# How to read a value-quantile diagram?

I've been reading an article on ACM queue, and got confused by figure 6 (latency distribution of a push-only workload), where the authors used the y axis to display the latency, and the x axis to denote the quantile. The problem is, I really don't get how to read that plot:

• What do the bars show? How can interpret a long bar at 0.95, for instance?
• Where do I find, e.g. the mean?

By the way, what is the name of this type of plots?

• It looks like an inverse cdf for a discrete distribution. Normally I'd expect data like that plotted the other way around (giving an empirical cdf). – Glen_b Jul 19 '13 at 8:44

It is a chart of quantiles, in effect the inverse of the cumulative distribution function.

The style is similar to a waterfall chart, which I suspect is not quite appropriate here: blocks which are separated horizontally as shown here suggests to me that certain cumulative probabilities do not exist, but blocks which are not separated vertically suggests to me that all latencies do exist. I am not certain that makes sense.

You can estimate the medians instantly by taking the values when the cumulative probability is $0.5$.

The means would be the areas under the curves, if the data were presented as curves rather than separated blocks. You would need the curve to reach, or to be arbitrarily close to, the right hand side (a cumulative probability of $1$) to calculate the mean.

The long bar at a cumulative probability of $0.95$ suggests to me that there is a 2.5% chance the latency is between about $7200$ and about $10200$, as there are about $40$ bars for each variable. There is also a chance it is above $10200$, possibly very much above.

• The main idea is right here, but the terminology used in the paper is backwards from statistical convention and indeed logic. The quantiles are ordered values or interpolated between them. Think of quantiles as including median, quartiles, deciles, etc. Numbers such as 0.5 or 0.95 or 1 are not quantiles but the associated cumulative probabilities. So, the median is the quantile associated with a cumulative probability of 0.5, and so forth. – Nick Cox Jul 19 '13 at 8:11
• @Nick Fair enough - I have changed some words to cumulative probability – Henry Jul 19 '13 at 8:14
• It's the paper that got it wrong. – Nick Cox Jul 19 '13 at 8:24
• Ok, that definitely helps! I start (slowly) to understand what it should show. Still, I really do not know why someone would choose such a weird chart to probably explain something much easier ;) - though it looks quite nice :) – navige Jul 19 '13 at 8:33
• @NickCox Too bad such things get edited and corrected when these papers get printed (the paper is in the printed version of the Communications of the ACM). This leads very much to misuse of statistical terms in other papers, too! – navige Jul 19 '13 at 8:36