# Spearmanâ€™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 Spearmanâ€™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, Spearmanâ€™s rank correlation coefficient is defined as

where, â€˜*R*â€™ is the Spearmenâ€™s rank correlation coefficient

â€˜*n*â€™ is the number of observations

*d _{i}* is the difference in ranks for the

*i*

^{th}observation

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

**Case 1**: *When ranks are given*

**Case 1**: *When 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 Spearmanâ€™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, *d _{i}* =

*x*-

_{i}*y*(difference in ranks)

_{i}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

â€˜*m*â€™ is the number of times the data repeats in the data series

*x*) and the marks in mathematics (

*y*) of 10 students in an examination. Find the coefficient of rank correlation.

The correction factor is:

**Note:** Spearmanâ€™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 Spearmanâ€™s rank coefficient also lies between â€“1 and +1

*Merits*

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

*Demerits*

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