# Displaying fine variation across multiple orders of magnitude

I have a sequence of numbers that grows super-exponentially:

0.993, 0.999, 1.037, 1.054, 1.195, 1.55, 2.953, 15.369, 815.687, 26492.118


I'd like to be able to plot the data in such a way that the viewer will be able to tell how the whole sequence increases: for my application, the first few values are especially important, but I'd still like to show the whole range. So I tried a log scale (which stretches out small values), but the first five values still look pretty indistinguishable.

I could take the log twice, but that would probably hurt interpretability and I couldn't display the two values that were less than one (because they'd be negative after the first log transformation).

Right now I'm thinking about splitting it into two plots with different Y axes, but I thought I'd see what the community suggested. Thanks in advance for your time.

• Another possible transformation here is arctan. It will give you more variability for smaller values; however, as x approaches infinity arctan(x) approaches pi / 2; therefore, it will look like the whole sequence is leveling off, so maybe this is not what you want. Nov 17, 2013 at 2:51
• A very careful choice on the Tukey ladder is one possibility, but it's still not particularly satisfying; with a bit of fiddling about one can do something like this; is that the kind of thing you wanted? Nov 17, 2013 at 4:09
• @Glen_b The choice of plotting the reciprocal actually works pretty well here and in any situation where the data are already rates or ratios it has a natural interpretation.
– whuber
Nov 18, 2013 at 17:04
• @whuber The reciprocal is actually pretty good (for myself, I'd have probably used it; it's a good compromise), but I don't think it quite gets the 'fine variation' in the title (which I assumed to imply being able to see the increase in the first two values), and it also loses the distinction in the two uppermost values; alternatively, $-1/\sqrt y$ helps at one end while making the other worse. If one shifts the result and takes a further transformation, things improve a lot (but you lose the ready interpretability of the reciprocal). It could well be that I interpreted the title wrongly. Nov 18, 2013 at 19:56

Let's step back, and think how to represent data instead of how to visualize data. I love data visualization but I'd say that graph is not a suitable solution here.

1. I'd like to be able to plot the data in such a way that the viewer will be able to tell how the whole sequence increases
2. [F]or my application, the first few values are especially important, but I'd still like to show the whole range.

The problem with graphs with original or log scale is that it's still practically impossible to name the first few values from those graphs, which is your objective #2.

A graph is on the only tool to show range, you can also tabulate your readings:

    0.993
0.999
1.037
1.054
1.195
1.550
2.953
15.369
815.687
26492.118


Most people can immediately notice that very high rate of growth, which should fulfill your objective #1. In the mean time, you got to keep the true data of the first few cases. And if you so wish, you can even plot those first few cases using the original scale.

Your idea of two plots with different scales is better than a broken axis since it gives you a perceptual view of how disparate the values are. Below is a quick mock-up. For presentation, the two graphs should have more separation and some cue or text that one is a zoomed in view of the other.

Log or some other transformation may still be best to show the power of the growth. Here's log.

Another option is to plot some derived value, such as the ratio yk+1/yk.

• Cleveland suggest that when breaking the scales like this to have more "forceful" visual separation (see a similar Q/A on the user experience site). So the idea is a good one, but I might suggest separating the panels so it is much more visually clear that there is a break in the axis. What do you think about a linear scale on the lower panel and a log scale on the upper? Nov 18, 2013 at 13:37
• I agree about the separation. Thanks for pointing it out. I'll update my answer. Hadn't thought about mixed scales, but that could work -- the log isn't really adding anything on the zoomed in view.
– xan
Nov 18, 2013 at 14:49