Graphs not drawn to scale
Source: Anonymous corporate annual report
Another way in which graphs can be misleading is if they are not drawn to scale. Typically, this will involve the use of nonlinear scales. A nonlinear scale can best be described by describing what a linear scale is not. A good example of a linear scale is a ruler. On a ruler, equal distances between points represent equal values. For example, the interval between the point labeled “2cm” and the point labeled “3cm” represents the same thing as the interval between the point labeled “3cm” and the point labeled “4cm”. They both represent 1cm. Another way of putting it is that on a linear scale equal values are represented by equal distances. So 1cm can only be represented by one distance between points on the ruler.
On a nonlinear scale, equal values are not represented by equal distances. The graph to the right shows an example of a nonlinear scale. Here the labels on the vertical axis are not spaced evenly. The interval between the point labeled “20,000” and the point labeled “30,000” represents 10,000, and the interval between the point labeled “30,000” and the point labeled “40,000” also represents 10,000. But the intervals are not of the same length. The distance between the points labeled “20,000” and “30,000” is smaller than the distance between the points labeled “30,000” and “40,000”. The use of such a nonlinear scale makes the graph misleading.
Source: http://blog.1000memories.com/
75-number-of-people-who-have-ever-lived
But while it was easy to notice that the above graph had a nonlinear scale, authors can be more subtle by not labeling the scale. The following bar graph, for example, shows the population of the world over time. But is it drawn to scale? You can tell that it is not by noticing that the bar representing 300 million is about half the length of the bar representing 1 billion when it should only be 30% of the length (since 300 million is 30% of 1 billion). Also, the bar representing 10,000 shouldn’t even be visible on a linear scale that reaches 7 billion, given that the bar representing 7 billion should be 70,000 times the length of the bar representing 10,000 (since 7 billion is 70,000 times greater than 10,000).
Source: www.forbes.com
The following figure shows the above graph lined up with how it would have looked if it was drawn to scale. As you can tell, the bars representing 10,000 and 3 million aren’t visible. You might be wondering right now why the graph being not drawn to scale is even a problem. After all, you wouldn’t be able to see the bottom bars if the graph was drawn to scale, and so the bottom section of the graph would have offered nothing useful to analyse. But if you think back to the purpose of a graph, it is to represent data as a picture. In order to do this, the bars must be proportional to the numbers they represent, otherwise they would be misleading. In this case, by not drawing the graph to scale, the author has made the population in the earlier years of the world appear much larger than they really were.
Discussion
The line graph shows the improvement in the math scores of Year 4 students on a national exam in the US from 1990-2000.
Source: Brookings Instituion report on American education
- What is non-standard about the vertical axis and what do you think was the effect of this?
- What is non-standard about the horizontal axis?
- What effect did this have on the steepness of the black lines?
- What misleading message is being sent about the increase in the scores over the time periods?
- Redraw the graph so that it is not misleading. (HINT: start the graph from zero on the vertical axis and space the years evenly on the horizontal axis)