Race Time Prediction Across Distances: VDOT, Riegel, and Their Failure Modes

Race time calculators are the most-used and least-understood tool in distance running. Punch in a 5k, get a marathon prediction. The number is plausible, the runner trains for it, the marathon comes in twenty minutes slower than predicted. The calculator wasn’t lying. It was answering a different question than the runner thought.

The two dominant prediction models (Riegel's 1.06-exponent power law and Daniels VDOT), the alternatives, and the specific failure modes that produce optimistic predictions.

The Riegel formula and what the 1.06 means

Pete Riegel's 1981 paper in American Scientist^[1] fit a power law to world-record times across distances from 400 m to the marathon. The result is one equation that has held up for four decades:

T2 = T1 × (D2 / D1)^1.06

  T1 = your known time over distance D1
  D2 = the target distance
  T2 = predicted time at D2
  1.06 = Riegel's fatigue exponent

The 1.06 exponent encodes pace decay as distance grows. 1.0 means same pace at any distance (impossible). 1.10 means steep decay (under-trained marathoners). 1.06 was the population-level best fit on Riegel’s world-record dataset.

Fit on world records means the model implicitly describes runners whose lactate threshold sits at a high fraction of VO2 max and whose training volume supports the predicted time at every distance. Reasonable for elites, less so for the rest of us.

The Daniels VDOT system

Jack Daniels' VDOT system^[2] takes a different approach. Empirical curves linking race times across distances to a fitness index (VDOT, a pseudo-VO2-max score). Each VDOT value corresponds to a row: 5k time, 10k time, half time, marathon time, plus recommended easy, marathon, threshold, interval, and repetition training paces.

Two consequences of the lookup-table approach:

1. VDOT was fit on trained runners with adequate volume. Daniels was a coach. The curves were calibrated against runners with aerobic-base training behind them. For that profile, VDOT is more accurate than Riegel because it implicitly assumes the training is there.
2. VDOT does not extrapolate beyond marathon. The tables stop at 42.195 km. No 50 km row, no 100 km row. Ultra-distance physiology departs from marathon physiology in ways a single-fitness-index model cannot capture.

The Race Time Predictor exposes both the Riegel exponent and a VDOT-style mode so you can compare predictions from the same input.

Cameron, Pugh, and fatigue-index alternatives

• Cameron's 1998 formula. A time-adjusted constant producing predictions slightly slower than Riegel at the marathon end. Useful when Riegel feels too aggressive but VDOT is unavailable. Accuracy gain over Riegel is 1 to 2 percent.
• Pugh / critical-power models (Monod-Scherrer). Two-parameter fits requiring multiple known race times. Three or four near-maximal results across a wide distance range beat Riegel by 1 to 2 percent. With one race time, no advantage.
• Personalised fatigue-index regression. Two recent race results solve for your own exponent: exponent = ln(T2/T1) / ln(D2/D1). Typical recreational runner: 1.07 to 1.09. Aerobically strong: 1.05 to 1.06. Under-trained marathoner: above 1.10.

For a single known race time, Riegel with a personalised exponent or VDOT is the starting point. Critical-power fits are more accurate when you have the data to feed them.

Where Riegel breaks: marathon prediction for under-volume runners

The most common Riegel failure is using a sharp 10k or half-marathon time to predict a marathon for a runner with insufficient weekly volume. The model assumes the runner can hold near-lactate-threshold pace for 42 km, which requires aerobic base, glycogen storage, and muscular durability that only volume builds.^[3]

Practical threshold: under 40 mpw (~64 km/week) of consistent volume in the 12 weeks before a marathon, Riegel overstates the achievable time by 10 to 30 minutes. A 3:00 marathon prediction from a 39:00 10k assumes 4:15/km for 42 km; runners training 30 mpw discover at kilometre 32 that the prediction was a fitness ceiling, not a current state.

Where VDOT breaks: ultras and variable terrain

VDOT beats Riegel for trained runners up to the marathon. It fails in two predictable ways:

• Ultras. Tables stop at the marathon. Ultra performance is dominated by factors VDOT doesn’t measure: GI tolerance, fueling strategy, heat acclimatisation, muscular damage, sleep deprivation in 12+ hour events. A 3:00 marathoner has a 50 km range, not a predictable time.
• Hilly and technical terrain. VDOT assumes a flat certified course. A trail 50k with 1,500 m of vertical gain is a different physiological event from a road 50k. Course corrections (vertical gain × 8 to 10 sec/m, plus 10 to 30 percent technical-terrain discount) matter more than the underlying VDOT score.

The training-volume confounder

A 5k time predicts a marathon time only if your weekly volume supports it. The cleanest way to see this is to compare two runners with identical 5k times but different volumes:

Runner A: 5k 22:00, weekly volume 25 km
Runner B: 5k 22:00, weekly volume 65 km

Riegel marathon prediction (1.06 exponent)
  Both:                              ~3:32

Realistic marathon time
  Runner A (under-volume):           3:55 to 4:10
  Runner B (volume-supported):       3:30 to 3:38

Same 5k input, two different realistic marathons. The 5k tests aerobic power and LT pace, both achievable on modest volume. The marathon tests sustained capacity at a high fraction of LT for three to four hours, which isn’t.

Trust the calculator only if your weekly volume in the 12 weeks before the race is at least 1.5 to 2.0× the marathon distance. For a 4:00 marathon target, 60 to 80 km/week. For a 3:00 marathon target, 80 to 110 km/week.

Worked example: 5k of 22:00 to half-marathon

A runner has a recent 5k PB of 22:00 and wants to predict a half-marathon. Compare predictions across volumes:

Inputs
  D1 = 5 km    T1 = 1320 s (22:00)
  D2 = 21.1 km

Riegel default (1.06)
  T2 = 1320 × (21.1/5)^1.06 = 6044 s = 1:40:44

VDOT lookup (5k 22:00 → VDOT ~46)
  Half-marathon prediction:           1:41:11

Realistic half time by training volume
  < 25 km/week (low volume):          1:48 to 1:55
  35 to 50 km/week (moderate):        1:42 to 1:46
  > 60 km/week (high volume):         1:40 to 1:43

Riegel and VDOT agree within 30 seconds because the half is close enough to the 5k that the under-trained correction hasn’t kicked in. The realistic time depends on volume; calculator predictions are only achievable at 50+ km/week. The runner training 25 km/week runs 1:50, not 1:40, and no formula tells them that.

Worked example: 10k of 45:00 to marathon

A more dangerous extrapolation: 10k to marathon. The distance ratio is 4.2x, near the edge of where Riegel was fit. Volume becomes the dominant predictor.

Inputs
  D1 = 10 km   T1 = 2700 s (45:00)
  D2 = 42.195 km

Riegel default (1.06)
  T2 = 2700 × (42.195/10)^1.06 = 12504 s = 3:28:24

Riegel with under-volume correction (1.10)
  T2 = 2700 × (42.195/10)^1.10 = 13080 s = 3:38:00

Riegel with severely-under-volume correction (1.15)
  T2 = 2700 × (42.195/10)^1.15 = 13830 s = 3:50:30

VDOT lookup (10k 45:00 → VDOT ~46)
  Marathon prediction:                3:29:05

Realistic marathon time by training volume
  < 40 km/week (under-volume):        3:50 to 4:10
  50 to 65 km/week (moderate):        3:35 to 3:45
  > 75 km/week (high volume):         3:28 to 3:38

Riegel default and VDOT both predict ~3:28 to 3:29 from a 45:00 10k. Achievable only at 75+ km/week. At 40 km/week the realistic marathon is 25 to 40 minutes slower. The 1.10-corrected Riegel (3:38) is honest for moderate-volume runners; the 1.15 correction (3:50) is the floor for first-time marathoners with thin volume.

Race-specific failure modes: heat, altitude, hills

Even calibrated to your fitness, race-day conditions move the actual time outside the prediction. Course-and-conditions corrections:^[6]

• Heat. Above 14 to 16 °C wet-bulb, marathon times degrade 1.5 to 2 percent per °C above optimum (~6 to 10 °C). A 3:00 target in 24 °C with humidity is closer to 3:10 to 3:15. Heat acclimatisation closes about half the gap with two to three weeks of training in similar conditions.
• Altitude. Above 1,500 m, oxygen availability degrades aerobic power. 30 to 50 seconds per 1,000 m on a marathon for sea-level-trained runners. Boulder (1,650 m), Mexico City (2,250 m), Albuquerque (1,620 m) all sit in the meaningful-correction zone. Derate predicted times 1 to 3 percent.
• Hilly courses. Vertical gain costs 8 to 10 seconds per metre climbed (recovers 30 to 40 percent on the descent). A 200 m vertical-gain marathon costs 1:30 to 2:00 over a flat course at the same fitness. Boston (~240 m gross gain) runs 4 to 8 minutes slower than a flat marathon.

These corrections are independent of the prediction model. A perfectly-calibrated VDOT estimate for a flat course in 8 °C still misses by 5 to 10 minutes on a hilly warm race.

The gold standard: adjacent-distance race results plus training load

The most accurate prediction isn’t from any formula. It’s from a recent race at a distance close to the target, plus an honest assessment of whether your training supports holding that pace longer.

• Target 5k: a recent 3k or mile time trial, plus current weekly volume.
• Target 10k: a recent 5k race, plus 4 to 6 weeks of 30+ km/week.
• Target half-marathon: a recent 10k race, plus 8 to 12 weeks of 40+ km/week with three runs above 14 km.
• Target marathon: a recent half-marathon race, plus 12+ weeks of 60+ km/week with three to five runs above 28 km.

Use Riegel or VDOT to project from the adjacent-distance race, then derate by 3 to 8 percent if your volume sits below the threshold for the target distance. The derated number is your honest target. The non-derated number is what you could run if your training had been adequate.

Reverse-calibrating your own exponent

Two recent near-maximal race results solve for your personal fatigue exponent:

exponent = ln(T2/T1) / ln(D2/D1)

Example
  10k    T1 = 45:00  (2700 s)
  Half   T2 = 1:40:00 (6000 s)

  ln(6000/2700) = 0.7985
  ln(21.1/10)   = 0.7467

  personal exponent = 0.7985 / 0.7467 = 1.069

For this runner, 1.069 is slightly steeper than Riegel’s default. Marathon prediction with 1.069 produces 3:32 instead of 3:28: a 4-minute difference that shifts pacing strategy.

Cross-link tools

• Race Time Predictor exposes the Riegel exponent as input: 1.06 (default), 1.10 (under-volume marathoner), or a personalised value from your race history.
• Running Pace Calculator converts a predicted time into kilometre and mile splits for race-day pacing strips.
• VO2 Max Estimator gives the underlying VDOT-style fitness number from a recent race.

Summary

• Riegel’s 1.06 is decent for 5k to 10k and 10k to half; overstates marathon time for under-trained runners.
• VDOT beats Riegel for trained runners up to the marathon; doesn’t extrapolate beyond.
• Cameron and critical-power models offer little over Riegel or VDOT for a single race time.
• Volume is the confounder. A 5k predicts a marathon only if weekly volume supports LT-pace for 42 km.
• Heat, altitude, and hills move race times outside the prediction independent of the formula.
• The honest target is the formula prediction, derated for volume and adjusted for course and conditions.