Is variation the same as variance?

Question

^{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 01 '14 at 07:36
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 01 '14 at 07:42
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 01 '14 at 07:46
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 01 '14 at 08:25
Variation reveals just only the dispersion of the values from their center. Where as variance quantifies the dispersion of the values from their center — , Sep 08 '15 at 18:19

score 23 · Accepted Answer · answered Mar 01 '14 at 08:34

23

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).

answered Mar 01 '14 at 08:34

David Marx

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but it is used interchangeably by Andrew Ng in this video https://www.coursera.org/learn/machine-learning/lecture/S1bq1/choosing-the-number-of-principal-components – Boris Mar 21 '18 at 04:20
Nice. See also "centrality" versus "mean", etc. :) – Alexis May 21 '20 at 15:49

score 14 · Answer 2 · edited May 21 '20 at 14:41

^{@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 variance $(\sigma^2)$ is only one. Levine and Roos ⁽¹⁹⁹⁷⁾ also consider standard-deviation $(\sigma)$ a variation concept, among others.

To demonstrate why the distinction might be important, compare also the coefficient-of-variation $(\frac\sigma\mu)$, 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}

Is variation the same as variance?

2 Answers2