I came across an interesting article the other day looking at the link between record breaking and the “harmonic series” .
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 .
Consider three numbers – we now have a one in three chance that the third is a record breaker, so the expected number is now .
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, 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, . We have already seen that (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!