Statistics - Field Sampling: Principles and Practices in Environmental Analysis - page 158

6.10. COMPARING AREAS
In some cases one might question if all the samples from a field or all the grids in a field
are the same or different. To answer this question one might think to carry out a t -test
comparing each grid with each other grid. This would be a tedious calculation. Rather
than do this, an analysis of variance (ANOVA) can be carried out. To do this the mean or
for all the areas or grids in the field is calculated. An for the two areas to be compared
is also calculated, * then as with previous calculations a hypothesis is needed. The null
hypothesis H 0 is that all the areas are the same.
The two areas represented by B3 through B6 and C3 through C6 in Figure 6.1 can be
compared. The null hypothesis would be that the two areas are the same; that is,
H 0 = µ B3−B6 = µ C3−c6 . An alternate hypothesis H a might be that the means are different
from each other.
The second step is to calculate what is called the sum of the squares for treatment
(SST). In this case we assume that grids B3 through B6 are one treatment and grids C3
through C6 are one treatment and that they are the same. The for all measurements is 600,
while the for area B3 through B6 is 796 and the for area C3 through C6 is 403. The
averages for these areas have been given as three digits without decimal for simplicity.
The general equation for calculating SST would be
In this equation, X i is the mean of the ith group (i.e., B3 through B6 would be a group),
is the overall mean, and n i is the number of observations in the ith group. (This would be
16 for the B3 through B6 group.) The answer is calculated as follows:
=16(796-600) 2 [phone] =1235600
The next step is to calculate what is called the sum of squares of error (SSE). This is
based on the variability around the sample means (i.e., 796 and 403). The general
equation for this calculation is: †
For area B3 through B6 the averages are 1090, 908, 724, and 463, and for area C3
through C6 they are 871, 586, 106, and 51. The averages for these
* All numbers have been truncated to whole numbers for illustration. Decimals are kept when
working with actual data.
† In this equation j is used in the same way i is used in previous equations.