# Is there a better way to graph a ratio over time than directly graphing the ratio?

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?

• How about using a logarithmic scale? – foxpal Oct 1 '19 at 2:40
• What is the range of values for the ratio that you need to plot? And can you add figures to illustrate the problem? – mkt - Reinstate Monica Oct 1 '19 at 5:51
• 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. – Nick Cox Oct 1 '19 at 7:03

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.