to attributes, the first thing is the classification of data. Here, data are classified on the basis of presence or absence of particular attributes. If only one attribute, say, blindness, is being studied the population would be divided in two classes-one consisting of those people in whom this attribute is present and the other consisting of those in whom this attribute is not present. Thus, one class would be of the “blinds” and the other of “non-blinds.” If more than one attribute are taken into account, the number of classes would be more than two. If, for example, the attribute of deafness is also studied, there would be a number of classes in which the universe/population would be divided; There would be “blinds”, “non-blinds”, “deafs”, “non:deafs”, “blind and not deafs”, “blind and deafs” “not-blinds and deafs” and “not-blinds and not-deafs”.

Classification by dichotomy : If only one attribute is being studied, the universe is divided in to two parts one in which the attribute is present and the other in which it is not present. These classes are mutually exclusive. Such a classification where the universe is divided in two parts is called "Classification by. Dichotomy". In actual analysis, usually there are more than two classes in which the universe is divided and such classification is called manifold classification.

NOTATION AND TERMINOLOGY

For the sake of convenience, in analysis, it is necessary to use certain symbols to represent different Classes and their frequencies. Usually, capital letters A, B and C etc., are used' to denote the presence or attributes and the Greek letters, α, β and ϒ etc. are used to denote absence of these attributes respectively. Thus if A represents the attribute of blindness, α would represent absence of blindness, if B represents deafness,β would represent absence of deafness'and if C represents insanity, ϒ would represent absence of insanity. The number of units possessing a particular attribute represented by A would be termed belonging to Class A, and similarly, those in whom this attribute is absent would be termed belonging to Class α.

If two attributes are being studied, their combination can be represented by the combination of the letters representing the two attributes. Thus, if blindness is represented by A and deafness by B, then AB would represent blindness and deafness ; Aβ would represent blindness and absence of deafness αB would represent absence of blindness and presence of deafness ; and αβ would represent absence of blindness and absence of deafness. '

Class Frequencies

The number of units in different classes are called "class frequencies. ” Thus, if the number of blind and deaf people is 20, the frequency of class AB is 20. Class frequencies are denoted by enclosing the class symbols by brackets. Thus, (AB) would represent the frequency of the class AB.

Number of Classes lf there is one attribute represented by A, the total number of classes is 3 (if the total or ‘N‘ is also taken as a class), they would be A, α and N. If the number of attributes is two, represented by A and B. the total number of classes (including N) would be 9. They would be N, A, B, α, β, AB Aβ, αB and αβ.

If the number of attributes is three,the total number of classes (including N) would be 27. The total number of classes is always equal to where n stands for the number of attributes. Thus, when there are three attributes, the total number of classes would be or 27 ; if the number of attributes is 4, the total number of classes would be or 81.

Order of Classes . The various classes and their frequencies are demarcated on the basis of an order.

Thus, N is a class of Zero order

Similarly, the frequencies of these classes are also known as frequencies of the Zero, First or Second order. If there are only two attributes under study, then the Second order classes and frequencies are called the classes or the fequencies of the ultimate order. Since these are the last set of classes and frequencies, the number of classes of the ultimate order is equal to where n stands for the number of attributes. Thus, in case of 2 attributes, the number of the classes of the ultimate order would be or 4 and in case of three attributes or 8 and so on.

Positive and Negative Classes The classes which represent the presence of an attribute or attributes are called positive classes. The classes which represent the absence of an attribute or attributes are called negative classes. The classes in which one attribute is present and the other absent are called pairs of contraries. Thus :

N,A, B and AB are positive classes

and β are negative classes

Aβ and αB are pairs of contraries

RELATIONSHIP BETWEEN CLASSES OF VARIOUS ORDERS

ln classifying statistical data according to attributes, the following simple rule should be kept in mind. Any class frequency can always be expressed in terms of class frequencies of higher order: Thus, the frequencies of first order can be expressed in terms of the frequencies of the second order which in turn can be expressed in terms of frequencies of the third order and so on. On the basis of this rule we can set up various types of relationships between the frequencies of different orders. If there is one attribute only, represented by A, the frequency of the universe or N can be divided into two classes (A) and (α). Thus,

N=(A)+(α)

Now, if one more attribute B is taken into account the first order classes, i.e., A and α can each be divided into two classes-one in which attribute B is present and the other in which it is not present.

Thus

The following examples would clarify the above rules.

Example I : Given the following ultimate class frequencies, find the frequencies of the positive and negative classes and the total number of observations :

(AB) = 250 (Aβ) = 120

(αB) = 200 (αβ) = 70

Solution