Correlation

In previous chapters you have learnt how to construct summary measures out of a mass of data and changes among similar variables. Now you will learn how to examine the relationship between two variables.

Let us look at various types of relationship. The relation between movements in quantity demanded and the price of a commodity is an integral part of the theory of demand, which you will study in Class XII. Low agricultural productivity is related to low rainfall. Such examples of relationship may be given a cause and effect interpretation. Others may be just coincidence. The relation between the arrival of migratory birds in a sanctuary and the birth rates in the locality cannot be given any cause and effect interpretation. The relationships are simple coincidence. The relationship between size of the shoes and money in your pocket is another such example. Even if relationships exist, they are difficult to explain it.

A scatter diagram visually presents the nature of association without giving any specific numerical value. A numerical measure of linear relationship between two variables is given by Karl Pearson’s coefficient of correlation. A relationship is said to be linear if it can be represented
by a straight line. Spearman’s coefficient of correlation measures the linear association between ranks assigned to indiviual items according to their attributes. Attributes are those variables which cannot be numerically measured such as intelligence of people, physical appearance, honesty, etc.

Scatter diagrams spanning over Fig. 7.1 to Fig. 7.5 give us an idea of the relationship between two variables. Fig. 7.1 shows a scatter around an upward rising line indicating the movement of the variables in the same direction. When X rises Y will also rise. This is positive correlation. In Fig. 7.2 the points are found to be scattered around a downward sloping line. This time the variables move in opposite directions. When X rises Y falls and vice versa. This is negative correlation. In Fig.7.3 there is no upward rising or downward sloping line around which the points are scattered. This is an example of no correlation. In Fig. 7.4 and Fig. 7.5, the points are no longer scattered around an upward rising or downward falling line. The points themselves are on the lines. This is referred to as perfect positive correlation and perfect negative correlation respectively.

A careful observation of the scatter diagram gives an idea of the nature and intensity of the relationship.

This is also known as product moment correlation coefficient or simple correlation coefficient. It gives a precise numerical value of the degree of linear relationship between two variables X and Y.

It is, therefore, advisable to first examine the scatter diagram of the relation between the variables before calculating the Karl Pearson’s correlation coefficient.

The standard deviations of X and Y, respectively, are the positive square roots of their variances. Covariance of X and Y is defined as

 

or

Let us now discuss the properties of the correlation coefficient

Substituting these values in formula (1)

 

Thus, years of education of farmers and annual yield per acre are positively correlated. The value of r is also large. It implies that more the number of years farmers invest in education, higher will be the yield per acre. It underlines the importance of farmers’ education. To use formula (3)

the value of the following expressions have to be calculated i.e.

where A and B are assumed means, h and k are common factors and have same signs.

Let A = 100; h = 10; B = 1700 and  k = 100

Spearman’s rank correlation was developed by the British psychologist C.E. Spearman. It is used in the following situations:

All the properties of the simple correlation coefficient are applicable here. Like the Pearsonian Coefficient of correlation it lies between 1 and
–1. However, generally it is not as accurate as the ordinary method. This is due the fact that all the information concerning the data is not utilised.

The first difference is the difference of consecutive values. The first differences of the values of items in the series, arranged in order of magnitude, are almost never constant. Usually the data cluster around the central values with smaller differences in the middle of the array.

The calculation of rank correlation will be illustrated under three situations.

Five persons are assessed by three judges in a beauty contest. We have to find out which pair of judges has the nearest approach to common perception of beauty.

The rank correlation between A and C is calculated as follows:

Once the ranking is complete formula (4) is used to calculate rank correlation.

where m1, m2, ..., are the number of repetitions of ranks and their corresponding correction factors.

The necessary correction for this data thus is

 

Substituting the values of these expressions

Thus, there is positive rank correlation between X and Y. Both X and Y move in the same direction. However, the relationship cannot be described as strong.

We have discussed some techniques for studying the relationship between two variables, particularly the linear relationship. The scatter diagram gives a visual presentation of the relationship and is not confined to linear relations. Karl Pearson’s coefficient of correlation and Spearman’s rank correlation measure linear relationship among variables. When the variables cannot be measured precisely, rank correlation can be used. These measures however do not imply causation. The knowledge of correlation gives us an idea of the direction and intensity of change in a variable when the correlated variable changes.