How should one define the sample variance for scalar input?

I was horrified to find recently that Matlab returns $0$ for the sample variance of a scalar input:

>> var(randn(1),0)   %the '0' here tells var to give sample variance
ans =
0
>> var(randn(1),1)   %the '1' here tells var to give population variance
ans =
0


Somehow, the sample variance is not dividing by $0 = n-1$ in this case. R returns a NaN for a scalar:

> var(rnorm(1,1))
[1] NA


What do you think is a sensible way to define the population sample variance for a scalar? What consequences might there be for returning a zero instead of a NaN?

edit: from the help for Matlab's var:

VAR normalizes Y by N-1 if N>1, where N is the sample size.  This is
an unbiased estimator of the variance of the population from which X is
drawn, as long as X consists of independent, identically distributed
samples. For N=1, Y is normalized by N.

Y = VAR(X,1) normalizes by N and produces the second moment of the
sample about its mean.  VAR(X,0) is the same as VAR(X).


a cryptic comment in the m code for var states:

if w == 0 && n > 1
% The unbiased estimator: divide by (n-1).  Can't do this
% when n == 0 or 1.
denom = n - 1;
else
% The biased estimator: divide by n.
denom = n; % n==0 => return NaNs, n==1 => return zeros
end


i.e. they explicitly choose not to return a NaN even when the user requests a sample variance on a scalar. My question is why they should choose to do this, not how.

edit: I see that I had erroneously asked about how one should define the population variance of a scalar (see strike through line above). This probably caused a lot of confusion.

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Did you mean to ask about the population variance definition or about the sample variance definition? There's no issue with the former. –  whuber Oct 29 '10 at 18:27
@whuber I think what Matlab is doing is the following. When it encounters a scalar it reports the population variance by default. When it encounters a vector it reports the sample variance by default unless you ask it to report population variance. In either context they use same function: var(). –  user28 Oct 29 '10 at 18:39
@Srikant That may be, but it does not address my request for clarification. I suspect there may be a typographical error in the last line. The var() function appears to be returning the correct values of population variance but not the correct values for sample variance. –  whuber Oct 29 '10 at 18:47
@whuber Actually, var(randn(1),0) weights the variance by 0 and hence the output is 0. The second element is used to do a weighted average if it is different from 1. See: Mathworks help for var –  user28 Oct 29 '10 at 18:52
@shabbychef: The page Srikant has referenced clearly documents the behavior you have noted. In particular, it does not maintain that var() returns the 1/(n-1) version of the variance when n=1. So the issue is not that Matlab is wrong, per se. (From a software design point of view this implementation of var() is terrible because it tries to do too much in a single function and consequently risks misunderstanding and erroneous results by unvigilant users.) –  whuber Oct 29 '10 at 19:17

Scalars can't 'have' a population variance although they can be single samples from population that has a (population) variance. If you want to estimate that then you need at least: more than one data point in the sample, another sample from the same distribution, or some prior information about the population variance by way of a model.

btw R has returned missing (NA) not NaN

is.nan(var(rnorm(1,1)))
[1] FALSE

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aha, I have not worked with R enough to spot the NA vs NaN distinction. Nice that it has both... –  shabbychef Oct 29 '10 at 18:22
Of course a single number has a variance! It's the mean square deviation from its mean, namely zero. Perhaps you meant to say a dataset with one number cannot have a sample variance? –  whuber Oct 29 '10 at 18:23
You'll notice that I said it can't have a population variance, not that it cannot have a variance. Clearly it has a sample variance, which is indeed trivially zero. –  conjugateprior Oct 30 '10 at 9:07
I have come to realize that our conversation derives from two different understandings of "population" and "sample" variance. The terms are confused on the Internet, so I think we're both blameless and that we may be in agreement. I was attempting to use the convention adopted by the OP, whose terminology indicates the "sample" variance divides by n-1 while the "population" variance divides by n. Incidentally, your comment seems (correctly) to allow for three variances: that of a random variable, the unbiased estimator thereof, and that of a set of independent realizations of it. –  whuber Oct 30 '10 at 20:55
@whuber that sounds like an awesome discussion to have. I will try to think of a question for it :) –  naught101 Apr 30 '12 at 1:52

I am sure people in this forum will have better answers, here is what I think:

I think R's answer is logical. The random variable has a population variance, but it turns out that with 1 sample you don't have enough degrees of freedom to estimate sample variance i-e- you are trying to extract information that is NOT there.

Regarding Matlab's answer, I don't know how to justify 0, except that it is from the numerator.

Consequences can be bizarre. But I can think of anything else related to the estimation.

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Actually, a single observation of a random variable often provides information about that variable's variance. It depends on the possibilities you allow. For instance, a single observation from a Uniform, Poisson, or Exponential distribution lets you estimate its single parameter, which gives you an estimate of the variance. –  whuber Oct 29 '10 at 18:34
@whuber thanks for the pointers. Actually, I was thinking more on the lines of sample variance. What I meant to say was, estimating sample variance may not be appropriate with one sample. It stands corrected now. –  suncoolsu Oct 30 '10 at 22:30
@whuber But this is not the case here; this function is documented to return variance/(n-1) and is returning 0 for 0/0 symbol. –  mbq Oct 31 '10 at 19:51
@mbq You missed the disclaimer in the documentation: "For N=1, V is normalized by N." That's an elliptical way of saying "For N=1, a value of 0 is returned." The point I was making with my comment actually is about something else: it addresses @suncoolsu's assertion that "you are trying to extract information that is NOT there" in the case N=1. On the contrary, a single observation from a random variable indeed does provide information about its variance in many practical applications. (However, I am not saying that the variance is best estimated by some kind of variance formula!) –  whuber Oct 31 '10 at 20:23
@whuber Fair enough; yet this does not justify this behavior. –  mbq Oct 31 '10 at 20:42
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I think Matlab is using the following logic for a scalar (analogous to how we define population variance) to avoid having to deal with NA and NAN.

$Var(x) = \frac{(x - \bar{x})^2}{1} = 0$

The above follows as for a scalar: $\bar{x} = x$.

Their definition is probably a programming convention that may perhaps make some aspect of coding easier.

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You have restated the problem without offering a resolution. The concern is that this convention can lead to wrong answers when the variance estimate is used later. It certainly delays the stage at which a failure is recognized. (Think about what happens if you try to use this "variance" in a t-test, for instance.) –  whuber Oct 29 '10 at 18:50
@whuber Not exactly. The OP is thinking that var` returns the sample variance for a scalar whereas it returns the 'population variance'. Whether it is a problem or not depends on what type of error handling routines they have internally. However, I agree that their convention is susceptible to errors. –  user28 Oct 29 '10 at 18:57
@Srikant In order to determine who is more confused, you or I, I have learned that we both are! Wikipedia terms the formula with 1/(n-1) the "sample variance" (at en.wikipedia.org/wiki/Variance ) whereas Mathworld clearly uses 1/n at mathworld.wolfram.com/SampleVariance.html . However, the OP uses "sample variance" in the first sense when remarking that "the sample variance is not dividing by 0=n−1". In this sense there is no such thing as "sample variance" for a dataset of one value (a "scalar"), so your characterization of what the OP is "thinking" looks incorrect. –  whuber Oct 29 '10 at 19:10
+1 As I know MATLAB, I think your hypothesis is (sadly) true. –  mbq Oct 31 '10 at 19:53