12.3. The Kruskal-Wallis Test and the Runs Test http://www.ck12.org
z=of observed runs−μ/σμ=expected number of runs= 1 +2 n 1 n 2
n 1 +n 2
σ^2 =variance number of runs=
2 n 1 n 2 ( 2 n 1 n 2 −n 1 −n 2 )
(n 1 +n 2 )^2 (n 1 +n 2 − 1 )When conducting the runs test, we calculate the standardz-score and evaluate our hypotheses just like we do with
other parametric and non-parametric tests.
Example:
A teacher is interested in assessing if the seating arrangement of males and females in his classroom are random. He
records the seating pattern of his students and records the following sequence:
MFMMFFFFMMMFMFMMMMFFMFFMFFFF
Is the seating arrangement random? Use aα=.05.
Solution:
To answer this question, we first generate the null hypothesis that the seating arrangement is random and indepen-
dent. Our alternate hypothesis states that the seating arrangement isnotrandom or independent. With aα=.05, we
set our critical values at 1.96 standard scores above and below the mean.
To calculate the test statistic, we first record the number of runs and the number of each type of observation:
R= 14
1.M 2 (n (^2) ↓ 1 ) = 13
2.F 2 (n (^2) ↓ 2 ) = 15
With these data, we can easily compute the test statistic:
μ=expected number of runs= 1 +
2 ( 13 )( 15 )
13 + 15
= 1 +
390
28
= 14. 9
σ^2 =variance number of runs=2 ( 13 )( 15 )( 2 ∗ 13 ∗ 15 − 13 − 15 )
( 13 ∗ 15 )^2 ( 13 + 15 − 1 )
=
390 ( 362 )
( 152100 )( 27 )
=. 0034
σ= 0. 05z= of observed runs−μ/σ=14 − 14. 9
. 05
=− 18. 0
Since the calculated test statistic is extremely high(z= 18. 0 )and exceeds our critical value we can reject the null
hypothesis and conclude that the seating arrangement of males and females is not random.