Mean Numbers

Years ago, when I was still in school, my chemistry teacher gave me a bollocking.  Not for blowing up the school, burning a hole in a textbook, or squirting people with a pipette but for implying an accuracy that was not there.


Yes, that was my first reaction too.  Here’s what I did.

I had taken five measurements as follows (don’t ask me what of, or the units):

3.4 / 4.2 / 3.9 / 2.8 / 3.1

And presented the average, calculated by Casio, as:


I got the bollocking for presenting the average to 2 decimal places when the accuracy of the input data was only to 1 decimal place.  The implied accuracy of the last digit was false, because the actual data might have been:

3.44 / 4.15 / 3.90 / 2.84 / 3.12

whose average is 3.49.  With the data rounded to 2 decimal places, I should have written the average to the same level of accuracy: 3.5

His point was a good one: the results derived from a series of numbers can never be more accurate than the input data.  Clearly I never forgot it, because I’m writing about it now.

What he didn’t quite say was that to present the result to a greater accuracy than the input data is in fact quite misleading, especially when the average is taken as a single number and used in a subsequent calculation.  Supposing somebody called Herbert wanted to know the height of the Tashkent Tower (and didn’t trust Wikipedia).  He calls up an engineering outfit in Uzbekistan to go and find out, and they take a series of triangulations from various points.  But their laser unit is broken and so they’ve had to rig up a piece of string, a stick, and a protractor in a system which is only accurate to the nearest 5 metres.  Nevertheless, after collecting enough data they reckon they have a pretty good idea and so they go back to the office, add it all up, take the average, and send an email back to Herbert stating confidently that Tashkent Tower is 378.24m high (along with a hefty invoice).  Herbert is impressed: this mob can measure a tower to the nearest centimetre!  He takes this number assuming they have used a highly accurate system of measurement, and plans his next base jump.

It’s not so much the inaccuracy of the number that is the concern, but the way it misleads regarding the methodology behind it.  The number as presented has made the recipient believe it has been carefully measured with a high-tech laser device, and not conjured up by a bunch of blokes winging it with bits of wood and string.  If they were being honest, our Uzbek engineers would have either presented the figure as 378m +/- 5m or (if they didn’t know the accuracy of their system) the range of their data.  But then they’d not have gotten their invoice paid.

I write this post more in preparation for a future post on management decision-making than a desire to teach my readers mathematics.  I’ll be referring to it shortly.


4 Responses to Mean Numbers

  1. Mark T says:

    Found your blog via your link on Tim Worstall and am enjoying reading through. On this particular issue it does make you wonder on the ability to measure average global temperatures to several decimal places does it not? Keep up the good work!

  2. Jake Barnes says:

    Thanks Mark, and welcome!

  3. pete says:

    Seeing as you’ve linked to your example over at Tim’s place, thought it’d be worth coming over & correcting your error 😉
    There is no height to the Tashkent Tower measurable to 1cm. Whatever method you’re using. Because the Tashkent Tower isn’t a stable height. It’ll be taller in the evening than it was at dawn. At 378m, likely several centimeters. Expansion. So no height, expressed any more accurately than half the expansion range of the tower, is of any greater use.
    Which is the other half of these spurious claims of accuracy. Few of the things being measured are dynamically stable to the accuracies claimed for the purpose of the claim.

  4. Jake Barnes says:

    Quite correct, Pete. I was struggling for a good analogy, and decided on that one for some reason. But yes, fair point.

    Welcome, by the way.