# What is the right notation for degree of freedom -- v or $\nu$?

How to denote the df (degree of freedom), particularly for $t$, $F$ and $\chi^2$ distributions in hypothesis testing?

Some references state it as the English letter $v$ such as this one, and in Miller and Freund's Probability and Statistics for Engineers, it is denoted as Greek letter $\nu$ (nu). Of course on typesetting, they look almost similar, but for the sake for teaching, I'd like to know. Are they both equally acceptable?

• Any notation is acceptable and I do not think we should loose sleep about it. Oct 15 '15 at 6:24

The most common convention in statistics is to use Greek letters for parameters ($\mu, \sigma$ for normal distributions, $\lambda$ for Poisson, $\beta$ when parameterizing the mean in regression and GLMS, etc). I'll assert this without any attempt to offer evidence.

You can define your notation is almost any convenient way as long as it's clear, but $\nu$, the Greek letter is probably the most traditional/widely used for the $t$ and $\chi^2$ distributions at least.

Where feasible, I think conventional notation is better, since it's likely to agree with more sources, and the Greek-letters-for-parameters is pretty well established.

[In no way should this be construed as me saying that any choice is 'right' or 'wrong'. The reason I mostly advocate following convention is because clearer/less ambiguous communication is facilitated. In a situation where there are larger benefits to choosing some other notation, convention be hanged.]

If you're using a text, I'd suggest that unless there's a good reason to do otherwise, you just use what the text uses. It will save some effort.

Presumably the intent in using $\nu$ is for the same reason we often use $\text{n}$ in our notation when dealing with sample size (presumably to stand for number), but transliterated to Greek since it's a parameter.

I expect $\text{v}$ mostly arises because some people are simply unaware that $\nu$ isn't $\text{v}$. In a few cases it could occur because people want to type $\nu$ but either can't or don't-know-how-to get it, and use $\text{v}$ as a visual approximation.

• I believe that in my answer when I say "you can define your notation is almost any convenient way" and explaining that $\nu$ is the most "common convention" I make it quite clear that we cannot say that it's "not right" to call it $\text{v}$ (as long as it is a $\text{v}$). It's no more or less right than any other defined notation. Convention is simply that -- convention. I spoke about things I thought were better or more convenient in particular circumstances, but made no mention of 'right' or 'wrong' because with a free choice of notation, "wrong" is absurd. I'll amend my answer. Oct 15 '15 at 6:21
• However, if you're asking "is it wrong to call $\nu$ '$\text{v}$'?" I'd say "of course", but its an error that may be worth making in some circumstances (such as when calling it $\nu$ is likely to cause great confusion). Oct 15 '15 at 6:29
• The spirit of this is entirely right, but we should perhaps warn that practice in statistical notation is full of historical hangovers, crazy conventions and inconsistent idiocies. At least as first encountered, the number of degrees of freedom is not a parameter in what I take to be the usual sense, namely an unknown constant in the specification the data generating process, and so something to estimate, but the Greek letter nu is indeed common; I would surmise that at least some uses of v arise when authors don't know any Greek or wish to avoid explaining Greek to students. Oct 15 '15 at 8:21
• $\chi^2$ for a sample statistic is hard to eradicate and indeed many statistical people would argue that $X^2$ for sample statistic (which goes back to 1975 at least) is objectionable on other grounds, namely that $X$ is so often a symbol for covariates, single or multiple. Oct 15 '15 at 8:23
• The terminology of "degrees of freedom" in statistics is carried over from classical mechanics; R.A. Fisher, who had much to do with emphasising degrees of freedom, was educated in the Cambridge tradition with heavy doses of mechanics as a major, if not the major, part of applied mathematics. I don't know what notation for d.f. was standard in mathematical and physical literature in the late 19th and early 20th century that he might have used. Oct 15 '15 at 8:28