Simple descriptive statistics with inter-quartile mean. What about standard deviation?

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?

• 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. Mar 24, 2017 at 12:25
• @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.
– PM.
Mar 24, 2017 at 12:33
• @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. Mar 25, 2017 at 5:10
• 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. Mar 25, 2017 at 15:59
• 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. Mar 10, 2020 at 1:59