After reading this term paper you will learn about:- 1. Measures of Correlation 2. Calculation of Correlation 3. Methods.
Term Paper # 1. Measures of Correlation:
Karl Pearson’s Coefficient of Correlation (Individual Observations):
To compute the degree or extent of correlation and direction of correlation, Karl Pearson’s method is the most satisfactory.
Symbolically, its formulation is as under:
where dx is the deviation of various items of the first variable from an assumed average and dy, the corresponding deviations of the second variable from the assumed average and N connotes the number of pairs of items.
The application of the formula is explained with reference to the following hypothetical data:
Term Paper # 2. Calculation of Co-Efficient of Correlation in a Continuous Series:
In case of a continuous series, the data are classified in a two-way frequency table. Computation of coefficient of correlation in respect of grouped data is based on the presumption that every item which falls within a given class interval is assumed to fall exactly at the mid-value of that class.
As an illustration, we shall compute the coefficient or of correlation with regard to following data:
The formula for the calculation of coefficient of correlation in this case will take the following form:
The only change in the above formula in comparison to the earlier one is the introduction of f which stands for frequency.
Applying the formula to the Table 18.50 we get:
Term Paper # 3. Rank Difference Method of Correlation:
Where the direct measurement of the phenomenon under study is not possible, for example, of characteristics like efficiency, honesty, intelligence, etc., rank-difference method is applied for finding out the extent of correlation.
The formula for computing rank correlation is:
where R denotes coefficient of rank correlation between paired ranks, D denotes the differences between the paired ranks and N stands for the number of pairs.
We shall, with the help of the following example, illustrate the application of the above formula:
Term Paper # 4. Calculation of the Coefficient of Correlation by Rank Difference Method:
(When there are two or more items having the same value):
If there are more than one item with the same value, a common rank is given to such items. This rank is the average of the ranks which these items would have got, had there been a slight difference in their values. Suppose the marks obtained by five students are 70, 66, 66, 65, 63, respectively.
If these marks are arranged in descending order the figure 70 would receive the first rank, 66 the second rank, 65 the third and 63, the fourth rank. Since, the two students in the example have an equal score their rank is 2. Now they will be given the average rank of those ranks which these students would have secured had they differed slightly from each other.
On this assumption, the rank of both the items would be 2+3/2. i.e., 2.5 and the rank of the next item (65) would be 4. Thus, the coefficient of rank correlation would need a correction because the above formula [R= 1 6ΣD2 /N(N2-1]is based on the assumption that the ranks of various items are different.
Where there is more than one item with the same value, a correction factor, 1/12(t3 -t) is added to the value of zd2, where t. stands for number of items whose ranks are common. This correction factor is added as many times as the number of items with common ranks occur.
This is explained in the following example:
Analysis of Data and Interpretation
Calculate the coefficient of rank correlation from the following data:
In the above data-set of the X series the number 60 occurs three times. The rank of all three items is 5 which is the average of 4, 5 and 6, the ranks which these items would have secured had they differed slightly from one another. Other numbers 68 in X series and 70 in Y series, have occurred two times. Their ranks are respectively 2.5 and 1.5.
The modified formula for coefficient of rank correlation would thus be:
where n stands for the number of items repeated. In regard to the above example the formula will be:
A caution relating to the meaning and implication of a coefficient of correlation is quite warranted. The coefficient of correlation, by itself a very useful estimate of relationship, should not be taken as an absolute proof of association among relevant variables in as much as its interpretation depends in a large measure on the size of the sample selected for the study, as also, on the nature of the data collected.
A seemingly high coefficient of correlation, say, of 0.80 (+) may really be quite misleading if the standard error indicative of sample fluctuation is relatively large, or to take a contrary example, a seemingly low coefficient of say 0.45 (+) may suggest that the relationship among the variables may well be ignored but on the plane of reality, this indication may again be erroneous, since the coefficient of correlation for certain variables may typically be so low that the above correlation coefficient, i.e., 0.45 in comparison would need to be considered relatively quite high for the class of data in question.
However, statistical convention decrees that the coefficient of correlation ranging from 1 to 0.7 (+) be taken as an indication of ‘high’ or significant correlation, that ranging from 0.7 to 0.4 (+) as substantial, that between 0.4 and 0.2 (+) as low and that below 0.2 (+) as negligible.
It also needs to be stressed that a high correlation among two variables does not in itself constitute a proof that they are casually related. A significant correlation between variables — for example, between income and size of family or the size of an educational institution and the performance of the students — hardly affords any indication of a casual relationship obtaining amongst them.
Suppose we were to find that higher income is inversely correlated with the number of issues (children), i.e., higher the income of parents the lesser their number of issues (the coefficient of correlation is, say, 0.8 which is statistically quite high), we shall be wrong and unjustified in saying that higher income is the cause of lower fertility.
It was pointed out earlier that an inference of causality is warranted only if three kinds of proof, concomitant variation, time order and elimination of any other variable as the determining condition of the hypothesized effect, can be secured.
In the instant case, following inferences may possibly be drawn in full consideration of the pronounced correlation evident among the variables of income and number of children:
(a) One might be causing the other,
(b) Both the variables might be the effects of some other cause or causes, and
(c) The association may be a mere chance occurrence. Causal inferences can obviously be very surely established in an experimental situation.
We have considered this when dealing with experimental designs. In social sciences, it is very difficult to set up experiments, so must of the studies are non-experimental ones. Analytical procedures have, however, been devised to draw inferences about causal relationship in non-experimental studies.
The social researcher is quite so often interested in estimating the degree of association between attributes, i.e., between variables that are defined qualitatively; for example, he may want to ascertain the degree of association between the sexual attribute and political preference or between nativity and attitude toward a certain social issue.
Basically, the problem of association is one of correlation but the association between attributes may not easily become amenable to mathematical treatment as in case of the quantitative measures of variables. A measure of such association among attributes is the coefficient of relative predictability (RP) which is, in fact, a qualitative correlation coefficient.