JLR wrote:
This is actually quite meaningless without knowing how many prior marathons the typical participant has run.
Um, yeah wrote:
The specific quote is:
"about 25% of the runners here set new personal bests"
So just off of gut feel, how many different marathons would you say the average marathoner has done? Maybe more importantly, what is the median number of marathons you think marathoners have done?
Maybe without doing any research you think most have only done one, two or three. Maybe you here enough from your buddies about the marathon they did last month and the one they are doing next year that you think most have done six, seven or more marathons.
I really have no idea. I would guess that in a typical field the largest group of participants are first-timers. All of these participants technically set PRs - but the quote implies that these runners are excluded from the 25% fraction (i.e., "new personal best" implies there was a prior personal best).
Since marathoning is still relatively rare (i.e., most Americans have not run one yet), I would guess that the second largest fraction of participants will be second time marathon runners.
What's really important is the distribution of the number of prior marathons.
Let's say the distribution is geometric. This is a plausible distribution because it is the distribution of outcomes from a set of independent trials in which the one parameter of the distribuion, p, is the probability that any trial (e.g., a marathon) will be your last.
If we set p to p=0.5, which corresponds to an average number of marathons of 1/p=2, and do the math, then we can calculate that the probability of PRing on a course of average difficulty is 0.39 (excluding runners with no prior marathons, all of whom technically PR). If we up p to p=1/3 (3 average marathons), then the probability of PRing on an average course goes down to 0.32. For p=0.25 (4 marathons on average) the probability is 0.28 and for 5 average marathons the probability of PRing on a course of average difficulty is approximately 0.25.
Of course, the distribution of number of prior marathons may not be geometric. It could be a mixture of two distributions: for example, a distribution for "one and done" marathoners who will run one or two marathons in total - and then a distribution of "life-long" marathoners, for whom the total number of marathons run increases as a function of their age. Nonetheless, this example illustrates the general point that "25% of runners PR" doesn't mean much without extensive additional information.