Statistics Review - Mathematics and Statistics for the Environment - page 138

7.16.1.3 Sums of Squares
There is a sum of squares associated with every source of variation. These SS are easily calculated
as follows. First we need what is known as a “correction term,” or CT . This is simply

2

n

∑

X
n

n

2

313
25

∑

CT = = = 3918 7.,where isthe sumof tems n

Then the total sum of squares is

n

( ) −=

∑

2 2 2 2

Total SS = X T − =+++ 15 14 … 11 CT 64 .24

24 df

The sum of squares attributable to treatments is

n

∑ (

2

treatment totals )

plotsper treatment

Treatment S df = − CT

No.of

4

2 2 2

= +++ −=

67 72 … 59 19 767

5

,

CT −= CT 34 64 .

5

Note that in both SS calculations the number of items squared and added was one more than the
number of degrees of freedom associated with the sum of squares. The number of degrees of free-
dom just below the SS and the numbers of items to be squared and added over the n value provided
a partial check as to whether the proper totals are being used in the calculation—the degrees of
freedom must be one less than the number of items.
Note also that the divisor in the treatment SS calculation is equal to the number of individual
items that go to make up each of the totals being squared in the numerator. This was also true in the
calculation of total SS , but there the divisor was 1 and did not have to be shown. Note further that
the divisor times the number over the summation sign (5 × 5 = 25 for treatments) must always be
equal to the total number of observations in the test—another check.
The sum of squares for error is obtained by subtracting the treatment SS from total SS . A good
habit to get into when obtaining sums of squares by subtraction is to perform the same subtraction
using degrees of freedom. In the more complex designs, doing this provides a partial check on
whether the right items are being used.

7.16.1.4 Mean Squares
The mean squares are now calculated by dividing the sums of squares by the associated degrees of
freedom. It is not necessary to calculate the mean squares for the total. The items that have been cal-
culated are entered directly into the analysis table, which at the present stage would look like this:

Source of Variation Degrees of Freedom Sums of Squares Mean Squares

Treatments 4 34.64 8.66

Error 20 29.60 1.48

Total 24 64.25

An F test of treatments (used to reject the null hypothesis) is now made by dividing the mean
square for treatments by the mean square for error. In this case,