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Spearman’s Rank Order Correlation Page 1
How to Compute and Interpret Spearman’s Rank Order Correlation
Objective: Learn how to compute, interpret and use Spearman’s rank order correlation.
Keywords and Concepts
1. Spearman’s rank order correlation 4. Rank order
5. Difference between ranks
2. rho
3. Ordinal data 6. Degrees of freedom (df)
Spearman’s rank order correlation (ρ or rho) determines the relationship between
two sets of ordinal data (usually paired) that initially appear in rank order or have been
converted to rank order. It uses the item’s position in a rank-ordered list as the basis for
assessing the strength of the association. Data in Kinesiology and sports competition
frequently appear as ranked data. For example, a coach may rank his players’ skill level
from 1 (highest skill), 2 (next best) on down to the last rank (lowest skill). Baseball
leagues and ladder and round-robin tournaments rank individuals or teams. Even
when data have been collected on a parametric variable, the raw data can be converted
to rankings and the rank order correlation method applied, although at the expense of
some loss in mathematical precision.
Rank Order Correlation Formula
6￥ 2d
ρ =1 - 2
N(N - 1)
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Spearman’s Rank Order Correlation Page 2
where, ρ (rho) is Spearman’s rank order correlation coefficient, d is the difference
between ranks for the two observations within a pair (see table 1 below), and N
represents the total number of subjects (i.e., number of data pairs ). The number in the
numerator is always 6. The degrees of freedom (df) for rho calculates as (df = Npairs - 2.)
Example
The data in Table 1 illustrate 10 major universities’ ranked for research dollars
awarded in health sciences and their football team’s conference ranking in 2001.
Table 1. Rankings of major U.S. universities (A-J) for research
dollars and football rankings.
School A B C D E F G H I J
Research
1 2 4 6 3 5 9 7 10 8
dollars
Football
4 5 3 1 9 7 6 8 2 10
rankings
3 3 1 5 6 2 3 1 8 2
d
9 9 1 25 36 4 9 1 64 4
d2
Solution
When two scores tie in rank, both are given the mean of the two ranks they
would occupy and the next rank is eliminated to keep N consistent. For example, if two
schools tied for 4th place, both would receive a rank of 4.5 (4 + 5 ÷ 2), and the next school
would be ranked number 6.
The following equation computes Spearman’s rank order correlation:
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Spearman’s Rank Order Correlation Page 3
6￥ 2d
ρ =1 - 2
N(N - 1)
6(162)
r =1 -
10(10 2 - 1)
972
r =1 -
990
r = 0.02
Interpretation
To determine if the rho coefficient is statistically significant (e.g., reject the Null
hypothesis that the real rho is zero), compare the magnitude of rho versus the value
found in Table 2. The degrees of freedom (df) equal:
df = Npairs - 2
= 10 - 2 = 8
From table 2, at the 0.05 level of significance, with df = 8, a rho correlation
coefficient of 0.74 is required for statistical significance. Thus, the observed rho of 0.02
indicates that there is no relationship between rankings of college football programs
and the amount of research dollars generated in the health sciences. Note that the table
only goes up to N = 30. If N > 30, then the following formula computes the critical value
to assess the statistical significance of the rho coefficient.
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Spearman’s Rank Order Correlation Page 4
ﾱz
ρ=
N- 1
where the value of z corresponds to the significance level. For example, if the
significance level is 0.05, z will equal 1.96. If rho exceeds the computed critical value, it is
statistically significant.