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I've read that when considering well distributed variables, median and mean tend to be similar, but can't figure out why mathematically this is the case.

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    $\begingroup$ What is a well-distributed variable? $\endgroup$
    – Dave
    Mar 20, 2022 at 23:36

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I'm not sure what a "well-distributed variable" is. Perhaps you could edit your question to provide a definition or a reference to what you were reading.

Many distributions that are commonly used for statistical modelling are symmetric and all symmetric distributions have the same mean and median (if the mean exists).

You can measure the extent to which a distribution is asymmetric by computing the non-parametric skew which uses the difference between the mean and median.

So a distribution where the mean and median are similar would have a low non-parametric skew. And a distribution where the mean and median are exactly the same would have a non-parametric skew of zero and be symmetric.

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For a lognormal distribution this is definitely not the case!

Many real world processes have normal distributions, for which mean, median, and mode are all equal. This is due to the central limit theorem. However, there are also lots of real world processes that are thoroughly non-normal.

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    $\begingroup$ You appear to have a common misconception about the central limit theorem. $\endgroup$
    – Dave
    Mar 21, 2022 at 17:03
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When you consider a well distributed dataset that means when you have a normally distributed dataset. If your dataset is normally distributed it must be symmetrical. In symmetrical datasets the mean, median and mode is equal. However, the mean is more reliable if the data is symmetric because it includes all the value of the dataset for its calculation and if there is any change in the dataset it will affect the mean the most as compared to median or mode. Mathematically-A symmetrical dataset looks like this {1,2,4,5,6,8,9} The mean, median and mode for this dataset will be 5.

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  • $\begingroup$ You could format your question so that it is easier to read (code format, paragraphs) $\endgroup$
    – hH1sG0n3
    Oct 12, 2022 at 7:25

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