Degrees of freedom in t-test

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What are degrees of freedom?

Degrees of freedom are the maximum numbers of logically independent values that have a freedom of varying in a sample dataset. Degrees of freedom are taken into consideration while studying various hypotheses testing methods in statistics like a t-test method. 

It is important to calculate degrees of freedom to justify your results if it either rejects or accepts a null hypothesis. Let us understand degrees of freedom concept through an example:

Say we have 5 integers determining the scores of students in maths out of 10. 4 of the numbers are as follows – 3,8,5,4. The average of these numbers is 6. This leaves us with the 5th number being 10 since it does not have freedom to vary. But why is it that there are five numbers in the sample but only one of them has no freedom to vary?

We can say it is because of the degrees of freedom formula:

Df​=N−1

Where, 

Df is degrees of freedom

N is sample size 

Now, the definition of degrees of freedom is the number of values that are free to vary. So when you are asked to choose three numbers whose average mean is 10 like: {9,10,11} {8,10,12} {7,10,13}

When you choose first two numbers, you are left with no choice but to choose the third number as 10. So once you choose 9+10 or 8+12, you have no choice but to select a number that will help you get an average of 10. 

Here the degrees of freedom for three numbers is 2.

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Degrees of freedom in t-test

When you are asked to select 10 values at random, they all have the freedom to vary. But this won’t be the case when you are going to estimate the means of population. Here each population sample will be of 10 and here your constraint will be the mean estimation. And this estimation brings limitations on the freedom of vary by the statement – “The sum of all values in the data must equal n * mean, where is the number of values in the data set.”

So for our sample size having 10 values, the sum of values of 10 values should be equal to mean * 10. So if the mean of 10 values is 3.5, according to the constraints, the sum of all values in the 10 values should be 3.5 * 10 = 35.

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So in the above example, the first 9 numbers have the freedom to vary. Any of the below examples can be a fit for the example:

34, -8.3, -37, -92, -1, 0, 1, -22, 99
0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9

But for the mean of the sum of all 10 numbers to be 35 and the mean of all 10 numbers to be 3.5, the last and 10th do not have the freedom to vary so it must be selected in a way that it meets the required constraints:

34, -8.3, -37, -92, -1, 0, 1, -22, 99  —–> 10TH value must be 61.3
0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 —-> 10TH value must be 30.5

Therefore, you have 10 – 1 = 9 degrees of freedom.

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