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# SPEARMEN’S RANK CORRELATION COEFFICIENT

The method of Karl Pearson is based on assumption population being studied is normally distributed. When it is known to us that population is not normal, or when distribution’s shape is not known, then there is a need for the measure of correlation that does not involve any assumption regarding parameters of population.

In 1904 Charles Edward Spearmen, Britishpsychologist, develop a method of finding out the covariabilityor lack of it between two variables. This method is useful when quantitative measure for certain factors cannot be fixed.

Spearmen’s rank correlation coefficient is defined as: Rs=1 – 6(SIGMA) D^2/N^3-N

Rs=spearmen’s rank correlation.

D=sum of difference between two ranks. i.e. (R1-R2)

N=no. of observations.

The value of coefficient ranges between -1 and +1.When coefficient is +1 then ranks are in same direction. When coefficient is -1 then ranks are in opposite direction.

Features of Spearmen’s Correlation Coefficient

The sum of difference between two ranks must be zero.(SIGMA)D=0

Rank correlation coefficient is non-parametric or distribution-free because there is no strict assumption is made about form of population from where sample is drawn.

The Spearmen’s correlation coefficient is same as Karl Pearson’s correlation coefficient between the ranks

Three types of problem arise in rank correlation:

When ranks are given

When ranks are not given

When ranks are equal

When ranks are given:

Steps required for computing rank correlation when actual ranks are given to us:

1. Calculate the difference between two ranks, i.e., (R1-R2) denote this difference by D.

2. Square that difference and calculate the total (SIGMA) D^2.

3. Then apply the formula

Rs=1 – 6(sigma) D^2/N^3-N

When ranks are not given:

when actual ranks are not given to us, the first step is to assignment ranks to the observations either taking highest value as 1 or lowest value as 1.The method of starting with either highest or lowest value must follow for both variables and must follow the step as followed in first method.

When ranks are equal:

In some cases two or more individuals have same ranks. In such case an average rank is given to each individual. For ex. If two individuals have same ranks for fifth place, then (5+6)/2=5.5 is a rank assigned to both individual. For ex. If there are three individuals having same ranks for fifth place, then (5 +6+7)/3=6 is the rank assigned to each individual.

This adjustment consist of adding 1/12(m^3-m) to (sigma) D^2, where m stands for no. of items with common rank, such value is added as many times the no. of such groups.

Where,

Rs=1 – {6(sigma) D^2+1/12(m^3-m) +1/12(m^3-m) +….}/N^3-N

Merits of rank method:

This method is easy to understand and to apply as compared to Karl Pearson’s method.

This method is also used where data is in qualitative nature i.e. honesty, intelligence. By allotting ranks to the individuals.

Demerits of rank method:

Correlation in a grouped frequency distribution cannot be finding out by this method.

When the no. of items exceeds 30 i.e. N>30, then calculation became difficult and requires lot of time.

When to use rank correlation coefficient?

When we are dealing with qualitative variables.eg. Honesty, intelligence etc.

When any assumption of Karl Pearson’s does not hold well.

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