Rank-Order Correlation:Example

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Contents

Here we illustrate use of the rank-order coefficient: we ask whether the grades in one course correlate with grades in a previous course.[1]

Initial Data and Value of r

We have grades for twelve students from a fall course and, for the same students, from another course taken the following spring. These data are given in Table 1; they are plotted in Figure 1.


Table 1. Final grades for twelve students in two courses.
Student Fall Grade Spring Grade
A 50.4 30
B 67.2 57
C 55.3 34
D 52.5 41
E 64.1 46
F 60.9 43
G 64.8 43
H 58.4 43
J 54.8 24
K 60.9 38
L 66.8 62
M 40.0 38
Figure 1. Grades from Table 1 for twelve students in a fall course and in a spring course. The linear correlation coefficient is r = 0.64.
Figure 1. Grades from Table 1 for twelve students in a fall course and in a spring course. The linear correlation coefficient is r = 0.64.

We compute the linear correlation coefficient for these 12 pairs of points and find r = 0.64. From this value we are hard pressed to say whether or not the grades are correlated. Figure 1 suggests that a correlation exists, but it is probably nonlinear; if it is nonlinear, then the linear correlation coefficient cannot be used to measure the strength of the correlation.

Value of ρ

So we turn to the rank-order coefficient. We rank the grades in the two courses, obtaining Table 2. Note in Table 1 that three students have the same grade of 43 in the spring term; these should fill ranks 4, 5, and 6, whose mean is 5. Therefore, the rank Q = 5 appears three times in Table 2.


Table 2. Rankings of student grades from Table 1.
Student R(fall) Q(spring)
B 1 2
L 2 1
G 3 5
E 4 3
F 5.5 5
K 5.5 8.5
H 7 5
C 8 10
J 9 12
D 10 7
A 11 11
M 12 8.5
Figure 2. Ranks of student grades from Table 2. The rank-order correlation coefficient is computed to be ρ = 0.81, which has a significance of better than 1%.
Figure 2. Ranks of student grades from Table 2. The rank-order correlation coefficient is computed to be ρ = 0.81, which has a significance of better than 1%.

The ranks in Table 2 are plotted in Figure 2. Using these ranks in the definition of the rank-order coefficient,


(1)
\rho \ = \ \frac{\sum_i^N{(R_i - R_m)(Q_i - Q_m)}}{\sqrt{\left (\sum_i^N(R_i - R_m)^2 \right )\left (\sum_i^N(Q_i - Q_m)^2 \right )}}


we find ρ = 0.81. From Table 3 on the page for ρ we see that for N = 12 points, we must have ρ ≥ 0.78 for 1% significance. Therefore we conclude that, with a high degree of confidence, the grades for these students in these two courses are correlated. This suggests (but does not prove; see Correlation vs Connection) that performance in the first course affects performance in the second. However, this result tells us nothing about any correlation of grades for other students. Nevertheless, the existence of a correlation for certain students might be important to those students and their instructors, even though we cannot say what form the correlation takes.

See Also

Reference

  1. ↑ J. M. Haile, Analysis of Data, Macatea Productions, Central, SC, 2003. ISBN 0-9728602-0-7.
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