Simple tests for two independent samples

The signed rank test described in the previous section is valuable for the study of single sets of measurements, and for paired sets that can readily be reduced to single sets. In many instances, however, it is necessary to compare two independent samples that cannot be reduced to a single set of data. Such samples may contain different numbers of measurements. Several non-parametric tests to tackle such problems have been devised. The simplest to understand and perform is the Mann-Whitney l/-test, the operation of which is most easily demonstrated by an example. 

A sample of photographic waste was analysed for silver by atomic absorption spectrometry, five successive measurements giving values of 9.8, 10.2, 10.7, 9.5, and 10.5 pg ml . After chemical treatment, the waste was analysed again by the same procedure, five successive measurements giving values of 7.7, 9.7, 8.0, 9.9 and 9.0 pg ml . Is there any evidence that the treatment produced a significant reduction in the levels of silver  

The Mann-Whitney procedure involves finding the number of results in one sample that exceeds each of the values in the other sample. 

In the present case, we believe that the silver concentration of the treated solution should, if anything, be lower than that of the untreated solution (i.e. a onesided test is appropriate). We thus expect to find that the number of cases in which a treated sample has a higher value than an untreated one should be small. Each of the values for the untreated sample is listed, and the number of instances where the values for the treated sample are greater are counted in each case. 

Untreated sample Higher values in treated sample Number of higher values 

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