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