Spearman’s Rank Correlation Coefficient
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.