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The goal is to find a term to describe a distribution that fits with the other "lofty"-sounding words like:

  • Skewness: mode!=mean / symmetry
  • Kurtosis: How "fat" the tail is
  • ???: how much spread is contained within one standard deviation

I guess it's just variance but if "variance" is the most "lofty" sounding word I will be vary disappointed. Ideally I want something that sounds more illustrious.

Question

Are there more technical/uber-statistical terms that could be used to refer to whether 1 standard deviation has "a lot" or "a little" spread?

Thanks

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    $\begingroup$ What do you mean by [has "a lot" or "a little"] spread? $\endgroup$ Commented Nov 2, 2022 at 8:03
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    $\begingroup$ In your example in the comment, the range of values is literally determined by the [size of] the standard deviation, is it not? So then are you looking for a precise synonym of standard deviation? $\endgroup$ Commented Nov 2, 2022 at 8:59
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    $\begingroup$ It's easy to get asymmetric distributions for which mode and mean are identical. Here is one 0,0,1,1,1,1,3, Many binomial distributions are such. Consider 5 trials with probability 0.2. $\endgroup$
    – Nick Cox
    Commented Nov 2, 2022 at 10:57
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    $\begingroup$ I really don't want to play this game. It's the numbers that count here, not extra wording introduced to impress (or, by accident, to obscure). $\endgroup$
    – Nick Cox
    Commented Nov 2, 2022 at 10:59
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    $\begingroup$ I do not understand 'spread'. But maybe you meant 'how much probability mass is contained within one standard deviation' $$P( \mu_X - \sigma_X \leq X \leq \mu_X + \sigma_X)$$ $\endgroup$ Commented Nov 2, 2022 at 11:27

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Some intuitive re-expression of the problem:

You could regard the quantile function of the standardized squared difference $\chi = \left(\frac{|X-\mu|}{\sigma}\right)^2$ from the mean

Let

$$F(\chi) = P\left(\left(\frac{|X-\mu|}{\sigma}\right)^2 < \chi \right)$$

Then the quantile function that we speak about is the inverse

$$Q(p) = \lbrace \chi: F(\chi) = p \rbrace$$

This needs to be a monotonically increasing function that integrates to 1.

An example of this function for the normal distribution is:

example

Taken from here: https://math.stackexchange.com/a/3781761/466748

From this view the kurtosis is equal to

$$ kurtosis = \int_0^1 Q(p)^2 dp$$

And your concept is the point $p$ where $Q(p) = 1$ or differently

$$ Z = \int_0^1 \mathbb{1}_{Q(p)\leq 1} dp$$

Where $\mathbb{1}$ is the indicator function.

Your measure is computing how often standardized values are close to 0, or spread out and away from 1, In some way similar as kurtosis, but kurtosis is assigning more weight to extreme values.

Your measure is similar to kurtosis (from Greek for bulging). But to come up with a term might be difficult since many different shapes can correspond to high/low values of your statistic. Like kurtosis it has a similar relationships with peakedness, but it is also not exactly the same as peakedness and only correlates with it.

Maybe you shouldn't try to condense this in a particular name and you could describe it with a few more words. Because of the binary nature, how you count values below and above $1\sigma$ as either 0 or 1, you might call this measure 'the degree of probability division around $1\sigma$' or (my favorite) 'bulge/tail ratio', a measure for how much probability mass is in the tails and how much in the bulge. Firebug's suggestion in the comments is also nice 'probability concentration'.

Distributions that have a high $Z$ will have most of the distribution being determined by a central part with the tails possibly having large influence on the kurtosis or other values, but in terms of amount of probability the tails will be small.

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  • $\begingroup$ Kurtosis does not measure how often standardized values are close to zero, nor is it correlated with peakedness. $\endgroup$ Commented Nov 11, 2022 at 1:56
  • $\begingroup$ @BigBendRegion I see it as relating to peakedness and values close to zero, as more values close to zero, and more peakedness allow for more values away from zero. (the bigger the green area in the image, the bigger the red area) $\endgroup$ Commented Nov 11, 2022 at 6:54
  • $\begingroup$ But you are stating things that simply are not mathematically true. Try proving something. $\endgroup$ Commented Nov 11, 2022 at 12:18
  • $\begingroup$ To elaborate, kurtosis can increase to infinity while the probability within a sd of the mean stays constant at .5; and kurtosis can decrease to 1 while the same probability increases to 1. See math.stackexchange.com/a/2523606/472987 . So any statements about relation of mean concentration and/or peakedness to kurtosis are only anecdotal, and not general. $\endgroup$ Commented Nov 11, 2022 at 16:13

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