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

2. rho

3. Ordinal data

4. Rank order

5. Difference between ranks

6. Degrees of freedom (df)

Spearman’s rank order correlation ( 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

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=1 −6 d2∑N(N2 −1)

where, (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

dollars

1 2 4 6 3 5 9 7 10 8

Football

rankings

4 5 3 1 9 7 6 8 2 10

d 3 3 1 5 6 2 3 1 8 2

d2 9 9 1 25 36 4 9 1 64 4

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

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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:

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

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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.

=±zN −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.

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Table 2. Critical values of Spearman's Rank Correlation

Coefficient (rho).

n alpha = 0.01

alpha = 0.05

alpha = 0.02

alpha = 0.01

5 0.900 --6 0.829 0.886 0.943 --7 0.714 0.786 0.893 --8 0.643 0.738 0.833 0.8819 0.600 0.683 0.783 0.83310 0.564 0.648 0.745 0.79411 0.523 0.623 0.736 0.81812 0.497 0.591 0.703 0.78013 0.475 0.566 0.673 0.74514 0.457 0.545 0.646 0.71615 0.441 0.525 0.623 0.68916 0.425 0.507 0.601 0.66617 0.412 0.490 0.582 0.64518 0.399 0.476 0.564 0.62519 0.388 0.462 0.549 0.60820 0.377 0.450 0.534 0.59121 0.368 0.438 0.521 0.57622 0.359 0.428 0.508 0.56223 0.351 0.418 0.496 0.54924 0.343 0.409 0.485 0.53725 0.336 0.400 0.475 0.52626 0.329 0.392 0.465 0.51527 0.323 0.385 0.456 0.50528 0.317 0.377 0.448 0.49629 0.311 0.370 0.440 0.48730 0.305 0.364 0.432 0.478

“Distribution of sums of squares of rank differences to small numbers of individuals," The Annals of Mathematical Statistics, Vol. 9, No. 2. Reprinted with permission of the Institute of Mathematical Statistics.