What's a logarithmic graph and how does it help explain the spread of COVID-19?

The coronavirus pandemic is undoubtedly the greatest challenge the world has faced in over a generation. But is the presentation of data about the outbreak leading us to overlook any cause for optimism?

There is an alternative to the commonly seen linear graph that could help give a more detailed picture. It’s called a logarithmic graph.

Interpreting information

The most common form of a line-graph has a linear scale. Along the Y axis, the numbers progress in a steady, linear form – 1, 2, 3, 4, or 10, 20, 30 and so on.

But infectious diseases don’t spread in an even, linear fashion. On a linear scale graph, the rate of growth keeps going up and up – the line can become almost vertical and appear to go on forever. That can create the impression measures like social distancing aren’t working.

On a logarithmic scale, numbers on the Y-axis don’t move up in equal increments but instead each interval increases by a set factor – it’s often 10 but could be a factor of 3 or 350 or 3,500, anything at all. It all depends on what is deemed to be the most effective way of interpreting the data in question. The Richter scale is logarithmic – an earthquake that measures 6 is 10- times more destructive than one that measures 5.

The logarithmic scale is ideal for measuring rates of change, particularly rates of growth, explains mathematician, teacher, and author of The Life-Changing Magic of Numbers, Bobby Seagull.

It “flattens out the rate of growth so it becomes easier to see,", he says. "On a logarithmic graph of COVID-19 infections, even though the overall numbers are still increasing, you can see the point at which the rate of growth starts to level off when that exponential growth has stopped."

At that point, the logarithmic scale makes it possible to see when public health measures are starting to have the desired effect.

This is what the COVID-19 situation in China looks like when plotted on a linear graph.

And this is how it appears when plotted on the logarithmic scale.

Same data, different perspectives

Both charts tell the same story – that China’s coronavirus cases have started to flatline. But the logarithmic graph shows a flattening of the line much earlier because of the way the scale has been compressed. It also makes it possible to fit a large or widespread set of results onto a graph that might otherwise not fit in a linear way.

A logarithmic graph can also help make it clear if the apparent evening-out of the curve started to change. While a linear curve would keep on pushing ever higher regardless, the logarithmic graph would highlight any substantial changes to the trend – whether upward or downward.

It’s an approach that is often preferred when there are huge numbers involved and a linear scale would just produce a dramatic-looking exponential curve.

The raw numbers aren’t affected by the choice of chart. The reality is that (as of 4 April 2020) there are more than 1 million confirmed cases globally and 55,000 deaths. But more than 219,000 people have also recovered – and that’s a part of the coronavirus story that’s just as important to tell.