# Statistics for Environmental Engineers

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2 = (13-16)7 + (17-16)7 + (20-16)7 + (14-16)7 _ 30 sB =    4-1    ‘3

2 = ( 1 8! — 1 8! )7 + (16 — 1 8! )7 + (2 1 — 18 )7 + ( 17 — 1 8 )7 _ 14 sc =    4 — 1    ’31

The pooled within-treatment variance is:

2 = (4 — 1)(3.33 + 10 + 4.67)

10.00

4.67

w

12 — 3

= 6.0

The between-treatment variance is computed from the mean for each treatment and the grand mean, as follows:

2 = 4(11 — 15 )7 + 4(16 — 15 )7 + 4(18 — 15)7

sb =    -1 —

3 — 1

52

Analysis of Variance Table for Comparing Treatments A, B, and C

 Source of Variation Sum of Squares Degrees of Freedom Mean F Square Ratio Between treatments 104 2 52 8.7 Within treatments 54 9 6 Total 158 11

Is the between-treatment variance larger than the within-treatment variance? This is judged by comparing the ratio of the between variance and the within variance. The ratios of sample variances are distributed according to the F distribution. The tabulation of F values is arranged according to the degrees of freedom in the variances used to compute the ratio. The numerator is the mean square of the “between-treatments” variance, which has v1 degrees of freedom. The denominator is always the estimate of the pure random error variance, in this case the “within-treatments” variance, which has v2 degrees of freedom. An F value with these degrees of freedom is denoted by FVi v ,a, where a is the upper percentage point at which the test is being made. Usually a = 0.05 (5%) or a = 0.01 (1%). Geometrically, a is the area under the FViv distribution that lies on the upper tail beyond the value Fv ,v,a.

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