Statistics for Environmental Engineers

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If we had two series of values for x and y and did not know they were ranks, we would calculate r = ~Tx=l, where xi is replaced by xi — x and yt by y{ y. The sums are over the n observed values.


VXx, Xy,


Knowing that the data are rankings, we can simplify this using d2 = (xt — yt)2, which gives xyyi = 2(x2 + y2 — d2) and:


rs



Z(x2+ y2 — d2) 2*fZx2Zy2



Ex? + Zy2 — Ed2 2jZxfZy2



Z y2 =


The above equation can be used even when there are tied ranks. If there are no ties, then Z x2 n (n2 — 1)/12 and:


rs = 1 —


6Zd2


n(n2 — 1)


The subscript S indicates the Spearman rank-order correlation coefficient. Like the Pearson product-moment correlation coefficient, rs can vary between -1 and +1.

Case Study: Taste and Odor


Drinking water is treated with seven concentrations of a chemical to improve taste and reduce odor. The taste and odor resulting from the seven treatments could not be measured quantitatively, but consumers could express their opinions by ranking them. The consumer ranking produced the following data, where rank 1 is the most acceptable and rank 7 is the least acceptable.


Water Sample


A


B


C


D


E


F


G


Taste and odor ranking


1


2


3


4


5

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