I want to compare how effective two potential solutions are, and I have "scores" for each solution such that I can graph the effectiveness of each solution over time by graphing the scores vs time for each potential solution. I want to directly compare the two in a graph by displaying the ratio of scores between the two, but the issue I have is that graphing ratios isn't visually proportional because below 1 a ratio tends towards zero, whereas above zero a ratio tends towards infinity.

Is there any precedent for graphing some derived value more akin to:

if a / b > 1
then graph (a / b - 1)
else graph (-b / a + 1)

This has the advantage of tending towards -inifity and +infinity, but I am afraid it is less intuitive to understand. Is something like this worth doing or is just graphing a direct ratio better?

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    $\begingroup$ How about using a logarithmic scale? $\endgroup$ – foxpal Oct 1 '19 at 2:40
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    $\begingroup$ What is the range of values for the ratio that you need to plot? And can you add figures to illustrate the problem? $\endgroup$ – mkt - Reinstate Monica Oct 1 '19 at 5:51
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    $\begingroup$ Is either quantity ever zero or negative? If not, ratios often benefit from a logarithmic scale. Consider that the entire interval $a < b$ is squeezed into (0, 1) whereas $b > a$ maps to (1, $\infty$). Logarithm corrects that asymmetry. But in practice the range of $a$ and $b$ is crucial for what works. $\endgroup$ – Nick Cox Oct 1 '19 at 7:03

As foxpal and Nick Cox mentioned the log scale will make a ratio symmetric.

Lets say you use a logarithm base 10. $\log_{10}(10)=1$ and $\log_{10}(\frac{1}{10})=-1$; similarly $\log_{10}(100)=2$ and $\log_{10}(\frac{1}{100})=-2$, etc.

The catch is, as Nick mentioned, that you cannot take a logarithm of zero or a negative number. As a consequence both the numerator and the denominator have to be larger than 0.

  • $\begingroup$ Thanks for the answer, I think using a log scale is probably the best solution because the domain of both data sets is greater than zero and a log scale is easily understood by the reader. $\endgroup$ – Chris Philip Oct 1 '19 at 17:31

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