How we change what others think, feel, believe and do

Degrees of Freedom

Explanations > Social Research > Statistical principles > Degrees of Freedom

The team choice analogy | Sample average | Estimating means | Multiple samples | See also

'Degrees of freedom' is a term that can be rather confusing. In fact it is, but there are several ways of explaining it that help to make sense of it.

The team choice analogy

A simple (though not completely accurate) way of thinking about degrees of freedom is to imagine you are picking people to play in a team. You have eleven positions to fill and eleven people to put into those positions. How many decisions do you have? In fact you have ten, because when you come to the eleventh person, there is only one person and one position, so you have no choice. You thus have ten 'degrees of freedom' as it is called.

Likewise, when you have a sample, the degrees of freedom to allocate people in the sample to tests is one less than the sample size. So if there are N people in a sample, the degrees of freedom is N-1.

Sample average

When you are calculating an average of a sample, you want the sample to have the same average as the population.

For example, if the average score for an entire population on a test is 3 and in a group of five people, four of them score 1, 2, 3 and 5, then for the sample average to be the same as the population average, then the last one must be scored as 4.

Estimating mean

There may be N observations in an experiment, but one parameter that needs to be estimated. That leaves N-1 degrees of freedom for estimating variability.

Multiple samples

Where there are multiple samples, then the degrees of freedom for each are N₁-1, N₂-1, etc. When the samples are combined, the total degrees of freedom is (N₁ + N₂ + ...) - Y, where Y is the number of samples. Thus combining two groups gives DF = N₁ + N₂ - 2.

As an example, if you have a table with a set of rows and columns, as below, and where you know the total of the rows and columns, then when you know the yellow squares, the blue squares can be calculated. You thus only have (R -)*(C - 1) choices (or degrees of freedom) in allocating numbers to cells.