# Statistics for Environmental Engineers

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and

2 = ntIk=1 (yt — y) sb    k — 1

The logic of the comparison between for s2W and sb goes like this:

1.    The pooled variance within treatments (sW) is based on N — k degrees of freedom. It will be unaffected by real differences between the means of the k treatments. Assuming no hidden factors are affecting the results, s2W estimates the pure measurement error variance a2.

2.    If there are no real differences between the treatment averages other than what would be expected to occur by chance, the variance between treatments (sb) also reflects only random measurement error. As such, it would be nearly the same magnitude as sW and would give a second estimate of a2.

3.    If the true means do vary from treatment to treatment, s7b will be inflated and it will tend to be larger than s7W .

4.    The null hypothesis is that no difference exists between the к means. It is tested by checking to see whether the two estimates of a (sb and sw) are the same. Strict equality (sw= sb) of these two variances is not expected because of random variation; but if the null hypothesis is true, they will be of the same magnitude. Roughly speaking, the same magnitude means that the ratio sllsW will be no larger than about 2.5 to 5.0. More precisely, this ratio is compared with the F statistic having к — 1 degrees of freedom in the numerator and N — к degrees of freedom in the denominator (i.e., the degree of freedom are the same as sb and sW).

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