# Comparing variability of likert scale answers beyond IQR

I am analysing a simple study where users were asked questions on 5-point likert scales. I use interquartile range (IQR) to assess the variability in the answers of the respondents. In particular, I want to

1. somehow compare the difference between treatment and control groups (especially the difference in count of positive/negative ratings) to the within-group variability, and
2. assess whether the two groups have similar variability.

Note that I am not doing hypothesis testing, I just want to let the reader have some intuitive understanding of the amount variability.

The problem I see is that the IQR is often the same even though intuitively the variability is different. For example:

Group A: 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4
Group B: 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3


In both cases, the lower quartile is 2 and the higher quartile is 3, so IQR is 1, although group A is intuitively more polarised on the subject.

My idea is to report something along those lines: "Group A and B have the same IQR, but in group A 62% of responses fall within IQR, while in group B it is 100%"

Is there any standard way of measuring/reporting ordinal data variability with better precision than IQR? The reader can always check bar charts of the data, but quantifying the variability seems useful.

• What is wrong with a boxplot? Include the FNS and then you have both. – mandata May 5 '15 at 14:19