# Is variation the same as variance?

This is my first question on Cross Validated here, so please help me out even if it seems trivial :-) First of all, the question might be an outcome of language differences or perhaps me having real deficiencies in statistics. Nevertheless, here it is:

In population statistics, are variation and variance the same terms? If not, what is the difference between the two?

I know that variance is the square of standard deviation. I also know that it is a measure of how sparse the data is, and I know how to compute it.

However, I've been following a Coursera.org course called "Model Thinking", and the lecturer clearly described variance but was constantly calling it variation. That got me confused a bit.

To be fair, he always talked about computing variation of some particular instance in a population.

Could someone make it clear to me if those are interchangeable, or perhaps I'm missing something?

-
Variation, unlike variance, is not the name of some specific quantity (however, Coefficient of variation is). It is a generic term, like variability. It is just amount of variability which can be measured by various quantities (most popular of them being variance). –  ttnphns Mar 1 '14 at 7:36
So basically you are saying that Variance is a real statistical term with a formal model standing behind it, but variation is just a word describing relation between expected & real data? –  ŁukaszBachman Mar 1 '14 at 7:42
Right - I changed that :) –  ŁukaszBachman Mar 1 '14 at 7:46
Variance has a formula. Variation has no one formula, it is a generic term. Both variance and variation can be 1) a statistic describing a sample, 2) a parameter describing a population, 3) a statistic as an estimate of the correstonding parameter –  ttnphns Mar 1 '14 at 7:46
Another analogue here is "spread." There isn't a formal equation for calculating "spread," although it's appropriate to say that "variance" is a measure of "spread." I think in this context "spread" and "variation" are equivalent. –  David Marx Mar 1 '14 at 8:25

Here's a full wikipedia article discussing this topic: http://en.wikipedia.org/wiki/Statistical_dispersion

As described by others in the comments here, the short answer is: no, variation $\ne$ variance. Synonyms for "variation" are spread, dispersion, scatter and variability. It's just a way of talking about the behavior of the data in a general sense as either having a lot of density over a narrow interval (generally near the mean, but not necessarily if the distribution is skewed) or spread out over a wide range. Variance is a particular measure of variability, but others exist (and several are enumerated in the linked article).

-
Thanks, this is what I was looking for –  ŁukaszBachman Mar 1 '14 at 16:55

@ttnphns is right, but since the info wasn't written as an answer, I'm going to attempt to steal the credit! :)

Variation may be understood best as a general term for a class of different concepts, of which $(\sigma^2)$ is only one. Levine and Roos (1997) also consider $(\sigma)$ a variation concept, among others.

To demonstrate why the distinction might be important, compare also the $(\frac\sigma\mu)$, and the mathematical concept, total variation, which has several definitions unto itself. Then there are all manners of qualitative variation, which are mentioned in the Wikipedia article @DavidMarx linked. These pages corroborate his answer BTW; statistical dispersion or variability are better synonyms for variation than variance, which is clearly not so synonymous.

BTW, here's a cool GIF of one kind of total variation: the length of the path on the $y$ axis that the red ball travels. Definitely not the same as variance!

Reference
Levine, J. H., & Roos, T. B. (1997). Description: Numbers for the variation. Introduction to data analysis: The rules of evidence (Volume I:074). Dartmouth College. Retrieved from http://www.dartmouth.edu/~mss/data%20analysis/Volume%20I%20pdf%20/074%20Description%20%20Numb%20for.pdf

-