Why statistics makes you feel old

Yesterday I was cycling up one of the local hills that is often on my route home when I realised that it’s increasingly unlikely that I’ll ever beat my personal record (PB) for the climb. Of course, I’m getting older and presumably less strong and that in itself makes it less likely, but what struck me was that even without a decline in athletic capability the chances of setting a new PB decrease after a certain time. Let me explain…

Imagine that you do the same event most weeks – say a park run or a TT circuit – over a number of years. Further, let’s suppose that your “true average time” for this is, say, 25 minutes but that your actual time on any particular week is affected by a number of varying factors. These could be, for example, the wind and weather, the volume of traffic you encounter, how well you slept, the state of your kit (bike, wheels, shoes…) and so forth. Perhaps we can summarise these into three separate variables for (1) environment (wind direction, traffic…), (2) personal condition (restedness, current form…) and (3) state of equipment (which bike and wheels you used…). It doesn’t matter what the variables actually are so long as we suppose, for the purpose of this thought experiment, that they are normal, trend-less and independent. Then, we can add all the random factors together into a single “net random factor” and this will also be normal.

Now in reality you may immediately object that in real life the “personal condition” variable is not trend-less because we initially get fitter as we start a new form of exercise and then over time we age and get less fit. However, my key point is that it will feel like this even if it’s not the case! To see this, let’s suppose that our fitness/form varies from week to week according to a variety of factors but, on average, doesn’t change over time.

Given these assumptions, our weekly time will always be 25 minutes plus or minus a varying amount. In the first few weeks it’s in the nature of statistical fluctuation that we will probably quite often set a PB. However, as time passes we’ll have those occasional “magic days” when the wind and air pressure are favourable, we have only green lights, we’re on our fast bike or in new shoes, and we’re rested and well trained. When all of those factors align we’ll set an exceptional time. In terms of our model, the net random factor will be at a rare and favourable extreme value. Because such values are rare, the time between PB’s will increase.

To illustrate this, I modelled it for an athlete who begins doing the regular weekly exercise at age 20 and continues each week for 50 years. This chart shows five sample random paths, which we can think of as corresponding to five athletes all starting at age 20 and performing the same weekly exercise until age 70:

As you can see, in each of the five cases, before the age of 50 (30 years on from the start at age 20) the athlete has attained a PB that isn’t seen again before age 70. For example, the red line represents an athlete who sets a couple of PB’s within the first five years, then a new one at around age 40 that isn’t matched again in the remaining 30 years. The green athlete has the most encouraging career and the orange athlete the least, while the violet athlete manages the latest PB at age 49.

Over 1,000 such random paths the average age of the last PB is 45, halfway through the athlete’s career. Remember, this is just an artefact of statistical randomness and happens even on the assumption of constant athletic ability!

The psychological impact of this is obvious. Most of us judge our “true” capability to be that recorded by our best times. As those PB’s recede into the past, even if our average times stay the same, there is a perhaps a tendency that it will induce a feeling of nostalgia and a worry of ageing. In truth, a new PB could (in this fantasy world of constant physiology) occur just as easily tomorrow as it did 20 years ago. The problem with ageing, viewed statistically and irrespective of physiology, is that we may not have enough tomorrows left in which to replicate the most extreme favourable conditions of our plentiful yesterdays.