Methodology: how these numbers are calculated

Every number on this site comes from one of four things: a definition, a published formula, a published table's parameters, or your own input. Nothing is estimated by guesswork, and nothing is hidden behind a black box. This page states the rules; /sources gives the citations.

Pace: the definition

Pace is time divided by distance. That is the whole of it:

pace = finish time ÷ distance

Two consequences that decide how the rest of the site is built:

• Distance and time must be converted exactly once. 42.195 km is 42.195 × 1000 metres, and one mile is exactly 1609.344 metres. We never convert kilometres to miles and back — a round trip accumulates rounding error, and the result looks plausible enough that nobody notices.
• Pace and distance convert in opposite directions. A distance of 42.195 km is 26.219 miles (smaller number, bigger unit's count is lower); a pace of 4:00 per kilometre is 6:26 per mile (bigger number — the same time spread over a longer unit). Confusing the two produces a pace that is merely "fast", not obviously wrong. The code keeps pace conversions separate from distance conversions for exactly this reason.

Rounding: display only

The site stores and compares integer milliseconds and rounds once, when displaying. Nothing is rounded before a comparison. Concretely:

If two people compute the same pace and get 4:59 and 5:00, the difference is almost always where the rounding happened — not the arithmetic.

Target times: which band a result reaches

The charts group finish times into target bands (sub-20:00, sub-3:00, and so on). The boundary is inclusive: a band sub-3:00 means a finish time of 3:00:00 or faster. 3:00:00.001 is not inside it, and the page says so in the table itself rather than leaving you to guess.

Bands exist per distance and are listed on each chart's page. They are editorial choices, not formula outputs: they are the round numbers people actually search for.

VDOT and training paces

The VDOT calculator uses the two equations published by Jack Daniels (with Jimmy Gilbert) in Daniels' Running Formula:

VO2(v)     = -4.60 + 0.182258·v + 0.000104·v²          v in metres per minute
%VO2max(t) = 0.80 + 0.1894393·e^(-0.012778·t)
           + 0.2989558·e^(-0.1932605·t)                t in minutes
VDOT       = VO2(race speed) ÷ %VO2max(race duration)

Training paces are the speeds at fixed percentages of that VDOT: easy 59–74%, marathon 75–84%, threshold 83–88%, interval 97–100%, repetition 105–110% of VO2max. Each range is shown as a pace range in both units.

Two limits, stated plainly:

• VDOT is back-calculated from a race result. It is not a measurement, and it is not the same thing as a laboratory VO2max test. A result from a hot, hilly or badly paced race produces a VDOT that describes that day.
• The printed VDOT tables round paces to 5 seconds. This site solves the equations directly and does not round to 5 seconds, so the same VDOT can differ from the book by a few seconds per kilometre. If you are following the book page by page, read the book's table; if you want the value behind it, this calculator shows the derivation. The detail is recorded in the data file behind the page.

Race time predictions

The predictor uses Riegel's formula:

T₂ = T₁ × (D₂/D₁)^1.06

The exponent 1.06 is empirical, fitted to record performances. It works well when the two distances are within a factor of about four and the result came from a comparable race effort. It drifts when you stretch it: for ultramarathon distances it tends to understate the slowdown, because it does not know about fuelling, terrain or the accumulation of fatigue. Never treat a predicted time as a target you are owed — treat it as a description of what your current fitness implies.

What we do not calculate

This site deliberately has no page for:

• maximum heart rate (including 220 − age)
• heart rate training zones
• VO2max as a medical or physiological measurement
• Karvonen or heart-rate-reserve prescriptions
• training plans, injury advice or fuelling plans

The reason is not modesty about arithmetic. These are physiological claims: they carry individual error bands of roughly ±10–15 beats per minute for the heart-rate formulas, and the 220 − age shortcut has a known systematic bias. Presenting a number like that without the qualifications that make it usable is misinformation, and qualifications alone are not enough — they need a named author with verifiable credentials, an error band stated next to the number, and no individual prescription. That is a different kind of site from this one, so the boundary is drawn here rather than halfway.

The dividing line we do work on is the mathematical layer (time, distance, pace, splits, unit conversion) and the training-science layer where a primary source exists and its version and assumptions can be stated (the training paces above). Every formula page links to its source.

Reproducing any number on this site

1. Take the finish time in seconds and the distance in metres.
2. Convert the distance to the unit you want (divide by 1000 for km, by 1609.344 for miles).
3. Divide the time by that distance figure.
4. Round only the final displayed value.

Worked example — a 20:00 5K:

1200 s ÷ 5.000 km = 240 s/km = 4:00 /km
1200 s ÷ 3.10686 mi = 386.24 s/mi = 6:26 /mi
VDOT = 47.46 ÷ 0.95296 ≈ 49.8