Spearman’s Rank Correlation Coefficient

Spearman’s Rank Correlation Coefficient

Spearman's rank correlation coefficient or Spearman's rho is named after Charles Spearman

This is used to determined the strength of relationship between two variables of ordinal type.

Used Greek letter ρ (rho)(non- parametric measure of statistical dependence between two variables)

The formula used is:

Where: r = rank correlation coefficient = sum of the squared differences between the sets of

ranks.n = total number of observation.

2d

2

2

61

( 1)

d

n n

Interpretation of Rank Correlation Coefficient (ρ)

The value of rank correlation coefficient, ρ ranges from -1 to +1.

The ‘+’ sign indicates a positive correlation

(the scores on one variable increase as the scores on the other variable increase)

The ‘-’ sign indicates a negative correlation (the scores on one variable increase, the scores on the other variable decrease)

-1 0 +1

Perfect Negative Correlation

No Correlation Perfect Positive Correlation

Correlation scale:

Value of ρ Interpretation

+/- .80 to +/- .99 High correlation

+/- .60 to +/- .79 Moderately high correlation

+/- .40 to +/- .59 Moderate correlation

+/- .20 to +/- .39 Low correlation

+/- .10 to +/- .19 Negligible correlation

Interpretation: The sign of the Spearman correlation

indicates the direction of association between X (the independent variable) and Y (the dependent variable)

If Y tends to increase when X increases, the Spearman correlation coefficient is positive

If Y tends to decrease when X increases, the Spearman correlation coefficient is negative

A Spearman correlation of zero indicates that there is no tendency for Y to either increase or decrease when X increases

Example#1The table below shows the rating

of a group of ten supervisors who have been evaluated independently for leadership on a scale of 1-10 by the production manager and by the workers whom they supervise. Calculate ρ to determine if there is a correlation between the two evaluation.

Supervisor

Workers’ Evaluation

(X)

Workers’ Evaluation

(Y)

A 4 3

B 2 4

C 2 5

D 1 1

E 7 7

F 9 8

G 3 6

H 5 8

I 2 5

J 7 3

Solution: Construct a table.Supervi

sorX Y Rank of

XRank of

Yd d2

A 4 3 5 8.5 -3.5 12.25

B 2 4 8 7 1 1

C 2 5 8 5.5 -2.5 6.25

D 1 1 10 10 0 0

E 7 7 2.5 3 -0.5 0.25

F 9 8 1 1.5 -0.5 0.25

G 3 6 6 4 2 4

H 5 8 4 1.5 2.5 6.25

I 2 5 8 5.5 2.5 6.25

J 7 3 2.5 8.5 -6 36

72.5

Compute for ρ:

There is a moderate positive correlation between evaluations of the personnel managers and the workers.

2

2

61

( 1)

d

n n

2

6(72.5)110(10 1)

435110(100 1)

4351990

1 0.44

0.56

Correlation scale:

Value of ρ Interpretation

+/- .80 to +/- .99 High correlation

+/- .60 to +/- .79 Moderately high correlation

+/- .40 to +/- .59 Moderate correlation

+/- .20 to +/- .39 Low correlation

+/- .01 to +/- .19 Negligible/No correlation

Example#2Five college students have the

following rankings in Mathematics and Science subject. Is there an association between the rankings in Mathematics and Science subject?Student Ashl

eyDavid Owe

nSteven

Frank

Mathematics class rank(X)

1 2 3 4 5

Science class

rank(Y)

5 3 1 4 2

Math Rank(X)

Science Rank(Y)

X-Y(d)

(X-Y) 2

(d2)1 5 -4 16

2 3 -1 1

3 1 2 4

4 4 0 0

5 2 3 9

30

Make a table:

Compute for ρ by substituting the values in the formula:

2

2

61

( 1)

d

n n

2

6(30)15(5 1)

18015(25 1)

1801120

0.5

Interpretation:There is a moderate negative

correlation between the Math and Science subject rankings of students.

Students who rank high as compared to other students in their Math subject generally have lower Science subject ranks and those with low Math rankings have higher Science subject rankings than those with high Math rankings.

Seatwork:1. Two judges rated each of the twelve Audio-Visual

Presentation, using a 10 point scale. The following table shows the result:

AVP Judge A Judge B

1 5 7

2 4 8

3 3 4

4 10 6

5 3 5

6 9 8

7 10 10

8 1 3

9 8 7

10 6 5

11 3 8

12 4 4

Merits Spearman’s Rank CorrelationThis method is simpler to

understand and easier to apply compared to karl pearson’s correlation method.

This method is useful where we can give the ranks and not the actual data. (qualitative term)

This method is to use where the initial data in the form of ranks.

Limitation Spearman’s Correlation

Cannot be used for finding out correlation in a grouped frequency distribution.

This method should be applied where N exceeds 30.