Statistical charts provide one of the most effective tools for conveying information (and misinformation) When a commentator presents a graph as “proof” of their argument, view it with great care.

Graphics can help describe patterns of behavior and provide evident of structural connections, but they never prove anything. Scientists must continually subject theoretical “proofs” to disconfirmation, or they do not qualify as science.

I frequently use charts to help explain statements, but I try not to present them as proof.

Charts seldom have any use without comparing at least two sets of data. I find line charts helpful to explain changes over time. I use line charts to compare either linear changes or percentage changes.

In a recent publication, I made a brief explanation of the difference between linear scales (measuring absolute changes) and semi-log scales (measuring relative changes). In this post, I want to clarify the explanation.

To illustrate the difficulty of visualizing relative changes, I offer this riddle from my files.

The Lily-Pad Riddle

A powerful illustration of the exponential effect of change comes not from a scientist or a futurist, but from a child’s riddle:

“On day one, a large lake contains only a single small lily pad.  Each day the number of lily pads doubles, until on the thirtieth day the lake is totally choked with vegetation.  On what day was the lake half full?”

The answer, of course, is the twenty-ninth day.  It takes twenty nine days for the first half of the lake to fill with lily pads, but only twenty-four additional hours for the lake to become overwhelmed.

Welcome to day twenty-nine.  Imagine that the proliferating lily pads represent the expanding array of changes that face the world. Suppose that the human resilience required to address these changes is represented by the lake’s capacity to accommodate the lily pads.  What happens as day twenty-nine approaches?

Daryl R. Conner
Managing at the Speed of Change

Comparing Charts

I will use two charts to help clarify the differences between a linear chart and a semi-log chart.

These two charts reflect the same data; the only difference between the charts consists of the vertical scales. I will explain the red box outlines after you have had time to examine the charts.

Linear Chart

Semi Log Chart

Although the charts represent precisely the same array of data points, they appear significantly different. Changing only the vertical scale changes the information conveyed in each chart.

The first chart (linear) shows the number of units of change regardless of the time interval. People use linear charts to demonstrate absolute changes (e.g. increase in the quantity of water in gallons regardless of time, or an increase in the number of dollars regardless of the length of time).

The second chart (semi-log) shows the number of units of change relative to the length of time. One might express these changes in quantities as rates of change (e.g., increase the quantity of water in gallons per hour, or an increase in the number of dollars per year).

One good way to visualize these changes consists of visualizing the length of time it takes for the absolute change to occur. I have drawn red boxes to help you visualize absolute change versus relative change.

Specific Charts—Excess Reserves vs. Money Supply (M2)

I wanted the readers of this publication to have a couple of real-life examples to see how changing the manner in which they have data presented to them can influence their interpretation.

I have chosen to present the same data streams (Excess Reserves vs. Money Supply (M2)) in two different formats: linear and semi-log. Does changing your chart change your interpretation of the data?

Linear

The linear chart clearly shows that the amount of excess reserves increased more in 2008 than in 2020.

In those same years, 2008 and 2020, respectively, M2 increased by a far greater amount in 2020 than in 2008.

Semi-log

The semi-log chart shows clearly that excess reserves increased by a dramatically greater percentage in 2008 than in 2020.

In those same years, 2008 and 2020, respectively, M2 increased by a greater percentage in 2020 than in 2008.

Which chart provides a more accurate representation of the relationship between excess reserves and M2? Should the larger percentage increase in excess reserves also have a larger percentage increase in M2?

Conclusion

How you have information presented to you will have a great deal to do with how you interpret that information. Charts provide an excellent way for people to see the same information from different perspectives. Writers should remember that different charts have different meanings to different people. When communicating, choose charts that clarify your meaning.

I have given an example of one graphic that I have used frequently: the semi-log chart. Semi-log charts provide a very useful tool for showing trends over a long period. Frequently, showing a rate of change carries more importance than the absolute numbers.

Charts never, by themselves provide any proof; they can, however, point the way.