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I will be grateful for your help in finding the logical meaning of each part of the formula of degrees of freedom, which are computed for a t-test when variances are unknown and are assumed to be unequal.

Please, take a look at the formula, the way I managed to understand some parts of it, and, please, help me to understand the rest of it.

The formula is as follows:

degrees of freedom =^{ [ s12/n1 + s22/n2 ] 2}/ [ ( s_{1}^{2}/n_{1}) / n_{1}+ ( s_{2}^{2}/n_{2}) / n_{2}]

where s_{1}^{2}is the variance of the first sample

s_{2}^{2}is the variance of the second sample

n_{1}- number of observations in the first sample

n_{1}- number of observations in the second sample

Here are parts I managed to decipher:

(1)s_{1}^{2}/n_{1}means that by dividing the variance of the first sample by the number of observations in this sample we get the mean variance of the first sample, that is the mean of all variances. Even if I am right, I don't understand what does the mean variance give us and its meaning.

Same for s_{2}^{2}/n_{2}but for the second sample.

(2)hence the numerator is the squared sum of two mean variances; why do we need to square them, if usually squaring in such contexts is used to avoid negative numbers; with variances negatives are excluded, as those are already squared numbers.

(3)the meaning of expressions in the denominator has skipped from me)

Here again we square mean variances, and then divide each by the number of corresponding observations, and then sum both results. What do all these mean?

Thank you very much.

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# I Degrees of freedom for t-test for 2 samples, 2 variances

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