Re: off-by-one in boxplot?

[Allin C. <cot...@wf...>]

On Mon, 2 May 2011, sfeam (Ethan Merritt) wrote:

> On Monday, 02 May 2011, Allin Cottrell wrote:
> > I'm wondering if there might be an off-by-one bug in gnuplot's
> > built-in boxplot functionality.
> >
> > I'm attaching a small data file which I've tried plotting using
> > the following commands (simplified from boxplot.dem):
> >
> > set style data boxplot
> > set pointsize 0.5
> > set border 2
> > set xtics ("wage" 1) scale 0.0
> > set xtics nomirror
> > set ytics nomirror
> > plot 'wage.dat' using (1):1 notitle
> >
> > I calculate quartiles 1 and 3 as 1345 and 2140, respectively, and
> > this seems to agree with what gnuplot shows for the central box. I
> > therefore get 795 for the inter-quartile range, and multiplying
> > this by the default range multiplier of 1.5 I get 1192.5. Adding
> > this to Q3 gives 3332.5 for the upper whisker limit. The greatest
> > y-value in the data less than or equal to 3332.5 is 3307, so I'd
> > expect the upper whisker to extend that far, but it extends only
> > to about 2600, which could correspond to the next largest y-value,
> > namely 2613.
> >
> > That is, I think that according to the docs one should see a
> > whisker extending to 3307 and three high outliers, but in fact I
> > see the whisker going to about 2600 and four high outliers. I can
> > get the result I'd expect to see with the default "range" value of
> > 1.5, if I do
> >
> > set style boxplot range 1.6
>
> 1st quartile == smallest index that encompasses 1/4 of the data points
>               = data[ceil(49/4)]
>               = 1433

Maybe I'm being dense but I don't see how you're getting that:
ceil(49/4) is 13, and data points 8 to 15 all have value 1345.
Also the plot from gnuplot 4.5 seems (by mouse-over) to have
Q1 at 1345.

> 3rd quartile == smallest index that encompasses 3/4 of the data points
>               = data[ceil(49*3/4)]
>               = 2115

OK, I can see that: the 2140 value that I gave for Q3 is an
average of two data points, which implies a different concept of
quartile -- and the rest follows.

> There are surprisingly many differing definitions of "quartile".
> I can sympathize if you prefer a different one than gnuplot uses,
> but I think gnuplot is at least consistent with its own definition.

OK, fair enough.

Allin Cottrell