Do time series charts really compare time series?
You can find time series comparisons in almost every newspaper, business journal and political magazine. Although they are frequently used in the media, these charts can be very misleading. What you should know about developments over time so that you don’t jump to the wrong conclusions.
Recently, it seems that I experience shocks like these quite often. Take the chart below. The lines suggest that the price for heating oil has exploded, while electricity and natural gas have increased moderately.
If you do the math yourself, however, you can also see that the blue line for natural gas has grown at about twice the rate as the green line for electricity and almost at the rate of the red line for heating oil. Apart from general inflation and the increase in value added tax over that period – honestly…do you see that?
| 1991 | 2007 | Δ | |
|---|---|---|---|
| Heating oil | 26,38 | 58,63 | +122 % |
| Electricity | 14,80 | 20,15 | +36 % |
| Natural gas | 3,55 | 6,51 | +83 % |
I don’t. As I recently sat with my son over a few geometry problems, I suddenly realized why this is. We observed two lines that run parallel to each other. We asked ourselves: Is ‘parallel’ equivalent to ‘similar’ – and learned that lines run parallel to each other when they have the same absolute slope. This is the case when the absolute changes per period have the same value and sign. Where they start is irrelevant.
When two lines with absolute values run parallel to each other, it means that they have the same absolute growth (in this case: +30).
A relative observation gives a completely different picture. Here, we can no longer talk about parallelism. And although it has higher values, the red line now runs below the blue one and at a much flatter angle. Why? The relative value growth is different in every period and is smaller due to the cumulative values from the previous period.
When we plot their relative growth, our parallel lines in the sample above don’t look parallel anymore. In this case, the +30 from 70 to 100 is 43 %, while the +30 from 370 to 400 is only 8 %.
And now for the shocking conclusion: When things change at the same rate, the development of the absolute values is anything but parallel. The same rate means the same relative growth. The greater the difference between the starting figures is, the greater the lines will spread apart:
The paradox of our shocking conclusion: When developments are very similar, the respective lines will break apart. They change at the same rate, or in other words, they have the same relative changes as we can see in the chart below.
The values from the previous chart in a relative comparison: Both lines are now similar. Since the growth is identical, the lines are on top of each other.
The results of this observation are surprising and illogical at the same time. The typical lines that are often drawn to show absolute values for revenues, stock prices, temperatures, etc. are only similar in a single case. And that is when they increase or decrease by a very similar absolute value from period to period. But that is not normally the case. If the finance crisis had an influence on the stock prices of Commerzbank (€ 16.82 on 5 September 2008) and Deutsche Bank (€ 56.69), we would never expect them to both drop by the same absolute value.
Instead, we would be much more likely to expect dependencies that show a similar relative influence on developments – which from an analytical perspective would also be much more valuable. To identify these, we must plot the relative changes. The closer the slope is, the more similar the values are.
If you plot the price values on a logarithmic plot, you can compare relative changes by comparing slopes, as I describe in Logarithmic Axis Scales, my follow-up to your evocative post.
Tuesday, September 16th, 2008, 2:51 pmGreat analysis. I read about it on Jon Peltier’s blog, and commented on the use of different units. Jon suggested I copy my comment to his post to you as well:
It would be even more useful if the data were all in the same units (i.e. cents per kWh) so that we can choose the most economic option.
Thursday, September 18th, 2008, 6:45 pmUsing a typical heating value, 1 euro/100 liters fuel oil converts to 0.092 cents/kWh. So, 26.38 euro/100l = 2.43 cents/kWh and 58.63 euro/100l = 5.39 cents/kWh. This lands fuel oil just below natural gas. However, it is somewhat less attractive because it doesn’t burn as efficiently, and not as clean.
Electricity is clearly the least attractive option.
Nicolas:
Very important post. Thanks for the refresher.
I’ve written a recent post on logarithmic scales that may be of interest.
Kelly O’Day
Thursday, September 25th, 2008, 10:31 pm[…] Nicolas Bissantz wrote in Do time series charts really compare time series? about time series being difficult to compare. The chart he discussed showed the cost of energy for […]
Tuesday, September 30th, 2008, 1:29 am