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I know this seems like a simple questions but I am trying to wrap my head around calculating variance for percentages (e.g. 15%, 16%).

I know that 15% is equivalent to 0.15, but when I try to calculate the variance using “15” versus “0.15” format, my variance is in the order of 100x different. I wanna say this is due to the fact that the variance is squaring the difference between the mean and each sample, but what is the correct way of calculating variance for percentages?

For example let’s say my dataset is the following: 16.34%, 16.11%, 16.02%, 15.32%, 18.13%, 15.58%, 18.17%, 19.01%, 17.03%, 18.79%, 17.97%, 18.36%

I get about 1.58% if I use “16.34” as the format but 0.0158% if I use “0.1634”. Or do I need to do additions conversions?

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    $\begingroup$ you should use the actual number so that you don't have to worry about things being percentages. think of them as just numbers and take out the % by moving the decimal over. So 16 percent becomes 0.16. you can always convert back later if you want. $\endgroup$
    – mlofton
    Commented Jan 26, 2020 at 5:49
  • $\begingroup$ Hi. So if I actually use the number then it comes out to be 0.0158%, which seems like too low of a variance. Since the percentage is derived from two separate values (one divided by the other), could I calculate the variance separately and then divide by one and the other? In order words, can I divide one variance by another? $\endgroup$
    – StealthEXE
    Commented Jan 27, 2020 at 0:19
  • $\begingroup$ StealthEXE: This is a late response because I only see this comment now. Basically, you should always convert percents into real numbers and then do the variance calculations on those. Don't worry about the large differences. It's caused by the variance being a squared result. It's not important ( atleast based on your question ) why it's so different. Note that Carol's solution using basis points which is the real number times 10,000. If you go that route, that's fine also but then you need to remember what a basis point represents as a real number. $\endgroup$
    – mlofton
    Commented Apr 3 at 5:29
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    $\begingroup$ The fact that these are percentages is not important to answer this particular question. If the numbers above were lengths you would get the same apparent discrepancy if you measured length in centimetres or meters. $\endgroup$
    – dariober
    Commented Apr 3 at 7:51

2 Answers 2

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Variance between percentages should be shown as basis points. For example when the interest rate changes from 4.0% to 4.25% we say that the interest rate went up 25 basis points, "Basis points, otherwise known as bps or "bips," are a unit of measure used in finance to describe the percentage change in the value or rate of a financial instrument. One basis point is equivalent to 0.01% (1/100th of a percent) or 0.0001 in decimal form."

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  • $\begingroup$ This may be relevant to analysis of the particular variable you're mentioning. I see no reason whatsoever why it applies generally. For example, many percentages don't have an inherent resolution (smallest possible step) of 0.01%. $\endgroup$
    – Nick Cox
    Commented May 8 at 9:32
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As @dariober points out, this is just a question of units of measurement, but it is easy to get confused.

A proportion -- a fraction scaled between 0 and 1 -- such as 0.1234 is also a percentage -- scaled or rather rewritten to be between 0% and 100% -- such as 12.34%.

Here the understanding should be that % is notation for /100.

You also need to remember that the units of variance are always the square of the units of the original variable. Usually we just ignore that, as the units are typically unusual, puzzling or worse. That is, if you care about units, you work with the standard deviation, which is on the same scale as the variable you start with.

An oddity here is that no statistical software I've heard of recognises % as a display format. You have to multiply by 100 to see results for percentages, even though a percentage and a proportion are sisters under the skin. (I am told that MS Excel supports this.)

But once you work in proportions there is no longer any question of working with 100 as a denominator to be applied again: it has already been applied. As said, depending on why you want this, SD may make more sense substantively than the variance. The variance may look suspiciously small, but that is not wrong, as it is just a side-effect of working with numbers between 0 and 1.

That is, with this example (I can't reproduce the OP's results)

16.34%, 16.11%, 16.02%, 15.32%, 18.13%, 15.58%, 18.17%, 19.01%, 17.03%, 18.79%, 17.97%, 18.36%

Variance is 1.731 (units %$^2$ or per 10000) and SD is 1.316 (units %). (I calculated first and rounded to 3 d.p. afterwards.)

The same numbers as proportions:

Variance is .0001731 = 1.731 $\times 10^{-4}$ and SD is 0.1316 = 1.316 $\times 10^{-2}$ (I calculated first and show 4 s.f. afterwards.)

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