Sources and formula references
Every formula used on this site is listed here with its source, the version we used, and what it does not cover. If a number on another page cannot be traced back to a row in this table, that is a bug — please report it.
Formula by formula
| What | Formula | Source | Version used here |
|---|---|---|---|
| Pace | pace = time ÷ distance |
Definition — no source needed | — |
| Unit conversion | 1 mile = 1609.344 m, 1 km = 1000 m |
International yard and pound agreement (1959); SI | Exact values, no rounding |
| Splits | segment time = total time × segment distance ÷ total distance |
Follows from the pace definition (even pacing) | — |
| VDOT | VO2(v) = -4.60 + 0.182258·v + 0.000104·v²%VO2max(t) = 0.80 + 0.1894393·e^(-0.012778·t) + 0.2989558·e^(-0.1932605·t)VDOT = VO2(race speed) ÷ %VO2max(race duration) |
Jack Daniels with Jimmy Gilbert, Daniels' Running Formula | 4th edition (2021) |
| Training paces | Speed at 59–74% (easy), 75–84% (marathon), 83–88% (threshold), 97–100% (interval), 105–110% (repetition) of the VO2max implied by VDOT | Same as VDOT | 4th edition (2021) |
| Race time prediction | T₂ = T₁ × (D₂/D₁)^1.06 |
Riegel, P. S. (1981), Athletic Records and Human Endurance, American Scientist 69(3) | Exponent 1.06 as published |
The VDOT equations in full
The two equations are worth spelling out because everything in the training-pace table is derived from them rather than looked up:
VO2(v)is the oxygen cost of running atvmetres per minute. It is a quadratic, so doubling your speed more than doubles the oxygen cost.%VO2max(t)is the fraction of VO2max you can sustain fortminutes. The two exponential terms are why a 5K and a marathon run at the same pace are not the same effort: the sustainable fraction falls as the duration grows.- Dividing the first by the second gives the VO2max that would be consistent with your result — the VDOT.
Two known deviations from print, both deliberate:
- We do not round paces to 5 seconds. The printed tables do. Expect differences of a few seconds per kilometre on some rows.
- We compute, we do not transcribe. A transcription error in a table is invisible; an equation can be checked by anyone with a calculator. The percentages and the coefficients are stored in the site's data file, so the exact numbers used are reviewable rather than buried in code.
Because of (1), if you are working through the book page by page, use the book's tables. This site's value is the derivation and the cross-distance arithmetic around it, not a reprint.
What is deliberately missing
There are no sources listed for maximum heart rate, heart rate zones, VO2max as a laboratory measurement, or Karvonen formulas, because this site does not calculate them at all. That decision is explained on methodology.
How to check any number here
Take a 20:00 5K:
race speed = 5000 m ÷ 1200 s × 60 = 250.0 m/min
VO2(250) = -4.60 + 0.182258×250 + 0.000104×250² = 47.4645
%VO2max(20) = 0.80 + 0.1894393×e^(-0.25556) + 0.2989558×e^(-3.86520)
= 0.80 + 0.146704 + 0.006254 = 0.952958
VDOT = 47.4645 ÷ 0.952958 = 49.81
The VDOT calculator shows 49.8 for that input; the training paces below it come from the same two equations with the percentages above.
Why publish the derivation instead of just the value?
Because a number you cannot check is a number you have to trust. Publishing the equations means a reader who disagrees with a training pace can find out exactly which assumption they disagree with — the coefficient, the percentage, or the input result.
Do you use any data that changes year to year?
Not yet. Race entry standards (for example the Boston Marathon qualifying times) change annually and belong on a page with their retrieval date attached. When such a section is added, it will carry the year, the effective window and the date the figures were retrieved — a chart without those three things is worse than no chart.