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place. We might even honour them with the label of ‘expert’ because they were able to produce a number to such precision. But people who produce ‘precise’ figures often abuse our trust, and accidentally or deliberately they imply a level of confidence that is not justified. When we read that 59,723 people attended an Arsenal football match, we are being led to believe that an exact tally was taken as fans went through the turnstiles. So when we discover that the actual number in the ground was closer to 50,000, we feel like we have been conned. 3

       When it comes to the use of numbers to interpret the world, accuracy is more important than precision. After all, a measurement that is accurate but not precise can still be helpful. But a measurement that is precise but inaccurate is not just unhelpful, it can sometimes be dangerous.

       One of the unintended consequences of calculators is that they will give answers to as many decimal places as will fit on the screen – and in doing so, they tempt us to work to a level of precision that is often not justified.

      DRESSING UP NUMBERS WITH DRESSAGE

      In 2012 London hosted the Olympics. People throughout Britain celebrated as gold medallists stepped up on the podium in all sorts of sports in which previous GB athletes had rarely excelled.

There was particular joy when Charlotte Dujardin, who had worked her way up from being a stable girl to becoming an elite equestrian, claimed Britain’s first ever gold medal with her horse Valegro in the dressage event.

      Dujardin’s winning score from the judges was an impressive 90.089%.

      Sportspeople often talk about ‘giving it that extra ten per cent’, but in this case it seemed as if Dujardin had fine-tuned this, so she could put in that extra 10.001%. What was it that made her performance better than somebody who got, say, 90.088% instead?

      To understand where her three-decimal-place score came from, we need to look at how the judges allocated marks in that competition.

      In the dressage event at the London Olympics, the competitors were required to take their horse through a series of movements, which would be assessed by seven judges seated around the arena so they could view from different angles. The judges were scoring under 21 headings: 16 of them were ‘technical’ marks given for how well specific movements were carried out, and five of them ‘artistic’, applying to the overall performance, with descriptions such as ‘Rhythm, energy and elasticity’ and ‘Harmony between horse and rider’ (yes, really).

      Each item was marked out of 10, with half-marks allowed, but some scores were then given more weighting, and the five artistic scores were all multiplied by four. In total, each judge could give up to:

      240 technical marks + 160 artistic merit marks

      = 400 marks total.

It meant each rider had a total of 7 × 400 = 2,800 points to play for.

      There is an element of subjectivity when assessing how good a horse’s performance has been, so it is hardly surprising that the judges don’t all give the same mark. For a particular technical movement, one judge might give it an ‘8’, while another spots a slightly dropped shoulder and reckons it’s a ‘7’. In fact, in Dujardin’s case, the judges’ total marks ranged from 355 to 370, and when added together she got a total of 2,522.5 points out of a maximum possible 2,800.

      And this is where the percentage comes in, because her 2,522.5 total was then divided by 2,800 to give a score out of one hundred, a percentage:

      2,522.5 ÷ 2,800 = 90.089%.4

      Well, actually that’s not the exact number. It was really 90.089285714285714 … %.

      Indeed, this number never stops, the pattern 285714 repeats for ever. This is what happens when you divide a number by a multiple of 7. So Dujardin’s score had to be rounded, and the authorities who were responsible for the scoring system decided to round scores to three decimal places.

      What would have happened if Dujardin had been awarded half a mark less by the judges? She would have scored:

      2,522 ÷ 2,800 = 90.071%.

In other words, the precision of her actual score of 90.089 was misleading. It wasn’t possible to score any other mark between 90.089% and 90.071%, Dujardin didn’t give it an extra 0.001%, but rather she gave it that extra 0.018%. Quoting her score to two decimal places (i.e. 90.09%) was enough.

      The second decimal place is needed to guarantee that two contestants with different marks don’t end up with the same percentage, but it still gives a misleading sense of the accuracy of the scoring. In reality, each judge ‘measures’ the same performance differently. A half-mark disagreement in the artistic score (which is then multiplied by 4, remember) shifts the overall mark by 0.072%. And the actual discrepancies between the judges were bigger than that. For ‘Harmony between horse and rider’ one judge marked her 8 out of 10 while another gave her 9.5 out of 10.

      A NUMBER IS ONLY AS STRONG AS ITS WEAKEST LINK

      There’s a time and a place for quoting numbers to several decimal places, but dressage, and other sports in which the marking is subjective, isn’t one of them.

By using this scoring system, the judges were leaving us to assume that we were witnessing scoring of a precision equivalent to measuring a bookshelf to the nearest millimetre. Yet the tool they were using to measure that metaphorical bookshelf was a ruler that measured in 10-centimetre intervals. And it was worse than that, because it’s almost as if the judges each had different rulers and, on another day, that very same performance might have scored anywhere between, say, 89% and 92%. It was a score with potential for a lot of variability – more of which in the next section.

      All of this reveals an important principle when looking at statistical measurements of any type. In the same way that a chain is only as strong as its weakest link, a statistic is only as reliable as its most unreliable component. That dinosaur skeleton’s age of 69 million years and 22 days was made up of two components: one was accurate to the nearest million years, the other to the nearest day. Needless to say, the 22 days are irrelevant.

      BODY TEMPERATURE A BIT LOW? BLAME IT ON SPURIOUS PRECISION

       In 1871, a German physician by the name of Carl Reinhold Wunderlich published a ground-breaking report on his research into human body temperature. The main finding that he wanted to publicise was that the average person’s body temperature is 98.6 degrees Fahrenheit, though this figure will vary quite a bit from person to person.

      The figure of 98.6 °F has become gospel,5 the benchmark body temperature that parents have used ever since when checking if an unwell child has some sort of fever.

Except it turns out that Wunderlich didn’t publish the figure 98.6 °F. He was working in Celsius, and the figure he published was 37 °C, a rounded number, which he qualified by saying that it can vary by up to half a degree, depending on the individual and on where the temperature is taken (armpit or, ahem, orifice).

       The figure 98.6 came from the translation of Wunderlich’s report into English. At the time, Fahrenheit was the commonly used scale in Britain. To convert 37 °C to Fahrenheit, you multiply by 9, divide by 5 and add 32; i.e. 37 °C converts to 98.6 °F. So the English translation – which reached a far bigger audience than the German original, gave the figure 98.6 °F as the human norm. Technically, they were right to do this, but the decimal place created a misleading impression. If Wunderlich had quoted the temperature as 37.0 °C, it would have been reasonable to quote this as 98.6 °F, but Wunderlich deliberately didn’t quote his rough figure to the decimal place. For a figure that can vary by nearly a whole degree between healthy individuals,

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