RANK SUM TEST THE  MANN WHITNEY U - TEST

Mann  Whitney u test is an alternative to the  samples  test. This  test is based  on the ranks  of the  observation of two samples put together. This test is more powerful. When compared to the sign tests. The alternative  name for this test is rank sum test.

The sign  test for  comparing two population distributions ignores the actual magnitude of the paired  observation  and thereby discards information that  would be useful  in detecting a departure from the  null  hypothesis.

Rank sum  test is a whole  family of tests. Here we shall  discuss only one  of the types the Mann  Whitney u test. With this test  we can  test the  null hypothesis μ=μ0 without  assuming whether the  population sampled have roughly the shape of normal  distribution.

This  test helps  us to determine  whether two  sample shave come from identical populations. If  it is true  that the  samples  have come from the same population it is reasonable to assume that the means  of  the ranks assigned to the values of the two sample are more or less the same. The alternative  hypothesis  is that  the means  of the  population are not  equal  and if this is  the case  most  of the  smaller rank  will go to  the values  of one  samples  while  most of the   higher ranks  will go to  those of the  other  sample.

The  test of the null hypothesis that the  two  samples  come  from  identical population may either be based on   R1, the  sum of the  ranks of the  values of first sample  or on R₂ the sum of the  ranks  of the  values of the second  sample. It  may  be noted  that in  practice it does not  matter  which  sample  we call  sample   1 and  which  we call  sample 2.

If the sample  sizes  are n₁ and n₂ the sum  of R₁ and R₂ simply  the sum  of first  n₁ + n₂ positive  integers which is known   to be.

(n₁+ N₂) ( n₁+ n₂ + 1) /2

This  formula  enables us  to find  R₂ if we known  R₁ and vice versa.

When  the use  of the  sums  was first proposed  as a nonparametric  alternative to the  two sample t test  the decision  was based on R₁ or R₂ but now the decision  is usually based on  either  of the  related statistics:

U ₁ = n₁ n₂ + n₁( n₁+ 1) / 2 - R₁

 U  = n₁ n₂ + n₂( n₂+ 1) / 2 - R₁

Where  n1 and n2 are the size  of the samples  and R1 and R2 are the rank  sums of the corresponding  samples. For  small  samples if both n  and n2 are less than 10 (some  statisticians say 8.) special  tables must be  used and if U is smaller than the  critical  value H0 can be  related to the  standard normal curve by the  statistic.

Z= u - n₁ n₂/ 2 / √n₁ n₂ ( n₁+ n₂) / 12

It is unimportant  whether  the larger  or smaller  value  obtained  from the  formulae  is used. The values  for Z  will be numerically  equal but opposite in sign.