Statistical Analysis for Education and Psychology Researchers

ideas just described because these formulas are easy to use with a pocket calculator.
Whichever computational method is used the results will be the same.

Sample Variance and Standard Deviation:

The variance, S^2 of n observations x 1 , x 2 ,..., xn is

Sample

Variance-

3.1

The standard deviation, S, is the square root of the variance.

Sample

Standard

Deviation

• 3.2

Degrees of Freedom

You may wonder why the denominator (bottom number) is n−1 in the formulae for both
the variance and the standard deviation. The standard deviation represents the average
deviation of each value form the mean. When we compute an average we divide by n, the
sum of all the values. For the standard deviation we divide by n−1 because the sum of the
deviations, (mean−xi) always equals 0. If you think about this, the last deviation can be
found when we know the first n−1 deviations because all deviations sum to 0. Only n− 1
of the squared deviations can vary freely (the last one is fixed) and we therefore take an
average by dividing the total by n−1. We call the number n−1 the degrees of freedom
(df) of a statistic.
Non-statisticians generally find degrees of freedom a difficult concept to understand
and given its importance in design and statistical analysis it is worth explaining this idea
in more detail. The meaning is best illustrated by considering further examples. First, it is
important to realize that every statistic has a certain number of degrees of freedom

associated with it. For example the mean, has n degrees of freedom. If we look at the
formula for the mean:

Sample

Mean 3.3

we can consider which components in this formula can vary at all, and which cannot.

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