When to use logarithmic and arithmetic scales

When data is plotted on a chart, basically two types of scales can be used: arithmetic (linear) or logarithmic. Using one or the other can completely change the shape of a graph and consequently make you draw different conclusions.

Arithmetic (linear) scale

On the linear scale (used in most charts), the distance between dots on the scale is equivalent to the same absolute value. In the figure below, the distance is always 100,000 (100K, 200K, 300K, etc.).

 

 

Logarithmic scale

When using the logarithmic scale, on the other hand, the distance between the dots on the scale is equivalent to the same proportion. In the figure below, the distance is always 10x the previous value (100, 1000, 10000, etc.).

 

The most common situation where using the logarithmic scale can be advantageous is when either the sample data or the range of values in a chart is very large. In these cases, a considerable portion of the data may be visually “flattened”, making it difficult to read, as depicted in the situation above. Note that in the first figure (arithmetic scale) the analysis on the blue line (expenses of the finance department of a fictitious company) is impaired. This situation is avoided with the use of a logarithmic scale. Given that in this example we are making a reading of a trend (hence the line chart and not the bar chart), it makes perfect sense to keep a log scale; This way we are making it easier to read the percentage changes (while still giving the absolute value , both on the Y axis and on the line’s hover states).

In the example below, the two charts show the same information: US GDP (blue line) and government debt (red line) since 1940. The first chart uses an arithmetic scale and the second uses a logarithmic scale.

 

Comparing the two charts, we can draw some quick conclusions:

 

1. It’s almost impossible to get much information out of the first 25 years on the linear scale. The higher values at the end of the chart end up dominating the lower values at the beginning, which makes the analysis of the first decades practically useless.
2. The logarithmic scale shows that the biggest rise in debt took place in the 1940s. From 1944 to 1948, acquired debt was greater than the GDP. From 1948 onwards, the GDP was consistently greater than acquired debt.
3. On a linear scale reading, the impression you get is that GDP grew faster than debt between 1965 and 1983. However, the logarithmic scale makes it clear that the growth of the two was practically the same during that period.
4. From 1983 to 1993, the perception is reversed. Although the difference between the lines has decreased, the linear scale shows the lines running almost in parallel. While GDP rose 90%, acquired debt tripled during this period. The logarithmic scale makes this comparison more evident.

It is important the following: if you are compelled to use a logarithmic scale, do not choose a bar chart for plotting, as the bars are interpreted using their height: a bar with twice the height gives the idea of having twice as much value, which probably will not be true on a logarithmic scale. Do use a Dot Plot chart (same as a bar chart, but with a dot instead of a bar) for category analysis, as the reading is done in relation to the position on the X or Y axis, but not by it’s height. If the chart is for logarithmic temporal analysis, it’s preferable to use lines over bars for the same reason.

 

So why is it then that the charts used to analyze stock exchange performance usually make use of the logarithmic scale?

Simple: because analysts look for percentage performance indicators. It doesn’t matter so much whether the share went from USD $1.00 to USD $2.00 in one day, but whether the gain (or loss) was 10%, 50% or 100%.

 

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