We are interested in predicting and evaluating times for long distance races between 3K and 50K. In general, empirical results have suggested that between the distance you run and the time it takes you to run the distance is a log-linear relationship (there seems to be a linear relation between the speed you ran and the log of the distance you run). Most pace-predictors found on the web( Race Pace Calculators and Predictor Web-Sources) are based on the formulas that will be explained below. The mentioned log-linear relation between speed and distance basically means the following: if you want to compare your 5K, 10K, and 20K performance, your percentage of slow down from a 5K to a 10K is the same as your slowdown from a 10K to a 20K, although the difference with respect to distances is 5000 meters in one case and 10000 meters in the other case. However, as you might remember from your math-classes:

log(20000)-log(10000)=log(20000/10000)=log(2)= log(10000/5000)=log(10000)-log(5000)The last paragraph mentioned the concept of a slow down rate; in our particular discussion slow down rate refers to the amount you slow down if you double the distance. For example, if you ran a 5K in 20 minutes and 10K in 42 minutes your slow-down rate is 42/20 over 2, which is 2.1 over 2 which is 1.05 (5% slow down, you were expected to run 40 minutes, but it took you 2 minutes (5%) longer). That is, in this particular case, if you double your distance your time will be 2.1 times your time over the shorter distance. The assumption, stated in the previous paragraph, is that your slow down rate remains more or less constant if you double the distances: this is, no matter if you double the distance from 5K to 10K or from 10K to 20K you slow down by the same percentage...

The next question is what is your slow down rate? In general, it seems that picking a number between 5% to 9% is a good approximation for your slow down rate; it has been observed that 5% is a good approximation for elite runners for predicting shorter distances and that 5.5% is a good approximation for predicting marathon time of elite runners; however, non-elite runners seem to have higher slow down rates (e.g. your slow down rate might be 7%) which can be explainted due to the lack of traning, speed, and individual differences, if compared to elite runners.

This relationship can now be used to predict your times using times that you ran over other distances. Let:sl-rate be your slow-down rate

dist1 be a distance you ran before

time1 the time you ran over the distance dist1

dist2 be the distance for which you want predict your time

Based on this information your time time2 over distance dist2 can be predicted based on your time time1 over distance dist1 as follows(** denotes power; e.g. 2**4=16):

time2:= time1*(dist2/dist1)*((sl-rate)**((log(dist2/dist1)/log(2))For example, if you run a 5K in exactly 19 minutes, and you want to predict you 10K time assuming a slow down rate of 1.05 (5%). You obtain for your 10K time:

19*2*(1.05**(log(2)/log(2))=19*2*(1.05**1)=19*2.1=39.9which translates to 39 minutes and 54 seconds. Now, let us assume you want to predict you 8K time from your 5K time. In this case we receive:

19*1.6*(1.05**(log(1.6)/log(2)))=19*1.6*(1.05**0.67)=19*1.6*1.0336=31.42which corresponds to running a 8K-race in 31 minutes and 25 seconds. If you want to predict your 20K time based on your 5K time you obtain:

19*4*(1.05**2)=19*4*1.05*1.05=83.79which corresponds to running a 20K-race in 83 minutes and 47 seconds. The above formula can also be used to predict times for shorter distances from longer distances, e.g. your 3K time (from your 5K time). In this case, you receive a negative exponent for the slow down rate, which has to be expected: because the distance is shorter, your slow down rate becomes a speed up rate.

In my case, I observed that a slowdown rate of close to 5.5% (1.055) is a very good predictor for distances between 800 meters to 25000 meters, but not for my marathon time (at least I never got close to the times, I was supposed to run, perhaps due to the lack of total mileage in my training and due to the bad weather conditions I faced in my last two marathons).

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