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If one summarizes a data set with an inter-quartile (iqm) mean, rather than the mean calculated using the complete data set, is a standard deviation (sd) calculated using the complete data set then inconsistent with presenting the iqm?

Put another way, for consistency, should the sd be computed using the exact same data ( inter-quartile or complete ) that was used to compute the mean?

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    $\begingroup$ The inter-quartile mean in a box plot is the midpoint of the inter-quartile interval. It does take all the data into account. The standard deviation is a commonly used description of variability but there are probability distributions for which it does not exist. $\endgroup$ – Michael R. Chernick Mar 24 '17 at 12:25
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    $\begingroup$ @MichaelChernick I am using IQM as defined by the mean of the data excluding the upper and lower quartiles statisticshowto.com/interquartile-mean-iqm-midmean It is supposed to be more robust against outliers. $\endgroup$ – PM. Mar 24 '17 at 12:33
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    $\begingroup$ @scitamehtam For what it's worth, I agree with your definition -- and so does wikipedia -- though in my (perhaps limited) experience it's more often called the midmean. What Michael is talking about is usually called the midhinge. $\endgroup$ – Glen_b Mar 25 '17 at 5:10
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    $\begingroup$ I've never seen the SD of the data between the quartiles being used as a measure. I can't see any reason to calculate it. $\endgroup$ – Nick Cox Mar 25 '17 at 15:59
  • $\begingroup$ If you like mid-means, you might perhaps consider a quartile-winsorized standard deviation -- rather than eliminate data outside the quartiles, move it to the quartiles and then calculate a form of standard deviation. $\endgroup$ – Glen_b Mar 10 '20 at 1:59
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If you use the interquartile mean (or midmean), the maybe you should also report its standard error ? But, if what you want is some robust measure of spread of the data, then choose such a measure (maybe MAD?), preferably a known one. Your proposal of using the standard deviation of the data between the first and third quartile doesn't seem to have much meaning.

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