How Race Time Predictors Work — The Maths Behind the Magic

You ran a half marathon in 1:52:30. Your friend wants to know: "What's your marathon time going to be?" You look it up on a race predictor website. It says 3:54:12. Is that number trustworthy?

The honest answer is: it depends entirely on which model generated it. Race time predictors are not magic — they are mathematical formulas built on specific assumptions about human physiology. Some assumptions hold broadly. Others fall apart at distance. Understanding the machinery helps you interpret the output intelligently.

The Riegel Power Law

The most widely used prediction model was published by Peter Riegel in a 1977 paper in American Scientist. The formula is elegant:

$t_2 = t_1 \times \left(\frac{d_2}{d_1}\right)^{1.06}$

Where:

• $t_1$ is your known race time at distance $d_1$
• $t_2$ is the predicted time at distance $d_2$
• $1.06$ is Riegel's empirically derived fatigue exponent

The exponent $1.06$ encodes a fundamental truth: running endurance degrades predictably with distance. If you double the distance, you do not simply double the time — you add a fatigue penalty. The longer the race, the larger that penalty compounds.

Example: You ran a 5K in 22:00.

$t_{\text{marathon}} = 22:00 \times \left(\frac{42.195}{5}\right)^{1.06}$

$= 1320 \times (8.439)^{1.06} = 1320 \times 9.588 \approx 12,656 \text{ sec} \approx 3:30:56$

The Fatigue Exponent Is Not Fixed

Riegel's original $1.06$ was derived from performance data across a general athletic population. But individual runners don't have identical fatigue profiles.

This is why two runners with the same 10K time can have very different marathon times. The athlete who trains primarily for speed and 5K races will have a higher effective exponent; the ultramarathon-focused athlete a lower one. The Critical Speed and Hybrid models attempt to capture this individual variation directly.

The VDOT Approach

Jack Daniels' VDOT model approaches prediction differently. Rather than extrapolating from a time-distance power law, it:

1. Derives an oxygen-cost function for running at velocity $v$:

$\dot{V}\text{O}_2 = -4.60 + 0.182258v + 0.000104v^2$

2. Computes your maximal aerobic capacity (VDOT) from any known race performance by solving for the value that makes the equation consistent.

3. Uses that VDOT to calculate the velocity at any given fraction of VO₂max — and therefore the time for any distance.

The key advantage: VDOT predictions are physiologically grounded. They assume the same metabolic machinery is at work across distances, calibrated through exhaustive experimental data published in Daniels' research.

Limitation: Like Riegel, VDOT is calibrated from short-to-medium distance race data. Marathon predictions from a 5K VDOT are less reliable because marathon performance depends heavily on glycogen management, pacing execution, and training volume — factors VDOT doesn't directly capture.

Confidence Intervals

A good race predictor doesn't just give you a point estimate — it tells you the uncertainty around that estimate.

For a runner using the Riegel model to predict a marathon from a half marathon:

$t_{\text{predicted}} \pm k \cdot t_{\text{predicted}}$

Where $k$ depends on the distance ratio and the model's empirical error:

A prediction of 3:45:00 for a marathon with a large distance ratio might honestly mean "somewhere between 3:30 and 4:00." Understanding this range prevents over-reliance on a single number during race planning.

The Critical Speed Extension

The Critical Speed (CS) model adds a third parameter to the prediction — $D'$, the finite anaerobic work capacity available above CS. CS-based predictions are especially accurate in the 3–40 minute range and for runners with multi-distance data:

$t = \frac{D'}{v - CS} \quad \text{for } v > CS$

We cover the CS model in depth in Critical Speed: The Training Metric Serious Runners Use.

Which Model Should You Use?

There is no universally best model. The right choice depends on what data you have and what distance you're predicting.

The race predictor tool below runs multiple models simultaneously and shows you where they agree and where they diverge. Wide divergence is useful information — it signals that your training history is asymmetric (strong at short efforts, weak at long ones, or vice versa).

The best prediction you can make for a marathon is still a 32-kilometre training run where everything goes exactly right. But the models get you to the start line with a defensible target.