# Understanding “variance” intuitively

What is the cleanest, easiest way to explain someone the concept of variance? What does it intuitively mean? If one is to explain this to their mom or child how would one go about it?

It's a concept that I have difficulty in articulating - especially when relating variance to risk. I understand it mathematically and can explain it that way too. But when explaining real world phenomena how do you make one understand variance and it's applicability in the 'real world', so to speak.

Let's say we are simulating an investment in a stock using random numbers (rolling a die or using an excel sheet, doesn't matter). We get some 'return on investment' by associating each instance of the random variable to 'some change' in the return. Eg.:

Rolling a 1 implies a change of 0.8 per \$1 in investment, a 5 a change of 1.1 per \$1 and so on.

Now if this simulation is run for about 50 times (or 20 or 100) we will get some values and the final value of the investment. So what does 'variance' actually tell us if we were to calculate it from the above data set? What does one "see" - If the variance turns out to be 1.7654 or 0.88765 or 5.2342 what does this even mean? What did/can I observe about this investment?? What conclusions can I draw - in lay man terms.

Please feel free to augment the question with that for standard deviation too! Although I feel it's 'easier' to understand, but something that would contribute to making it also 'intuitively' clear would be greatly appreciated!

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Shouldn't we merge this question with the same one asked last year? – whuber Oct 26 '11 at 15:10
@whuber I think these should be merged. Having several time the same question (even if here the context is different) reduces the average quality of answers. – robin girard Oct 26 '11 at 15:19
I'm okay with it being merged but I know how to calculate variance and it's use in statistics too. I want to be able to articulate this concept to people who wouldn't know anything about it and it takes a long while to do so and hence the question. The intent is rather different from the question on SD, IMHO – PhD Oct 26 '11 at 19:10
I don't think any of you are doing a very good job of answering this in a way that a Layman would understand. I see a lot of assumptions being made and almost every answer ends with something that needs to be interpreted. I'm not complaining, just trying to point that out. I too can't answer the question simply. Maybe it's too difficult? – user98013 Dec 12 '15 at 18:16

I would probably use a similar analogy to the one I've learned to give 'laypeople' when introducing the concept of bias and variance: the dartboard analogy. See below:

The particular image above is from Encyclopedia of Machine Learning, and the reference within the image is Moore and McCabe's "Introduction to the Practice of Statistics".

EDIT:

Here's an exercise that I believe is pretty intuitive: Take a deck of cards (out of the box), and drop the deck from a height of about 1 foot. Ask your mom/child to pick up the cards and return them to you. Then, instead of dropping the deck, toss it as high as you can and let the cards fall to the ground. Ask your mom/child to pick up the cards and return them to you.

The relative fun they have during the two trials should give them an intuitive feel for variance :)

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So what does it 'mean'? If someone would see the the statistical variance of the darts on the board, what would they conclude? What does it mean to have low/high variance intuitively speaking... – PhD Oct 26 '11 at 3:36
I'd say something like: Let's say we threw 4 darts. The number of hands required to remove the darts from the board all at once increases as variance of the dart positions increases (Note: very informal argument here as there a number of counterexamples, such as when 3 darts are grouped together and the last dart is on the wall 3 feet from the darboard). – Mike Wierzbicki Oct 26 '11 at 21:05
@Nupul I didn't care for my comment above, so I added a related example to my answer. See my edit. – Mike Wierzbicki Oct 26 '11 at 21:42
AAAAAAAAAAAH! Nice exercise! Good way to show someone what it means to have low/high variance! The average distance from the average value (mean) of the data points :) – PhD Oct 26 '11 at 21:56
(+1) The dartboard-analog to demonstrate the difference between bias and variance is simply brilliant – steffen Dec 1 '11 at 15:16

I used to teach statistics to a layman by jokes, and I found they learn a lot.

Suppose for variance or standard deviation the following joke is quite useful:

Joke

Once two statistician of height 4 feet and 5 feet have to cross a river of AVERAGE depth 3 feet. Meanwhile, a third statistician comes and said, "what are you waiting for? You can easily cross the river"

I am assuming that layman know about 'average' term. You can also ask them the same question that would they cross the river in this situation?

What are they missing that is 'variance' to decide "what to do in the situation?"

It's all about your presentation skills. However, jokes help a lot to the layman who wants to understand statistics. I hope it helps!

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Maybe I'm not good with statistical jokes (I'm am quite good with the others though :). But I don't think I understand what is meant by "what to do in the situation"? What 'exactly' should one do if they have an idea of the variance? How should one interpret it? – PhD Oct 26 '11 at 3:38
@Nupul: Actually, "what to do in the situation" means either they cross a river or not? If you know the variance (or SD) then you could decide it easily. Suppose variance is 0.25 (SD=0.5) then they can cross the river safely because range of interval (dont confuse this with confidence Interval (CI)) is 3+0.5 or 3-0.5, and their heights are 4 and 5. If variance is 4 then better to not cross the river. By the way, just enjoy jokes here stats.stackexchange.com/questions/1337/statistics-jokes – Biostat Oct 26 '11 at 7:43
Perfect! I got it! :) That makes a lot of sense. In fact combining the answers from various people helps me frame the understanding better... – PhD Oct 26 '11 at 21:44
Or, if sharks don't 'on average' eat people, that's little comfort if they are very moody (highly variant behavior). In the river analogy it's about whether you will take a step that will put you over your head. – Dean Radcliffe Nov 26 '12 at 6:28

I would focus on the standard deviation rather than the variance; the variance is on the wrong scale.

Just as the average is a typical value, the SD is a typical (absolute) difference from the average. It's not unlike folding the distribution over at the average and taking the average of that.

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Agreed. Let's say we focus on SD. My question still stands as to how to make someone understand SD intuitively other than 'high SD doesn't seem good'...how would I explain SD to a lay person since it's the square root of variance!!! – PhD Oct 26 '11 at 3:39
@Nupul - Read my second paragraph: I would explain the SD as the typical difference from the average. – Karl Oct 26 '11 at 15:39
"It's not unlike folding the distribution over at the average and taking the average of that." That comment, like the rest of your post, seems to describe mean absolute deviation, not standard deviation. – Macro May 15 '12 at 18:59
@Macro - yes; in trying to explain the SD, I would approximate it by the MAD. I think it's best not to quibble over root-mean-square vs mean absolute value. – Karl May 19 '12 at 6:53

Maybe this could help. I apologise in advance that as complete amateur I may get this wrong.

Imagine you ask 1000 people to correctly guess how many beans are in a jar filled with jelly beans. Now imagine that you are not necessarily interested in knowing the correct answer (which may be of some use) but you wish to get a better understanding of how people estimate the answer.

Variance could be explained to a lay person as the spread of different answers (from highest to lowest). You could continue by adding that if enough people were to questioned the correct answer should lie somewhere in the middle of the spread of 'guestimates' given.

I now refer to some of my more esteemed colleagues for adjudication

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I was sitting down trying to puzzle out variance and the thing that finally made it click into place for me was to look at it graphically.

Say you draw out a number line with four points, -7, -1, 1 and 7. Now draw an imaginary Y axis with the same four points along the Y dimension, and use the XY pairs to draw out the square for each pair of points. You wind up with four separate squares consisting of 49, 1, 1, and 49 smaller squares, each. Each of them contributes to an overall sum of squares which, itself, can be represented as a large 10 x 10 square with 100 smaller squares overall.

Variance is the size of the average square contributing to that larger square. 49 + 1 + 49 + 1 = 100, 100/4 = 25. So 25 would be the variance. The standard deviation would be the length of one of the sides of that average square, or 5.

Obviously this analogy does not cover the full nuance of the concept of variance. There are a lot of things that need explained, such as why we often use a denominator of n-1 to estimate the population parameter, instead of simply using n. But as a basic concept to peg the rest of a detailed understanding of variance to, simply drawing it out so I could see it helped immensely. It helps understand what we mean when we say that variance is the average squared deviation from the mean. It also helps in understanding just what relationship SD has to that average.

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Welcome to Cross-Validated! I like the approach, but it might be even more helpful to emphasize that the points are spread 'around' zero (i.e., they have zero mean) and you're measuring the spread relative to an "atom" located there. (+1) and I look forward to seeing more answers from you! – Matt Krause Apr 25 at 2:20

I disagree with a lot of the answers advocating people to purely think of variance as spread. As smart people (Nassim Taleb) have pointed out, when people think of variance as spread they just assume it is MAD. A serious number of statistically fluent peers "fudge" it. It's needless.

Variance is a description of how far members from the mean, AND it judges each observation's importance in deciding this by how far away it is. This means observations far away are judged more importantly. Hence squares.

I think the variance of a continuous uniform variable is almost easier to picture than any other. Each observation can have a square drawn to it. Stacking these squares creates a pyramid. Cut the pyramid in half so half the weight is in one side and half is in the other. The face where you cut it is the variance.

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I don't know why this answer wasn't upvoted more. The point made in the second paragraph is crucial to understanding variance and differentiating it from MAD, which as correctly pointed out is what people intuitively think of when told about "measure of spread". And it's not beyond a layman to understand the idea that the weight given to a point's distance from the mean doesn't grow linearly, even if they don't understand squares mathematically. – jeremy radcliff Jul 6 at 23:05

I think the key phrase to use when explaining both variance and standard deviation is "measure of spread". In the most basic language, the variance and standard deviation tell us how well spread out the data is. To be a little more accurate, although still addressing the layman, they tell us how well the data is spread out around the mean. In passing, note that the mean is a "measure of location". To conclude the explanation to the layman, it ought to be highlighted that the standard deviation is expressed in the same units as the data we're working with and that it is for this reason that we take the square root of the variance. i.e. the two are linked.

I think that brief explanation would do the trick. It's probably somewhat similar to an introductory textbook explanation anyway.

Actually, I am mistaken. It would also be necessary to make some gestures with the hands while talking to the layman whilst making the explanation! The style of communication is probably just as important as the words being used to explain the concepts on this occasion. Some professors / tutors have yet to crack this method! :)

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