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This calculator predicts the time required to run a specified distance, using known times for other distances. Generate your own prediction by filling in the following table. Using a least-squares fit of your input, the time required to run the designated distance will be calculated. In addition, the same calculation will be done for several commonly-raced distances.

Entries left as zeroes are ignored, as are nonsensical entries. If entries are conceivable but unlikely, they are used but flagged by regurgitating in red (too fast) or blue (too slow). If the output seems outrageous, check whether your inputs were all correct.

Desired prediction:
Data (known times for various distances):
in (HH:MM:SS)
in
in
in
in
in
in
in
in
in
in
in
in
in
in

Mathematical details

The technique used is a least squares fit to various curves:

  1. t = A*d + B*d*log(d)
    (a formula suggested by the data.)
  2. t/d = A + B*log(d)
    (same equation, but least squares is on t/d, so slightly different result)
  3. log(t) = log(A) + B*log(d)
    (For t=A*d**B, but least squares is on log(t) for tractability)
Least squares is straightforward on the above curves -- it amounts to summing on d*d*log(d) and the like, and then solving a 2x2 linear system of equations. I can handle that! But remember, these equations are merely simplistic models of a runner's endurance. The predictions should be regarded as approximate, and very approximate if
  • The input data is approximate.
  • The input data is not reasonably close to being the runner's capabilities at a single point in time, e.g. best times from a single season. If you are improving every year, and the mile time is from 2 years ago but the 5K time is last week, the mile time is probably too slow.
  • The requested prediction is too far from the input data. An example of this would be predicting a marathon time based on a 400m and an 800m.

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