it is a nonparametric measure of the relationship between 2 sets of ordinal (ranked) values

calculation

where d is the difference between ranks

example

significance

only has meaning in a sample, if it is say the set of preferences for people it has no meaning

if it is a sample case

H₀: there is no rank correlation

H₁: may be directional or nondirectional

a nondirectional test is a 2 tail test

a directional test is a 1 tail test

the df is the number of of pairs of ranked values

you can use a table if the sample size is small <100

but more generally the test statistic is

where N is the number of observations

this might be a case where a one tailed test is desirable since most researchers have an idea of the direction of the sign of the coefficient

example

we reject H₀, 2 tailed c.v.=2.57

H₀: no association between 2 variables

H₁: association between 2 variables - 2 tailed

H₁: +/- association between 2 variables - 1 tailed

correction for tied ranks

from a practical viewpoint it is often not worth correcting for ties

use of correction is advised if

1) when 3 or more observations are tied equally

2) when the number of pairs of ties is more than 1/4 of the number of observations

the formula is

A=(n³-n)/12

where t_x is the number of values of variable x tying at a given rank

t_y is the same for y

the effect of the correction for ties is to increase the value of r_s making it easier to reject the null hypothesis