This can be used to prove I've gone insane. A couple of scientists will hate the imprecision and everyone else will stop reading once they hit the word "kurtosis."
[Pause, waiting for people to click away]
My definition for training: 1/24 x fourth partial derivative (w/ respect to time) of kurtosis minus 1/6 third partial of skewness plus 1/2 second partial of variance minus first partial of mean is positive.
Here's how it works in laymen's terms. Imagine your training as a set of data points that form a curve not too different from a normal distribution. Your races are outliers in performance, far removed from your average days, but you want to increase the possibility of a performance better than your personal record.
There are several ways to do this. First, you could simply increase the number of data points. This means running a lot more and, keeping the same percentage of races, racing a lot more. This is probably how you improved when you first started running. The likelihood of an unusually good race came from not knowing what "good" actually is. This is also the high volume approach to training, which is what was popularized in the 1960's and 1970's.
Second, you can increase the mean (average). This is what people commonly try to do, to nudge all their workouts just a bit, so the average improves and, theoretically, their best performances move as well. The problem with this is that it's far easier to improve your easy days than your hard days, so you end up having no truly easy days and, after a brief improvement, you fall apart.
Third, you can increase the variance (the square of the standard deviation, if you prefer). This can be done by removing the average runs, doing just easy and hard runs, creating a bimodal distribution.
This method is akin to the current idea of polarized training.
Fourth, you can increase the skew [bear with me on this]. A properly skewed distribution would look like this:
This can be done in a number of ways. You could eliminate the easiest runs. You could make the easiest runs harder and make the average runs easier. You could increase the number of hard runs. You could combine any or all of these approaches.
This is similar to the approach commonly advocated for masters runners, to do mostly high-quality workouts, even if it means doing far fewer workouts. It is also what is seen in the peaking/tapering phase of many workout schedules.
The fifth way is to increase the kurtosis, which is a measure of how many outliers there are, which would seem to be exactly what we want. This can be done by racing more, by having a few extremely easy runs (the Long Slow Distance method) or by pushing almost all workouts to close to the average, making anything else an outlier. A properly leptokurtotic (yes, that's the word) distribution looks like this:
So what am I proposing?
All of these approaches should be used, in sequence. Have a period in which you do a lot of average runs, then a period when you don't. Have a period when you drop your easy runs and have one when you introduce one very long extremely easy run. Have a period when you try to improve each element at the same time and a period when you focus on one specific element. And, when in doubt, race more.
1 day ago