## 15. Degree of freedom

The concept of degrees of freedom is central to the principle of estimating statistics of populations from samples of them. "Degrees of freedom" is commonly abbreviated to df. df is a mathematical restriction that needs to be put in place when estimating one statistic from an estimate of another.

Ex 1: Imagine you have four numbers (a, b, c and d) that must add up to a total of m; you are free to choose the first three numbers at random, but the fourth must be chosen so that it makes the total equal to m - thus your degree of freedom is three.

Ex 2: Take data that has been drawn at random from a normal distribution. In order to estimate standard deviation (sigma), we must first estimate mean (mu). Thus, mu is replaced by x-bar in the formula for sigma i.e., we work with the deviations from mu estimated by the deviations from x-bar. At this point, we need to apply the restriction that the deviations must sum to zero. Thus, degrees of freedom are n-1 in the equation for s below:

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Sources:
http://www.statsdirect.com/help/content/basics/degrees_freedom.htm