Among the most important problems faced by the scientist is the problem of measuring and describing relationships between things. Most scientific laws are. statements of such relationships. For example, the law of gravitation tells us that the attraction between bodies depends on their masses and on the distance between them, and describes the nature of that relationship by the equation.
F=X. mim2 . Or taking a simpler and more common example we say that the circumference and the radius of a circle are related by the equation c = 2nr. In these examples, the relationship is definite and unalterable. For any given radius there is one and only one circumference, and this relationship of r to c remains the same^ century after century, without end.
Correlation Defined
What, then do we mean when we say that there is a "relationship" between the height and the age of children ? We may mean any one of a number of things. We may mean, for example, that the average height increases (or decreases) with age, so that if we divide children into groups according to age, we shall find changes into the average height accompanying changes in the age. We may mean that the dispersion of heights differs with age, so that heights are more widely scattered at some ages than at others. If the frequency distributions of height vary (more than they would vary as the result of chance) from one age to another, we should say that the ages and heights are related. -
Thus, the knowledge of a child's age does not make it possible to estimate his height exactly, but it does help in estimating the height with less error. In such cases, where a knowledge of the value of one variable helps in estimating the values of another variable, we say that the two variables are "related". This does riot mean that one of them "causes" the other, but merely that the knowledge of the value of one is an aid to us in estimating the value of the other.
Definition of Correlation. When two variables are so related that a change in one is accompanied by a change in the other. In such a way that an increase in one is accompanied by an increase or decrease in the other, or decrease in one by a decrease or increase in the other, and the greater the magnitude of the change in one, the greater the magnitude of the change in the other, then the variables are said to be correlated
' The correlation is said to be positive or direct when an increaseor decrease in one corresponds to an increase or decrease respectively in the other. It is said to be negative or Inverse when an increase in one corresponds to a decrease in the other or a decrease in one corresponds to an increase in the other.
Consider the relationship between the quantity of money in circulation arid the price. If, other things being equal, the quantity of money in circulation is doubled, the price is also doubled, and vice-versa. This is an instance of direct or positive correlation between the quantity of money in circulation and the price.
- On the other hand, consider the relationship between the value of money and the price. If the price is doubled, the value of money is halved. The relationship between the price and the value of money is, therefore, inverse or negative correlation.
We say that correlation is of a high degree, if the relationship between any two variables is, more or less, of a permanent nature, is well-defined and tends to last over a long period of time. In the reverse circumstances, on the other hand, correlation will be of a low degree.
The student is warned that although sometimes there may appear to be a mathematical correlation, it may be fortuitous, or it may be the result of some other cause. For example, these days the cost of living index is increasing and the number of deaths from tuberculosis is decreasing, but there is no connection between the two. No serious statistician will ever make the mistake of supposing that a high correlation necessarily implies that there is a cause and effect relationship.
Degree of Correlation
Correlation may be perfect or imperfect. When the changes in the corresponding values of two variables are proportional, directly or inversely, the correlation between them is said to be perfect. It is perfect positive if the increase (or decrease) in the values of one variable is accompanied by a proportional increase {or decrease) in the values of a second variable, e.g., the correlation between the circumferences of circles and their radii is perfect positive. If the reverse is the case, i.e., if the increase (or decrease) in one is accompanied by a proportional decrease (or increase) in the other variable, the correlation between the two variables is said to be perfect negative, e.g., if a rectangle has constant area, the correlation between the lengths of its sides is perfect negative.
Such perfect (positive or negative) correlations are met with only in exact sciences like Mathematics, Physics, Chemistry, etc., but not in social and economic phenomenon. In such a phenomenon, the changes in one variable are not generally proportional to the changes in the other. In this case, the correlation between them, if it exists, is said to be imperfect positive or negative depending upon its nature. Imperfect correlation again may be high, moderate or low. The degree of imperfect correlation lies between perfect correlation and no correlation. Thus, we may have high positive correlation, e.g., between incomes and standard of living or we may have negative correlation, e.g., between supply and price of commodity.
Similarly, we have situation where correlation may be moderate (or low), negative or positive.
The various methods to determine whether two variables ae correlated or not are :
(i) Scatter diagram method
(«") Karl Pearson''s coefficient of correlation (iii) Rank method (Spearman's and Kendall's coefficient)
We shall now discuss each of these methods in detail.
Scatter Diagram
Scatter diagram is a graphic device for finding correlation between two variables. One variable normally, an independent variable or time is plotted on the horizontal axis. This is also called a predicting variable. The other variable known as the dependent variable' or one to bis predicted is shown on the vertical axis. The movements of the pairs of these variables shown by dots on the graph reveal whether they move in the same or the opposite direction.
If the points form a band of some width, it will indicate imperfect correlation between the two variables. The direction of the band indicates the nature of correlation. If the band slopes upward, it indicates positive correlation and if it slopes downward then it indicates negative correlation. The width of the band gives an idea of the degree of correlation. The narrower the band the greater is the degree of correlation. 1 ;
When the points do not form a band, i.e., they are scattered in all directions it indicates that there is no correlation between the variables. ,
In the case of perfect correlation, the points will be on a straight line.
This method is mainly used when we are interested in finding out whether there is correlation and only in getting a rough idea about its nature and degree. It does not give us any measure of correlation. The following diagrams illustrate the various cases.
