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I have a list of time-series variables which are all in different scales. The variables are not normally distributed and some values are negative. What are some ways to normalize these variables so that I can compare their standard deviations?

I thought about normalizing by the average of each variables. What are some other ways?

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  • $\begingroup$ Could you explain what a comparison of standard deviations of any kind of normalized values would mean? What are you hoping it could tell you? $\endgroup$
    – whuber ♦
    Commented Jul 31, 2015 at 12:48
  • $\begingroup$ Hi Whuber, I wish to find the variable which gives me the lowest standard deviation. $\endgroup$
    – A1122
    Commented Jul 31, 2015 at 13:09
  • $\begingroup$ Then pick the smallest standard deviation! What is the purpose of normalizing? $\endgroup$
    – whuber ♦
    Commented Jul 31, 2015 at 15:03
  • $\begingroup$ the levels for the time series are different. $\endgroup$
    – A1122
    Commented Jul 31, 2015 at 15:06
  • $\begingroup$ Now we're going around in circles--It's impossible to tell what you're actually looking for, apart from the impression you want to measure something about the spread of the data. Let me end this conversation, then, just by pointing out that the answer you have accepted does not address this. By comparing the SD to some other estimate of spread, such as the range, IQR, or MAD, you would actually be getting some information about the shape of the distribution, but not about any actual measure of spread of the data. $\endgroup$
    – whuber ♦
    Commented Jul 31, 2015 at 15:10

2 Answers 2

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You can't normalize by the average when some values can be negative. For example:

-20, -10, 0, 10, 20

now, add 20 and multiply by 10:

0, 100, 200, 300, 400

The first set has mean of 0, so if you normalize by the mean, you get infinity.

One alternative is to normalize by the range or some variation of it, such as the interquartile range. ALso, instead of the standard deviation, you might want to use the MAD (median absolute deviation) if the variables are very non-normal.

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  • $\begingroup$ The comment by @silverfish applies here, too: what exactly would a comparison of SDs mean after carrying out any such normalization? What would be the pros and cons of the three approaches you describe? $\endgroup$
    – whuber ♦
    Commented Jul 31, 2015 at 12:50
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simply use coefficient of variation which is SD/mean.

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    $\begingroup$ Coefficient of variation is useless when some values can be negative. $\endgroup$
    – Peter Flom
    Commented Jul 31, 2015 at 11:49
  • $\begingroup$ But my variables are not very close to normal distribution. Would this bias the final standard deviation in some way? $\endgroup$
    – A1122
    Commented Jul 31, 2015 at 11:54
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    $\begingroup$ This answer is potentially a valid response to the original question, but would be much better if it compared some of the pros and cons of using the coefficient of variation. $\endgroup$
    – Silverfish
    Commented Jul 31, 2015 at 12:27

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