# Degrees of Freedom (dF)

Objective:

The term Degrees of Freedom (dF) is used in statistics to calculate the number of measurements that are needed to make an unbiased estimate of a statistic. It is the number of independent pieces of information available to  estimate a statistic.  Also defined as the number of values that may vary in the final calculation of a statistic.

An "unbiased estimate" is when the mean of the sampling distribution of a statistic can be shown to equal the (population) parameter being estimated.

The dF is represented by the lowercase Greek letter nu (v).

dF  = n - x

where n is the sample size or observations and x is number of parameters to be estimated.

An examples of a "parameter to be estimated" is the sample mean when calculating the sample standard deviation as shown below.

A simple way to generalize it is as the number of samples minus the number of calculated (estimated) parameters.

The dF in this case is equal to n-1 since only the sample mean is being estimated. In other words, the sample variance contains n-1 degrees of freedom. There a "n" random values minus the sample mean which is estimated.

dF formulas:

dF = n-1 when using the Paired t-test and 1-Sample t test.

dF = n1 + n2 - 2 when using the 2-Sample t test with assumed equal variances

dF = (# of Rows - 1) * (# of Columns - 1) when using chi-square tests

Notation:

There are a few variations used to denote degrees of freedom:

v = lower case Greek letter nu which is most commonly found in equations

n = however this is usually applied to represent sample size

d.f. or dF

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