# [R] Q: Mean, median and confidence intervals with functions "summary" & "boxplot.stats"

S Ellison S.Ellison at lgc.co.uk
Thu Aug 30 14:17:17 CEST 2007

```If you look at ?boxplot.stats, you will find that the confidence interval it reports is centred on the median and :
"The notches (if requested) extend to '+/-1.58 IQR/sqrt(n)'."

If you have skewed data it is very possible (as you have found) that the mean is outside median+/-1.58 IQR/sqrt(n).

All that is happening is that the majority of the data are around 1 or 2  and you have a substantial number near zero. Result: mean much lower than median. And with a high n, the boxplot notch is very narrow and excludes the mean.

But it does sound very much as if you are doing something questionable at best. I would not trust IQR as a dispersion measure on discrete data with few possible values even if they were on an interval scale; too much risk of getting the same IQR for many different distributions. On an ordinal scale it is worse; the only points that are valid at all are the valid scale values, so a CI that uses intermediate values is formally meaningless (what is a shoe size of 7.2, for example? Answer: Not a shoe size at all). It is of course entirely meaningless to talk about an IQR on a categorical scale.

It sounds like boxplot.stats is an inappropriate tool for summarising your data.

>>> Tom Willems <Tom.Willems at var.fgov.be> 30/08/2007 10:00:50 >>>
Dear R ussers,

My question is, " How can my mean be outside the confidence intervals ?!"

I think i have the answer for it, but i would like to hear some other
ideas on it.

First my data is not continuose but categorical, it is a titre calculated
on a dilution serie.
It is stored as a column of values, and a column indicating the phase of
the trail.
Theoreticaly it is possible to have a value ranging from 0 to 4, but in
practice, only sertain values will occure, and they will repeat.
So it are frequencies.

This is why i belief that it is better to work with a median than with a
mean, because it represents the cluster of values wich occure most.
Below I only give one example, but the mean being below the lowest
confidence limit occures several times over different tests.

does my answer seam reasonable, or should i perhapes use an other methode,
any sugestion?

summary_1d  = summary(subset(eda_data,  phase=='1' & test=='test
1' ,select=lg_value), na.rm = T)
conf_1d  = boxplot.stats(subset(eda_data,  phase=='1' &
test=='test 1' ,select=lg_value))

Mean            Median             95% Confidence     Int. StDev.
Variance
1.198            1.681                   1.441 >  < 1.922 0.931
0.866

Kind regards,
Tom W.

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