Statistics for Environmental Engineers

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6


7


Chemical added (mg/L)


0.9


2.8


1.7


2.9


3.5


3.3


4.7

The chemical concentrations are converted into rank values by assigning the lowest (0.9 mg/L) rank 1 and the highest (4.7 mg/L) rank 7. The table below shows the ranks and the calculated differences. A perfect correlation would have identical ranks for the taste and the chemical added, and all differences would be zero. Here we see that the differences are small, which means the correlation is strong.


Water Sample


A


B


C


D


E


F


G


Taste ranking


1


2


3


4


5


6


7


Chemical added


1


3


2


4


6


5


7


Difference, di


0


-1


1


0


-1


1


0

The Spearman rank correlation coefficient is:


24


1 — -24- = 0.93


336


r = , 6Z(-1)2+12+12 + (-1)2 s    7(72-1)


From Table 31.2, when n = 7, rs must exceed 0.786 if the null hypothesis of “no correlation” is to be rejected at 95% confidence level. Here we conclude there is a correlation and that the water is better when less chemical is added.

Comments


Correlation coefficients are a familiar way of characterizing the association between two variables. Correlation is valid when both variables have random measurement errors. There is no need to think of one variable as x and the other as y, or of one as predictor and the other predicted. The two variables stand equal and this helps remind us that correlation and causation are not equivalent concepts.

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