The Maths Behind Record Breaking

I came across an interesting article the other day looking at the link between record breaking and the “harmonic series” 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + .....

If you went through a list of 100 random numbers, have a guess as to how often would you expect to break the record for the largest number so far? Let’s take a simpler example.

Here are 10 random numbers:

3, 8, 5, 7, 2, 5, 3, 9, 1, 7

Clearly the first number is a record; the second number, 8, is also a record; and then the eighth number 9 is also a record. So there were three record numbers in the set of 10. But how does the harmonic series fit in?

Consider just one number – clearly it’s a record breaker.
Consider two numbers; there’s a half chance the second is a record breaker, so the expected number of record breakers is 1 + \frac{1}{2} = 1{\textstyle{1 \over 2}}.
Consider three numbers – we now have a one in three chance that the third is a record breaker, so the expected number is now 1 + \frac{1}{2} + \frac{1}{3} = 1{\textstyle{5 \over 6}}.

I think you can see where this is going! The harmonic series isn’t like an arithmetic or geometric progression – there is no formula for the sum of n terms, nor does it converge in the same way that, say, 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... converges to 2. It diverges, but very slowly.

Let’s use the function R(n) to represent the number of times we would expect record to be broken in a list of n random numbers. So, R(n) = 1 + \frac{1}{2} + \frac{1}{3} + .... + \frac{1}{n}. We have already seen that R(3) = 1{\textstyle{5 \over 6}} = 1.83 (to 3SF). For our set of 10 random numbers R(10) = 2.93, so the three records we got is about what we would expect. But R(100) is not much more: just 5.19 (was your guess anywhere close?). To try this out, I created a list of 100 random numbers in a spreadsheet and wrote a short bit of code to count the records. I did this several times, and here are the results:

7, 3, 7, 6, 3, 7, 3, 8, 4, 4, 6, 6, 4, 5, 6 – and the mean of these is 5.27, very close to our expected value of 5.19.

R(1000) = 7.49 and R(5000) = 9.09. Can we therefore say that, in 5000 years of temperature records, we might only expect a record high (or record low, for that matter) about 9 times. Sounds ridiculous, doesn’t it? It would be true if temperatures were random numbers, but of course they aren’t – and we might expect temperature records to be broken more frequently as global warming continues. However, I have just looked up the temperature records for London in the second half of the 20th century. I selected the maximum temperature for January every year, and the record was broken only 3 times (R(50) = 4.50). Since I had the figures, I also counted how often the record for the lowest maximum temperature was broken, and that came out to 5 times. Isn’t it great when the practical results match the theory!

 

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