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I have a dataset where I want to describe (e.g.) for each city the distribution of the building volumes.

If I had only one city, I could make a histogram and it would nicely capture that. I have many cities though, which is why I need some indicator(s) that capture the distributions.

Of course there are the basic summary statistics (mean, max, mode etc.) as well as Gini-Index comes to mind. I struggle to find alternatives to them that can capture the distributions in a "complete" way.

Are there any other useful metrics that can capture the distribution (disparities)?

If it's relevant, I'm working with R.

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    $\begingroup$ Boxplots (which summarise distributions by five "points", 1st, 2nd, 3rd quartile, two "non-outlier region" borders plus outliers) are quite popular for this. $\endgroup$ Aug 30, 2022 at 12:59
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    $\begingroup$ Note that you can hardly ever be "complete", as there is so much flexibility in distributions. For example they can be multimodal, which without severe restrictions cannot be easily summarised in one or few numbers. $\endgroup$ Aug 30, 2022 at 13:00
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    $\begingroup$ Mean, variance, skewness, kurtosis would also be a candidate set of useful statistics (not capturing multimodality either, and affected by outliers though). $\endgroup$ Aug 30, 2022 at 13:02
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    $\begingroup$ Skewness is more likely to be interesting than kurtosis, though both are harder to interpret than mean and standard deviation. But if you are willing to use a picture for one city, why not have superimposed histograms for several cities? $\endgroup$
    – Henry
    Aug 30, 2022 at 13:41
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    $\begingroup$ A small set of statistics is pretty much useless unless you have specified a low-dimensional parametric family of distributional shapes in the first place. Histograms are inefficient and vague. Instead, select a better graphical method, such as boxplot, beanplot, violin plot, probability plot, etc. $\endgroup$
    – whuber
    Aug 30, 2022 at 14:33

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Some popular sets of statistics to summarise a distribution are

  1. Mean, variance (or standard deviation), skewness, kurtosis (these are potentially affected by outliers though).

  2. The boxplot set of statistics, namely the two "hinges", the "whisker extremes", and the median. The "hinges" are essentially the first and third quartile (the median being the second), the "whisker extremes" are the smallest and biggest non-outlying observation according to an outlier identification rule based on the pretty robust interquartile range statistic (basically upper minus lower hinge, indicating variation in the central part of the data). These are given out by R's boxplot function. One may add something to summarise the outliers, like number or proportion of outliers, maybe differentiating between upper and lower outliers.

  3. Connected to the boxplot, Tukey suggested as a five number summary (R function fivenum) minimum, lower hinge, median, upper hinge, maximum. It depends on the situation whether minimum and maximum are more relevant than the whiskers mentioned above; obviously one could give all these statistics but of course there is something to be said for constraining oneself to as few numbers as possible for efficient communication.

Note that distributions can have so flexible shapes that it is impossible to give a "complete" characterisation by few numbers. Particularly multimodality may be of interest on top of what the statistics listed above indicate, but multimodality is a can of worms that you may not want to open. Just to give an impression why, real data are not continuous, and therefore modes of an empirically discrete distribution of potentially continuous data are not even well defined, and the number (and location) of modes in a density estimator depends on (difficult) user tuning. Furthermore any number of modes may be of interest, so if you want to indicate where the modes are, there is no bound on the number of statistics you will need. For such reasons, even "the mode" (global max of the density ignoring potential further local modes) is troublesome as a statistic, as long as you're not dealing with, say, natural numbers only.

I should add that generally outlier identification also depends on user tuning; the way boxplots do it is fairly well established, but there are alternatives, so when using statistics regarding boxplot outliers, these should be called something like "boxplot outliers" rather than suggesting that what the boxplot indicates are always real, objective, and meaningful outliers.

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