Monday, 2 April 2012

arbitrary doubling

One reason for the news report about temperatures (amongst other things) falling is that it has been quite warm this last week. The average daytime temperature here in the UK for March is usually about 10°C, but it has been 20°C and more.

This warmth has been the source of some irritating newspaper headlines:
The Mirror has a headline with: “temperatures soar to twice the average for March
BBC New reports that the Daily Telegraph (which possibly should know better) says : “a barbecue weekend is in prospect, with temperatures double the usual for March.” 
No, the temperature hasn’t doubled. If it had doubled, if would be nearer 300°C!

If something doubles in size, units don’t matter. Something can double in length from 1 metre to 2 metres, or equally from 100 cm to 200 cm, or even from 3.28 feet to 6.56 feet.
0°C, 0°F and 0K are all different temperatures
(yes, there is no degree symbol when
using the absolute kelvin units
But let’s look at changing the temperature units, say to Fahrenheit (an obsolescent temperature scale where the freezing point of water is 32°F, and the boiling point is 212°F).  10°C is 50°F, so if the doubling was real, 20°C would be 100°F. But 20°C is only 68°F (a seemingly much more modest rise!), whereas 100°F is 38°C (or nearly quadruple that 10°C, by newspaper headline “logic”!).

Things are even weirder if we used the Delisle scale (an obsolete and peculiar temperature scale where the freezing point of water is 150°D, and the boiling point is 0°D, so higher numbers are colder temperatures). 10°C is 135°D; “doubling” this gives 270°D, which is -80°C, whereas 20°C is 120°D. (Although it may seem weird for numbers to go down as temperatures go up, Delisle was not alone: this is the direction the original Celsius scale went, with freezing being 100, and boiling being 0.)

So why does doubling work for lengths, but not for temperatures? What’s different about the temperatures is that they have an arbitrary zero point. 0 metres means no length, but 0°C doesn’t mean no heat. There’s nothing special, temperature-wise, about the freezing point of water; it’s an arbitrary zero point (as evidenced by the fact that the Fahrenheit scale chooses a completely different arbitrary zero point).

To be able to do the multiplication and get a meaningful doubling, we need a true zero point for temperature, the absolute temperature, measured on the Kelvin scale (or the Rankine scale if you prefer those good old small Fahrenheit-sized degrees). There is another condition: the scale needs to be linear (rather than say logarithmic, like acidity, sound loudness, or star brightness); temperature is a linear scale, so that’s okay.  (If you want to play around with temperature scales, there’s a nice temperature units converter on the web, although it says °K, when it should be just K).

Let’s look at that doubling calculation again, now using an absolute scale. 10°C is 283K (rounded to the nearest degree, since that 10°C isn’t supposed to be very precise). Then 283K x 2 = 566K – this time, a meaningful calculation. And 566K is 293°C, which is worryingly hot for March!

The same data, graphed with
an arbitrary zero and a true zero on the y axis

This trick of an arbitrary zero leading to nonsensical comparisons like “temperatures double” is used all the time in “presentation graphics” (often in newspapers!) with misleading y axes. If the y axis doesn’t show a true zero (or worse yet, is unlabelled), then beware!

Note that we can do this kind of multiplication if we are talking about temperature differences, no matter what the scale. A rise of 20°C is twice a rise of 10°C. (A rise of 10°C is a rise of 18°F; a rise of 20°C is a rise of 36°F ). Rises work this way because a rise of 0°C, which is also a rise of 0°F, is a true zero.

However, newspapers can even mangle this! Several decades ago (although I doubt things have improved today) I saw a newspaper report that included the immortal phrase: “a rise of 1°C (33°F)”.

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