The variance of a t-distribution is given by df/(df-2), hence the t-distribution with 1 and 2 degrees of freedom have no defined variance. Yet these distributions do exist and can be drawn, so one would say that their variances can be calculated. I hope somebody can explain this without using too much mathematics, as my skills in mathematical statistics are poor.

  • $\begingroup$ Have you seen Douglas Zare's nice (physical) explanation at stats.stackexchange.com/questions/36027/…? The way you ask your question also suggests you might be conflating two meanings of "variance": one is the variance of a hypothetical data-generation distribution while the other is the variance of a dataset. The former can be infinite while (obviously) the latter is always finite. $\endgroup$ – whuber Jan 11 '17 at 23:04
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    $\begingroup$ @Michael The Cauchy distribution is the Student t with 1 df. $\endgroup$ – whuber Jan 12 '17 at 0:01
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    $\begingroup$ When you "draw the distribution" are you drawing all of it? Or only the middle 99.something percent of it? It's not that part that makes the variance not-finite. The finiteness or otherwise of the variance is essentially to do with the way the tail behaves in the limit as the variable approaches $\infty$ and $-\infty$ $\endgroup$ – Glen_b Jan 12 '17 at 0:41
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    $\begingroup$ Related: stats.stackexchange.com/questions/94402/… $\endgroup$ – kjetil b halvorsen Aug 14 '17 at 21:00
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    $\begingroup$ I'm not convinced this is an exact duplicate because I think it would be helpful to see an answer that covers the case with two degrees of freedom. Clearly one degree of freedom is already covered. $\endgroup$ – Silverfish Aug 15 '17 at 10:20