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4.9 Rank Sum Test
Two means may be compared using a t-test if the data described by the means is normally distributed.
If not, either a transformation should be applied to the data in order to obtain normal distributions, or a
distribution-free test should be used (Davies and Goldsmith, 1977 p. 74). The rank-sum test is a
distribution-free test that uses the rank-order of the data :
For 2 data sets with n1 and n2 number of observations
• The data sets are combined and ranked in order of increasing value
• Rank 1 is assigned to the lowest value, rank 2 to the next etc. until the highest value which is
assigned rank n1+n2.
• n1 = smaller sample size; n2 = larger sample size; n = n1+n2
• R is the sum of ranks of the smaller sample
′ 1
• R = n1
⋅ ( n + ) − R
• The critical value M is available in statistical tables or can be calculated from
and
u = 2.58 for α =
( n + 1)
n1
n1n
2
M = ( n + 1)
− u
2
12
where u = 1.96 for α = 0.05 (double - sided test)
0.01 (double - sided
192
test)
• If R or R’