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Spearman’s Rank Correlation Coefficient

In 1904, a British psychologist Charles Edward Spearman developed a method to measure the statistical association between two variables, say x and y, when only ordinal or ranked data are available. This implies that Spearman’s rank correlation coefficient method is applied in a situation, where quantitative measure of qualitative factors, such as judgement, brand personalities, beauty, intelligence, honesty, efficiency, tv programmes, leadership, etc. cannot be fixed but individual observations can be arranged in a definite order or rank.

 

 

Mathematically, Spearman’s rank correlation coefficient is defined as

 

 

 

where, ‘R’ is the Spearmen’s rank correlation coefficient

 

n’ is the number of observations

 

di is the difference in ranks for the ith observation

 

Note: Highest value of the observations is given the rank 1 and then following ranks are given in decreasing order of observations.

 

Case 1When ranks are given
 

Example
Following are the ranks of 10 students in Botany and Zoology projects given by 2 professors. Find the rank correlation coefficient.
Solution
 

 

Case 1When ranks are not given

Highest value of the observations is given the rank 1 and then following ranks are given in decreasing order of observations.

 

Example: Following are the scores of 10 students in Botany and Zoology projects given by 2 professors. Find the rank correlation coefficient.

 


 

The above example shows that, how to find the Spearman’s rank coefficient when the ranks are all different.

 

When the ranks of certain observations are same we can use the below formula to find the rank correlation coefficient:

 

 

 

In the above relation, di = xi - yi (difference in ranks)

 

It represents the number of times a certain rank is repeated.

 

Case 3: When ranks are equal

If more than one observation in the data are equal, then the ranks given are called tied ranks. The ranks to be assigned for the individual observations are an average of the ranks and these ranks deserve individual observations.

 

 

 

Where, C.F. is correction factor and

 

m’ is the number of times the data repeats in the data series
 

Example
The following are the marks in statistics (x) and the marks in mathematics (y) of 10 students in an examination. Find the coefficient of rank correlation.
 
Solution
The students are awarded ranks according to their marks in the two subjects.
 

The correction factor is:

 

Note: Spearman’s coefficient of rank correlation can be calculated even if the characteristics under study are qualitative. It can be calculated in the case of ordinal data also.

 

The value of Spearman’s rank coefficient also lies between –1 and +1
 

Merits

  1. As mentioned earlier, this method can be used as a measure of degree of association between qualitative data.
  2. This method is very simple and easily understandable
  3. It can be used when the actual data is given or when only the ranks of the data are given

Demerits

  1. We cannot calculate the ranks coefficient for a frequency distribution, i.e., grouped data
  2. When a large number of observations are given, the calculation becomes tedious





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