Basics of Environmental Sampling and Analysis - Fundamentals of Environmental Sampling and Analysis - page 24

independent estimates of two population variances.

s 1 ;

Population 1 ¼ Nðm 1 ; s 1 Þ!Sample 1 : n 1

s 2 Þ!Sample 2 : s 2 ; n 2
The F-value is defined as the ratio of the dispersion of two distributions:

Population 2 ¼ Nðm 2 ;

F ¼ s 1 = s 2 ð2 : 22Þ

where s 1 > s 2 , otherwise switch s 1 and s 2 . A comparison of F-value with tabulated

F critical value ðAppendix C3Þ will show whether the difference in variances is
significant or not. The difference is significant when calculated F-value is greater
than F critical value . F distribution is the basis for the analysis of variance (ANOVA)
and is useful, for example, (1) to compare one method of analysis to another
method, (2) to compare the performance of one lab to that of another lab, and (3) to
compare contaminated levels at two locations (are they statistically significantly
different?). An alternative of F-test is to use Excel with One-Way ANOVA as
described below:

Select ToolsjData Analysis
Select ANOVA: Single Factor from Analysis Tool list box in the Data
Analysis dialog box. Click the OK button
In the ANOVA: Single Factor dialog box. Enter the Input Range edit box
Select the Grouped by Column option button and select the Labels in First
Row check box
Select Output Range and enter 0.05 in the Alpha edit box
Click the OK button

EXAMPLE 2.4. A groundwater sample was colleted from a benzene-contaminated site.
This sample was split into two subsamples, which were then sent to two labs for independent
analysis. Lab A results were as follows: 1.13, 1.14, 1.17, 1.19, 1.29 mg/L (s 2 ¼ 0:0639 mg/L,
n 2 ¼ 5); LabB resultswere as follows: 1.11, 1.12, 1.14, 1.19, 1.25, 1.34 mg/L (s 1 ¼ 0:0893 mg=L;
n 1 ¼ 6). Are these results significantly different between two labs at a 5% level?

SOLUTION:

2 2

(a) Calculate F-value: F ¼ s 1 =s 2 ¼ð0:0893Þ ¼ 1:95

(b) df 1 ¼ n 1 1 ¼ 5 and df 2 ¼ n 2 1 ¼ 4, use Appendix C3, F 5,4,0.05 ¼6.26
(c) Because 1.95 is less than 6.26, the difference in the test results between the two labs
is not significantly different.

=ð0:0639Þ

For this example, one should also try the alternative method with one-way ANOVA
using Excel. This method is more conservative because it differentiates the variance
between the test groups (labs) and within the group (Details are omitted here, but
the answer is: F ¼ 0 : 0257 < F critical value ¼ 5 : 117, or p-value¼0.876 > 0.05, not

significant).