How Accurate Is the Riegel Formula? Testing Against Real Race Data

In 1981, Robert Riegel published a short paper in American Scientist that most runners have never heard of, using data from athletic records that most runners use every day. The formula he derived — T₂ = T₁ × (D₂/D₁)^1.06 — is behind every race time predictor you've used. It's in Garmin. It's in Strava. It's in every "what could I run?" calculator on the internet.

What none of those tools explain is how accurate it actually is, under what conditions it fails, and what the alternatives do differently. This guide examines the formula analytically and empirically.

The paper behind the formula

Riegel analysed world record progressions from 1 mile to 100 miles, fitting a power law model to the relationship between distance and time. The model assumes:

T = a × D^b

Where T is race time, D is distance, a is a constant (different for each athlete), and b is the exponent that governs how time scales with distance. Riegel found that b ≈ 1.06 fit the world record data well, and — critically — that b was similar across many runners, even though their absolute speeds differed.

Rearranging the model to eliminate the constant a:

T₂ / T₁ = (D₂ / D₁)^b

Therefore: T₂ = T₁ × (D₂/D₁)^1.06

This is the Riegel formula. It predicts T₂ from any known T₁ and D₁, for any target distance D₂.

What the exponent 1.06 means

The exponent greater than 1 is the formula's core insight. If b were exactly 1, doubling the distance would exactly double the time — you'd run twice as far in twice the time at the same pace. Every long-distance runner knows this isn't how it works. Longer races are harder per unit distance, not just harder overall.

At b = 1.06:

• Doubling the distance costs 2^1.06 = 2.085× the time (4.2% more than simple doubling)
• Quadrupling the distance (e.g., 10K to marathon) costs 4^1.06 = 4.33× the time

In concrete terms: if you run 10K in 40:00, the Riegel prediction for a marathon is:

T₂ = 40 × (42.195/10)^1.06 = 40 × 4.2195^1.06 = 40 × 4.637 ≈ 185 minutes = 3:05:00

Compare to naive doubling + extra: 40 × 4.2195 = 168.78 minutes = 2:48:47. The Riegel formula adds 16 minutes to account for the fatigue premium.

The calculator

Use a recent race result to test the Riegel formula predictions against your own experience:

Empirical accuracy: where does 1.06 actually perform?

The original 1.06 exponent was fitted to world record data — the fastest humans ever to race those distances, in optimal conditions. Applying it to recreational runners introduces systematic biases.

Well-trained runners, adjacent distances

For trained runners going from 5K to 10K, or 10K to half marathon, the formula is typically accurate within ±3–5%. This is its best use case:

• A 19:00 5K runner predicted to run 39:33 10K: realistic for a balanced runner
• A 42:00 10K runner predicted to run 1:32:47 half marathon: tends to be accurate within a few minutes

The marathon prediction gap

Marathon prediction from a half marathon is where the formula gets more scrutiny. The standard Riegel prediction for doubling from half to marathon multiplies the half time by approximately 2.112 (2^1.06). In practice, most recreational runners run the marathon in 2.10 to 2.18× their half marathon time, with the median around 2.14.

A runner who half-marathons in 1:45:00:

• Riegel prediction: 1:45:00 × 2.112 = 3:41:15
• Typical actual range: 3:41 to 3:48 for well-trained runners; 3:48+ for those undertrained in long runs

The formula slightly underestimates the fatigue penalty for most recreational marathon runners. Why? The 1.06 exponent was fit to competitive runners who typically have the endurance training, fuelling strategy, and pacing discipline to run close to their physiological potential. Recreational runners — especially first marathoners — more often run with suboptimal pacing (positive splitting) or insufficient long-run training, producing times slower than the formula predicts.

Large distance gaps

Predicting a marathon from a 5K amplifies the formula's error range. The 5K demands speed-power qualities that don't scale the same way as endurance. A runner who has trained specifically for 5K with lots of intervals may have a high 5K VDOT that produces an optimistic marathon prediction. Conversely, a runner with very high aerobic endurance but modest speed may outperform the Riegel prediction at marathon distance relative to their 5K time.

For any gap larger than two standard distances (e.g., 5K to marathon), treat the Riegel output as a rough order-of-magnitude estimate rather than a training target.

Alternative formulas

Riegel's 1.06 exponent isn't the only value researchers have fitted.

Cameron formula (1998)

Pete Cameron developed an empirical formula using a different functional form:

T₂ = T₁ × (ln(D₂ × 2 × π) / ln(D₁ × 2 × π))^1.07

Cameron's exponent (1.07) is slightly higher than Riegel's, predicting somewhat slower times at longer distances. For the half-to-marathon projection, Cameron gives a multiplication factor closer to 2.14 rather than 2.11, which is empirically closer to average recreational marathon performance.

Vickers-Vertosick study (2016)

The most data-rich alternative comes from a 2016 study by Vickers and Vertosick that analysed approximately 2.5 million finishes from the New York City Marathon alongside previous race results for the same runners. Their analysis found the optimal exponent varied by distance pair and by runner type, and that a factor between 1.07 and 1.09 better described the average recreational runner's marathon performance.

Their key finding: the population-average multiplier from half marathon to marathon is approximately 2.14, not the Riegel formula's 2.11. For predicting a personal marathon from a personal half, the Vickers-Vertosick model gave lower root mean square error than Riegel on their dataset.

When Riegel is still the right choice

Despite these alternatives, Riegel (1.06) remains the most common formula because:

• It's the best-known and most cited
• For trained runners at adjacent distances, accuracy differences versus alternatives are small
• Its simplicity makes it easy to apply and understand

The RunPaceLab finish time predictor implements Riegel by default. A comparison guide covering all four formulas (Riegel, Cameron, Purdy, Vickers-Vertosick) is available for runners who want to see how they diverge.

When the formula fails

Five specific scenarios produce significant prediction error:

1. The positive-splitting recreational marathoner: The formula predicts optimal performance. A runner who starts the marathon at 4:20/km when the Riegel prediction requires 4:40/km will blow up and run 4:00+ for a total well over Riegel's prediction. The formula isn't wrong — the runner's execution is wrong.

2. The undertrained first marathoner: The formula assumes endurance base proportional to the shorter-distance fitness. A fast 5K runner who hasn't done the long run work may predict a 3:15 marathon and run 3:45.

3. Temperature extremes: A 10K run in cool conditions may be the basis for predicting a marathon run in heat. The formula doesn't adjust for conditions. A 1°C increase in air temperature above 10°C is associated with approximately 1% increase in marathon time (Ely et al., 2007).

4. Injury or detraining: VDOT from six months ago predicts your fitness six months ago. Using an old race to predict an upcoming marathon — when fitness has declined — produces overoptimistic predictions.

5. Very short base distances: Predicting a marathon from a 1K time trial or 400m personal best compounds the distance gap error with the different physiological demands of sprinting.