How to calculate spearman's rank coefficient of correlation.

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PRELIMINARIES

This post is designed to help students in their higher learning units that include correlation as topic. The concept of correlation is tested in all examinations national and international. This post is prepared by a professional Educator and can be used as lecture notes.

What is Correlation?

Correlation is the degree of relationship between to variables say X and Y or Independent and Dependent Variables.

Whenever there is some definite connection existing between two or more groups, classes or series, there is said to be correlation.

Why is it important to study correlation?

1.      Correlation shows the degree of relationship between variables such as demand and supply, income and expenditure etc

2.      The knowledge of correlation helps to locate critically important variables on which others depend.

3.      Progressive development of methods of science and philosophy has been characterized by the increase in the knowledge of relationship among variable.

4.      It clearly shows the cause effect relationship which has been used to solve many problems and decision making.

Types of correlation

1.      Positive and negative correlations.

-          Correlation is positive if two series /variables move in the same direction. For example, supposing that Demand and supply are two variables, and an increase in demand causes an increase in supply, we can conclude that demand and supply are positively correlated.

-          However, correlation is said to be negative if two series / variables move in the opposite direction. For instance, the above mentioned scenario, an increase in demand leads to  a decrease in supply, then we can conclude that demand and supply are negatively correlated.

2.      Linear and non-linear correlations.

-          Correlation is linear if the amount of change in one variable bears a constant change in the amount of change in the other variable. However, correlation is Non–linear if the amount of change in one variable does not bear a constant ratio to the amount of change in the other variable.

3.      Simple, partial and multiple correlations.

-          When only two variables are studied it is a simple correlation problem.

-          When more than two variables are studied, the problem is either partial or multiple correlations.

 

     What are methods of studying correlation?

a)      Scatter diagrams method

b)      Karl Pearson’s coefficient of correlation.

c)      Spearman’s rank correlation coefficient.

d)     Method of least squares.

 

A)    SPEARMAN’S RANK CORRELATION COEFFICIENT

This method was developed by a British psychologist, Charles Edward Spearman in 1904.

- RANK COEFFICIENT OF CORRELATION is defined as the correlation between the ranks assigned to individuals in two characters

The formula is:  (insert formula)

 

Where R= rank coefficient of correlation.

            d = difference between ranks

            n = number of pairs of ranks.

- In rank correlation there are two types of problems.

   a) Where actual ranks are given.

   b) Where ranks are not given

1. WHERE ACTUL RANKS ARE GIVEN.

 ILLUSTRATION

Compute the rank coefficient of correlation and comment on the value.

Employee Rank 1             Rank 2

A                           10                                9

B                           2                                  4

C                           1                                  2

D                           4                                  3

E                            3                                  1

F                            6                                  5

G                           5                                  6

H                           8                                  8

I                             7                                  7

J                             9                                  10

 

SOLUTION

- Calculate the differences between ranks ie d= (rank 1(R1) – rank 2(R2))

- Square the differences to obtain d²

- apply the formula to calculate R

THUS.

(d = RI-R2)      d²

1                                                        1

-2                           4

-1                           1

1                            1

2                            4

1                            1

-1                           1

0                            0

0                            0

-1                           1

                  ∑d²= 14


B)    WHERE RANKS ARE NOT GIVEN.

-Ranks can be assigned by giving the highest value 1 or the smallest value 1. In our illustrations we have assigned the smallest value 1.

      ILLUSTRATION

1.      The following are the scores awarded by two assessors to eight employees based on their work ethics.

Employee        Assessor X      Assessor Y

            A         65                    50

            B         70                    48

            C         55                    72

            D         52                    80

            E          48                    62

            F          77                    70

            G         83                    65

            H         45                    59

i) Determine Spearman’s rank coefficient of correlation

ii) Interpret the results in (i) above.

SOLUTION

- Assign ranks to the scores by assigning the smallest value 1 (one)

Employee        Assessor    RANK 1        Assessor     RANK 2

                        X                (R1)                      Y         (R2)

            A         65                    5                      50          2

            B         70                    6                      48          1

            C         55                    4                      72          7

            D         52                    3                      80          8

            E          48                    2                      62          4

            F          77                    7                      70          6

            G         83                    8                      65          5

            H         45                    1                      59          3

 d (R1 - R2)           d²

3                            9

5                            25

-3                            9                   

-5                           25

-2                            4

1                               1

3                               9

-2                              4

                             ∑d²=  86



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