Example number 1 seems to be nice if you have different minimum thresholds among the categories.

As pointed by Glen_b and whuber, it seems that examples number 2 and number 3 do not show the ranges of your categories, but just one unique statistic (it could be the median, or the maximum values) at the top of the horizontal bars.

The example number 4 is a little bit strange because the bell curve does not represent the distribution of the bars (for example, the blue light dot 'average paid' is the average of the bell curve, not the average of quantities shown in the bars). It is not "visually compelling yet immediately understandable" to me.

As you asked for another option, I would suggest the boxplot, which shows:

- outliers (the dots),
- minimum and maximum values without considering outliers (the end of the whiskers)
^{1},
- first and third quartiles (the edges of the box), and
- median (the horizontal bar inside the box).

Each box is a category. Order the boxes from left to right starting with the category with greatest median.

The example number 1 is simpler to understand, so it will depend if a boxplot will really help.

_{1: see whuber's comment for clarification.}

Your difficulty is that in order to find IQR you must first find the
two quartiles. And there are many
different formulas for quantiles (including quartiles) in common use.

In particular, major statistical software packages disagree on which methods to implement as their default: (a) SAS, (b) Minitab and SPSS, and (c) R (and its parent S) use three different methods. Furthermore, these methods differ from methods found in reputable elementary texts. (Adding to the confusion: Tukey's 'fourths', sometimes used in making boxplots and often considered essentially the same as quartiles, use yet other criteria.)

Generally speaking the differences among these methods become negligible for large sample sizes. However, there can be marked differences for small samples. Fortunately, it is for large samples that quantiles make the most sense. (Roughly, quartiles are intended to divide a sample into four 'chunks' of equal size: how do you do that with a sample of size 10?)

In R, you can type `? quantile`

to see the nine different types of
quantiles supported by R (using an extra argument), mentioned just now
by @EdM. The default result from quantile is min, Q1, med, Q3, max, so once you have selected a `type`

you could define your own IQR function based on the idea in the @Glen_b Comment and code like `as.numeric(diff(quantile(x, type=5)[c(2,4)]))`

.

## Best Answer

As Greenparker pointed out, understanding what the IQR is generally is 90% of your answer here.

What it describes is the range between the 1st and the 3rd quartile (i.e. the point at which 25% of values are lower and the point where 75% of values are lower)

In the movie case, the range alone tells you exactly that: 50% of movies lie within price differences of 6.50 and 3.50 respectively.

Together with concrete values for your quartiles (and additionally the median) it describes roughly how your DVD prices are distributed.

Let's say the median price of a DVD is 10, the 1st quartile is 8 and the 3rd quartile is 14.50 (IQR = 6.50). Then that means that 25% of your DVDs cost less than 8, 25% cost between 8 and 10, 25% between 10 and 14.50 and 25% are more expensive than that.