Non-parametric tests for more than two samples

The probability of obtaining any particular sum of squares can be determined by using the chi-squared statistic (see Chapter 3). If the samples are referred to as A, B, C, etc. (k samples in all), with numbers of measurements nJ, n, Uc, etc. and rank totals Ra, Rg, Rc, etc., then the value of is given by  

The latter are identical to the usual values when the total number of measurements 

We have already seen (Sections 3.4 and 6.3) that when paired results are compared, special statistical tests can be used. These tests use the principle that, when two experimental methods that do not differ significantly are applied to the same chemical samples, the differences between the matched pairs of results should be close to zero. This principle can be extended to three or more matched sets of results by using a non-parametric test devised in 1937 by Friedman. In analytical chemistry, the main application of Friedman s test is in the comparison of three (or more) experimental methods applied to the same chemical samples. The test again uses the statistic, in this case to assess the differences that occur between the total rank values for the different methods. The following example illustrates the simplicity of the approach. 

The levels of a pesticide in four plant extracts were determined by (A) high-performance liquid chromatography, (B) gas-liquid chromatography, and (C) radioimmunoassay. The following results (all in ng ml ) were obtained  

Do the three methods give values for the pesticide levels that differ significantly  

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