Statistics - Field Sampling: Principles and Practices in Environmental Analysis - page 152

the calculation carried out. Finally, all the numbers so obtained are added together. This is
summarized in the following equation:

This calculates the population variance (δ 2 ) . In mathematical terms what has been done in
the numerator is shown by the term

This term says that the difference

between the mean and the value is
calculated then squared and the numbers added together to give the answer (numerator).
The final step is to divide this sum by n . This type of calculation is programmed into most
handheld calculators and computers. It is such a common term that it is part of the
standard keypad of many handheld calculators and the tool bar of Excel.
In calculating the sample variance n −1 (or in this example, 3) would be used as the
divisor. The number n −1 is sometimes called the degrees of freedom. The reason for
using this number rather than n will not be covered here, but can be found in all standard
statistics topics, such as those given at the end of this chapter. When the sum of the
deviations is divided by n −1 the number obtained is frequently represented by the symbol
s 2 , which is called the sample variance. The sample variance is used as an estimator of the
total population variance. The mathematical formula for carrying out the sample variance
calculation is

A shortcut or direct way to carry out this calculation would be to use the following
equation:

Although this equation looks complicated and is, using a computer it can easily be
programmed into the spreadsheet and used to make calculations. In most cases the
spreadsheet will already have the capability to calculate the standard deviation, and so
entering the equation is not necessary.
To calculate the standard deviation, represented by s, the square root of the variance
(i.e., ) is determined. Thus

At this point we have the standard deviation, which is 15.87. This is also expressed as and
has the units ppm, just as the original data do. This is interpreted to mean that the actual