Degrees of Freedom Calculator Two Samples

Instructions: This Degrees of Freedom Calculator will indicate the number of degrees of freedom for two samples of data, with for two given independent samples:

Sample 1 Data (comma or space separated) =
Sample 2 Data (comma or space separated) =
Pop. Variances Assumption:

Degrees of Freedom Calculator for two samples

The concept of of degrees of freedom tends to be misunderstood. There is a relatively clear definition for it: The degrees of freedom are defined as the number of values that can vary freely to be assigned to a statistical distribution.

When there is one sample, the degrees of freedom are simply computed as the sample size minus 1.

How To Compute Degrees of Freedom for Two Samples?

The general definition of degrees of freedom leads to the typical calculation of the total sample size minus the total number of parameters estimated. Often times that will correspond to

\[df = n_1 + n_2 - 2\]

which is the same as adding the degrees of freedom of the first sample (\(n_1 - 1\)) and the degrees of freedom of the first sample (\(n_2 - 1\)), which is \(n_1 -1 + n_2 - 1 = n_1 + n_2 -2\).

Other ways of calculating degrees of freedom for 2 samples

The independent two-sample case has more subtleties, because there are different potential conventions, depending on whether the population variances are assumed to be equal or unequal. Even, there is a "conservative" estimate of the degrees of freedom for this case.

Example of computing degrees of freedom for the two-sample case

Example: How many degrees of freedom are there for the following independent samples, assuming equal population variances:

\(n_1\) = 1, 2, 3, 3, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8

\(n_2\) = 3, 1, 3, 2, 1, 2, 2, 4, 1, 6

Well, first we compute the corresponding sample sizes. In this case, the sample sizes are \(n_1 = 14\) and \(n_2 = 10\). Consequently, assuming equal population variances, the degrees of freedom are:

\[df = n_1 + n_2 - 2 = 14 + 10 - 2 = 22\]

Degrees of Freedom calculator for the t-test

Is this only valid for a two-sample t-test ? The answer is yes. You can compute the degrees of freedom for a two-sample z-test, but for a z-test the number of degrees of freedom is irrelevant, because the sampling distribution of the associated test statistic has the standard normal distribution.

The degrees of freedom take relevance for the case of the t-test, because the sampling distribution of the t-statistic actually depends on the number of degrees of freedom.

Observe that the calculation of degrees of freedom differs for the case of two-independent samples and for the case of paired samples , where the calculation is much easier.

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