Explained: How to draw the right conclusions from Covid-19 graphs

This article is written by Jennifer M. Miller, PhD, in Public Policy from University of North Carolina at Chapel Hill and Apolitical Insider Fellow. This article is edited by Sebastian Muermann, Apolitical Insider Fellow and part-time consultant for the World Bank.

During the Covid-19 crisis, public servants have been called on to interpret a lot of graphs and data to inform decisions.

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Interpreting data helps us to know how well our own community is doing in fighting the coronavirus pandemic. Data from other places can also help us decide which practices we want to adopt in our pandemic response.

Unfortunately, our intuition about graphs doesn’t always help us get it right.

My experience teaching quantitative analysis to public managers has shown me that hands-on practice helps them develop intuition about data

One particular area of confusion has been the difference between graphing a trend over time on a linear scale, and graphing that same trend on a logarithmic, or log, scale. A recent study by researchers at the London School of Economics showed that people’s understanding and predictions were less accurate when viewing graphs with log scales, possibly due to the lack of the visible upward slope we associate with growth. Further, the researchers found that these inaccurate interpretations resulted in different policy preferences.

In this article, I will use examples to explain how these two types of graphs differ. I will then walk you through an activity you can do in your favourite spreadsheet program to improve your intuition when interpreting graphs like these.

My experience teaching quantitative analysis to public managers has shown me that hands-on practice helps them develop intuition about data.

Log and linear scales, explained

The key difference between a log scale graph and a linear scale graph is the pattern of numbers on the vertical axis.

On a linear scale, evenly spaced lines or ticks on the vertical axis represent the same increase in the number of cases. For example, evenly spaced ticks on a linear scale might be numbered 100, 200, 300, along the vertical axis. On a log scale, evenly spaced lines or ticks on the vertical axis represent increasingly large increases in the number of cases based on multiplication.

For example, evenly spaced ticks on a log scale might be numbered 100, 200, 400 based on multiplying by 2 or 100, 1,000, 10,000 based on multiplying by 10.

Covid-19 case trend examples

Let’s look at some examples from Our World in Data.

This first graph uses a linear scale to show the total number of cases that have been diagnosed in nine countries starting in January, 2020. Notice that the distance between each of the equally spaced lines on the vertical axis represents 500,000 cases. That is how we know this graph uses a linear scale. This type of graph matches our intuition fairly well by clearly showing the upward slope we naturally associate with growth.

We know that we don’t want the line to reach a high number of cases. We also know that we don’t want a steep upward slope, because that would mean the diagnosis of many new cases in a short period of time. If a line on this type of graph flattens, it means the virus is no longer spreading in that country.

This second graph uses a log scale to show the same data ⁠— the total number of cases that have been diagnosed starting in January 2020 for the same nine countries. Notice that the distance between each of the equally spaced lines on the vertical axis represents ten times the number of cases as the distance between the previous pair of lines. That is how we know this graph uses a log scale.

This type of graph confuses our intuition. Curves visibly flatten, even when the virus is still spreading widely. Near the top of the graph, even a small upward slope of the line represents a large increase in the number of cases. Similarly, near the top of the graph, distances between country lines represent large differences in the pandemic’s impact.

To help develop your intuition around interpreting graphs with linear and log scales, it may help to practice in a familiar spreadsheet program using data that follow simple growth patterns

Compare the distance between the lines for Australia and Canada on the two graphs. The log scale makes it clear that ten times the number of cases have been diagnosed in Canada compared to Australia. Compare the distance between the lines for Brazil and India. If we are not careful, the log scale tends to obscure the difference in number of cases that was clearly visible with the linear scale.

If linear scales are more intuitive, why use log scales? There are at least two reasons. First, log scales can make some relationships more clear, such as the difference between Australia and Canada. Second, the spread of the virus itself is not linear. Its spread from one person to many is based on multiplication, like the intervals on the log scale.

Develop your intuition

To help develop your intuition around interpreting graphs with linear and log scales, it may help to practice in a familiar spreadsheet program using data that follow simple growth patterns.

Graphing two scenarios with a virus spreading in a community of 100 people helps to illustrate the difference on a small scale. I have created a spreadsheet here (Google sheets | Excel), which you can use to follow.

We will call the first example Cases Climb. In the Cases Climb scenario, the virus spreads through linear growth, with 10 new people infected each day until all 100 people catch the virus. Enter these numbers into your spreadsheet and create a line graph. Your graph should show a straight line with a steady upward slope.  Make a copy of this graph. On the new copy, use the settings for your vertical axis to convert it to a logarithmic (log) scale. On this graph, the trend will appear as a curve that gradually flattens.

We will call the second example Cases Multiply. In the Cases Multiply scenario, the virus spreads through exponential growth, or multiplication. Each day has twice as many people infected as the day before. This is what would happen if each person infected two people. Notice how the case numbers stay low at first, but manage to infect all 100 people even faster than in the previous case. Sometime between Day 7 and Day 8, the whole community has caught this virus!

As you did before, enter the data into your spreadsheet and create a line graph. Your graph should have an upward sloping curve. You may find it helpful to set the maximum of the vertical axis to 100 to represent the community’s total population. Now, make a copy of this graph. On the new copy, use the settings for your vertical axis to convert it to a logarithmic (log) scale. On this graph, the trend will appear as a straight line with a steady upward slope.

In summary, it can be difficult to understand trends that start slowly then rapidly increase in speed — like the exponential growth of a spreading virus. Graphs with log scales can be helpful for understanding these trends. Our initial intuition may lead us to misinterpret these graphs, downplaying the importance of small distances near the top of the graph. By practising with examples where we understand the growth patterns, we can strengthen our intuition for drawing the right conclusion from graphs. — Jennifer M. Miller

Work cited

Romano, Alessandro and Sotis, Chiara and Dominioni, Goran and Guidi, Sebastian, COVID-19 Data: The Logarithmic Scale Misinforms the Public and Affects Policy Preferences (April 29, 2020). Available at SSRN: https://ssrn.com/abstract=3588511 or http://dx.doi.org/10.2139/ssrn.3588511

(Picture credit: Unsplash)