**Type II error - An error caused by failing to reject a null hypothesis that is not true**

** RANK CORRELATION**

It is a method of ascertaining co variability or the lack of it between the two variables.

Rank correlation method is developed by the British Psychologist Charles Edward Spearmen in 1904. Gupta S.P has stated that "the rank correlation method is used when quantitative measures for certain factors cannot be fixed, but individual in the group can be arranged in order thereby obtaining for each individual a number of indicating

his/her rank in the group".

**The formula for Rank Correlation**

**Rank – Difference Coefficient of Correlation (Case of no ties in ranks)**

Rank Correlation,

= 1 – (6 x 38) / 5x(5^{2}-1) = 1 – 228/(5x24) = 1 – 1.9 = -0.9

Relationship between X and Y is very high and inverse. Relationship between Scores on Test I and Test II is very high and inverse.

**Procedure for Assigning Ranks**

First rank is given to the student secured highest score. For example, in Test I, student F is given first rank, as his score is the highest. The second rank is given to the next

highest score. For example, in Test I, student E is given second rank. Student A and G have similar scores of 20 each and they stand for 6th and 7th ranks. Instead of giving either 6th or 7th ranks to both the students, the average of the two ranks [average of 6 and 7] 6.5 is given to each of them. The same procedure is followed to assign ranks to the scores secured by students in Test II.

**Calculation of Rank Correlation when ranks are tied**

**Rank – Difference Coefficient of Correlation (in case of ties in ranks)**

= 1 – [(6x24) / 10(10^{2}-1)] = 1 – [144/990] = 0.855
**APPLICATION OF CORRELATION**

Karl Pearson Coefficient of Correlation can be used to assess the extent of relationship between motivation of export incentive schemes and utilization of such schemes by exporters.

**Motivation and Utilization of Export Incentive Schemes – Correlation**
**Analysis**

Opinion scores of various categories of exporters towards motivation and utilization of export incentive schemes can be recorded and correlated by using Karl Pearson

Coefficient of Correlation and appropriate interpretation may be given based on the value of correlation.

**Testing of Correlation**

't' test is used to test correlation coefficient. Height and weight of a random sample of six adults is given.

It is reasonable to assume that these variables are normally distributed, so the Karl Pearson Correlation coefficient is the appropriate measure of the degree of association between height and weight.

r = 0.875

**Hypothesis test for Pearson's population correlation coefficient**
H0: p = 0 - this implies no correlation between the variables in the population

H1: p > 0 - this implies that there is positive correlation in the population (increasing height is associated with increasing weight)

5% significance level

= 0.875 x [(6–2)^{1/2}] / (1–0.875^{2}) = 0.875 x 2 / 0.234 = 3.61
Table value of 5% significance level

4 degrees of freedom (n-2) = (6-2) = 2.132

Calculated value is more than the table value. Null hypothesis is rejected. There is significant positive correlation between height and weight.

**Partial Correlation**

Partial Correlation is used in a situation where three and four variables involved. There variables such as age, height and weight are given. Here, partial correlation is applied.

Correlation between height and weight can be computed by keeping age constant. Age may be the important factor influences the strength of relationship between height and weight. Partial correlation is used to keep constant the effect age. The effect of one

variable is partially out from the correlation between other two variables. This statistical technique is known as partial correlation.

Correlation between variables x and y is denoted as rxy

Partial correlation is denoted by the symbol r123. This is correlation between variables 1 and 2, keeping 3rd variable constant.

where,

r123 = partial correlation between variables 1 and 2 r12 = correlation between variables 1 and 2

r13 = correlation between variables 1 and 3 r23 = correlation between variables 2 and 3

**Multiple Correlation**

Three or more variables are involved in multiple correlation. The dependent variable is denoted by X1 and other variables are denoted by X2, X3 etc. Gupta S. P. has expressed that "the coefficient of multiple linear correlation is represented by R1 and it is common to add subscripts designating the variables involved. Thus R1.234 would represent the coefficient of multiple linear correlation between X1 on the one hand, X2, X3 and X4 on the other. The subscript of the dependent variable is always to the left of the point".

The coefficient of multiple correlation for r12, r13 and r23 can be expressed as follows:

Coefficient of multiple correlations for R1.23 is the same as R1.32. A coefficient of multiple correlation lies between 0 and 1. If the coefficient of multiple correlations is 1, it shows that the correlation is perfect. If it is 0, it shows that there is no linear relationship between the variables. The coefficients of multiple correlation are always positive in sign and range from +1 to 0.

Coefficient of multiple determinations can be obtained by squaring R1.23.

Multiple correlation analysis measures the relationship between the given variables. In this analysis the degree of association between one variable considered as the dependent variable and a group of other variables considered as the independent variables.

**SUMMARY**

This chapter outlined the significance in measuring the relationship. This chapter discuss the factors that affecting correlation. The different applications of correlation have been dealt in detail.

**KEY WORDS**

· Measures of Relationship

· Correlation

· Simple correlation

· Partial correlation

· Multiple correlation

· Regression

· Simple regression

· Multiple regressions

· Association of Attributes

· Scatter Diagram Method

· Graphic Method

· Karl Pearson's Coefficient of Correlation

· Concurrent Deviation Method

· Method of Least Squares Karl Pearson's Coefficient of Correlation
**REVIEW QUESTIONS**

1. What are the different measures and their significance in measuring Relationship?

2. Discuss the factors affecting Correlation.

3. What are the applications of Correlation?

4. Discuss in detail on different types of Correlation.