Race Time Predictor: Riegel Formula, Mile to Marathon

A race predictor answers one question: if you raced one distance all out, what could you run at another? Riegel's formula is the standard answer. Your time grows a little faster than the distance does, because pace slows as races get longer. With the usual exponent of 1.06, doubling the distance multiplies your time by 2.085, which slows your pace by about 4 percent.

Most predictors stop at one number. This one lists the whole ladder of standard distances with paces per mile or per kilometer, and it says how far each prediction can be trusted. For the marathon, it adds a model that Vickers and Vertosick built from 2,303 recreational runners, which uses your weekly mileage. In their data, Riegel's formula worked well up to the half marathon and ran too fast at the marathon.

How the Riegel formula works

Peter Riegel described the formula for runners in 1977 and published the analysis behind it in American Scientist in 1981 (Riegel 1981). He fitted world records from several endurance sports to one curve, in which time rises with distance raised to a power a little above 1. For your own races, the formula is:

T2 = T1 × (D2 ÷ D1)^1.06

• T1 is your recent race time and D1 its distance.
• D2 is the distance you want to predict and T2 the predicted time.
• 1.06 is the exponent, which Riegel called the fatigue factor. It makes your pace slow as the distance grows.

To run it by hand for a 25:00 5K and a half marathon:

1. Divide the target distance by your race distance. A half marathon is 21.0975 km and a 5K is 5 km, so the ratio is 4.2195.
2. Raise the ratio to the power 1.06. 4.2195 to the power 1.06 is 4.600. A phone calculator does this with the x^y key.
3. Multiply your time by the result. 25:00 times 4.600 is 115 minutes, a 1:55:00 half marathon.

For the common jumps, multiply your time by these numbers. To predict a shorter race, divide by them instead.

The exponent carries the whole idea. At exactly 1, pace would never slow. Riegel's own values came from world records. Reading his paper, Vickers and Vertosick give about 1.08 for elite runners and 1.05 to 1.06 for men aged 40 to 70, and 1.06 is the value most online predictors use, this one included. Each 0.01 on the exponent moves a 5K-to-marathon prediction by about 2 percent.

Race time predictor chart: 5K to marathon

This chart runs Riegel's formula from seven 5K times, with the pace for each race under the time. The last column is the Vickers and Vertosick model for a runner who logs 30 miles48.3 km a week, which shows how much slower the marathon comes out for a recreational runner.

Each doubling of distance slows the pace by the same 4 percent, so marathon pace is about 14 percent slower than 5K pace at every level. At 30 miles a week, the mileage model adds 19 minutes to the 18:00 runner's marathon and 6 minutes to the 35:00 runner's. It moves by about 5 to 6 minutes for every 10 miles a week, so put your own mileage in the calculator above. Our couch to 10K plan uses the same formula: a 30:00 5K predicts a 1:02:33 10K, and a sub-60 10K takes about a 28:47 5K.

Half marathon time predictor

The half marathon is where Riegel's formula stands on its firmest ground. In the Vickers and Vertosick data, its half marathon predictions were well calibrated, with no sign of running fast or slow on average. From a 10K, multiply by 2.206: a 45:00 10K predicts 1:39:17, a 50:00 10K predicts 1:50:19 and a 60:00 10K predicts 2:12:23. From a 5K, multiply by 4.600.

Two cautions apply. The prediction assumes your long runs have prepared you for 13.1 miles, and a hot or hilly course costs time the formula cannot see. Our half marathon training plan covers the long-run build, pacing by goal time and the taper for a first race.

Marathon time predictor: the recreational model

For the marathon, the calculator adds Model 1 from Vickers and Vertosick (2016), published in BMC Sports Science, Medicine and Rehabilitation. They surveyed 2,303 recreational runners about their recent races and training and found that Riegel's marathon predictions were too fast for most of them. Model 1 starts from Riegel's formula and corrects it with your typical weekly mileage:

1. Riegel's marathon speed. Take your longest recent race from 5K to the half marathon, predict the marathon with the study's exponent of 1.07 and turn that time into a speed in meters per second (v).
2. Correct the speed. Predicted marathon speed = 0.16018617 + 0.83076202 × v + 0.06423826 × (weekly miles ÷ 10).
3. Turn it back into a time. Predicted marathon time = 42,195 m divided by that speed.

The correction takes a share off Riegel's speed and gives some back for every mile you run a week. More mileage brings the prediction closer to Riegel's, and for a 1:55:00 half marathoner it beats Riegel above about 55 miles a week.

Marathon predictions from a 1:55:00 half marathon. Read them with the study's limits in mind. Use your longest recent race from 5K to the half marathon, run all out on an average course: the authors adjusted times from hilly, hot or windy courses and from fast ones, and left out very difficult and very fast races. Their marathon runners typically ran 30 to 50 miles a week, and the calculator accepts 10 to 120, the range in their published data. Only 21 of the marathon runners were over 60.

We reproduced the authors' published Model 1 and Model 2 predictions from their public dataset to check the calculator's arithmetic. Model 2 adds a second race and uses the exponent between your two races. It was a little more accurate, and the calculator leaves it out to keep the inputs short.

How accurate is the Riegel formula?

It is right on average up to the half marathon and too fast at the marathon for most recreational runners. The clearest test in everyday runners is the Vickers and Vertosick survey of 2,303 recreational runners, with a median age of 35 and a median of 30 miles a week. Only 7 of them reported a time that met an Olympic qualifying standard.

Where the exponent holds: up to the half marathon

The study tested the formula with an exponent of 1.07, between the values Riegel reported. Its predictions for the 10K and the half marathon were well calibrated: on average, runners finished close to the predicted time, with no lean toward fast or slow. This calculator keeps the more common 1.06, which is a little more optimistic. From a 25:00 5K, 1.07 predicts a 1:56:40 half marathon and 1.06 predicts 1:55:00.

The shape of the curve holds up too. Fitted to the race results of 2,571 mostly recreational British runners, a power law like Riegel's described efforts shorter than 2 minutes and longer than 15 minutes better than the critical-power model, and the two matched in between (Drake et al. 2024).

Where it fails: the marathon from shorter races

At the marathon, the formula ran fast. For the median runner, Riegel's prediction was 10:09 faster than the actual finish. Half of the runners finished 10 minutes or more slower than predicted, and nearly a quarter finished more than 20 minutes slower. The table shows the spread of misses in the 156 validation runners who reported a marathon and two shorter races. A negative number means the prediction was too fast.

The mean squared errors were 381 for Riegel and 228 and 208 for the two models. Their square roots, a typical miss, are about 19.5, 15.1 and 14.4 minutes. Model 1 is the one this calculator uses for the marathon.

Why your own exponent is probably above 1.06

Riegel fitted his exponents to world records set by different athletes, and Drake and colleagues point out that such values may not suit any one runner. Blythe and Király (2016) modeled 1,417,432 performances by 164,746 British athletes in ratified races from 100 m to the marathon. Individual exponents had a median of 1.12, with 90 percent between 1.10 and 1.15, higher than the world-record values. Elite distance runners sat lower, at 1.08 for Haile Gebrselassie and Mo Farah. If your exponent were 1.12, a 2:00:00 half marathon would become a 4:20:49 marathon instead of Riegel's 4:10:12.

Why the calculator does not fit an exponent from two of your races

Some predictors take two of your races and fit your own exponent. In Blythe and Király's out-of-sample test, a power law fitted to each runner predicted slightly worse than Riegel's fixed exponent, an error of 0.103 against 0.098 in log time across events from 100 m to the marathon. Vickers and Vertosick's two-race model uses the exponent between your two races as one input next to weekly mileage, and it beat their one-race model only narrowly.

How to use your prediction on race day

Treat the prediction as an even-effort target. In the power-law model behind Riegel's formula, an even pace gives the best finishing time, and starting too fast costs more than starting too slow (Drake et al. 2024). Vickers and Vertosick scored fast predictions as twice as costly as slow ones for the same reason: a runner who starts too slowly can speed up, but one who starts too fast slows dramatically.

For a half marathon or marathon, a small negative split is the practical version. Our guide to negative split running turns a goal time into first-half and second-half targets, and how to pace yourself running covers the mile, the 5K and the long run by feel.

If the prediction becomes your goal, the VDOT calculator turns the same race into training paces. To see how your time compares with runners of any age and sex, use the age-graded calculator.

What moves your prediction: mileage, tempo runs and intervals

Riegel's formula sees only a distance and a time. Vickers and Vertosick also asked about training. After adjusting for sex, age and the rest of each runner's training, these went with faster race times:

• Weekly mileage. Running 50 rather than 30 miles a week went with a marathon 25:32 faster for men and 31:41 faster for women, and the link was similar at every distance. Our guide to how many miles a week to run covers how to build it safely.
• Tempo runs. Runners who did tempo runs most weeks were 6.0 to 7.2 percent faster at 5K and 10K and 4.2 percent faster at the marathon. The tempo run guide shows how to find the pace without a lab.
• Intervals. Interval training went with times about 2.5 to 3.8 percent faster at every distance. The interval running workouts guide has six sessions by level.
• Age. Being 50 rather than 30 went with a marathon 16:18 slower for men and 21:37 slower for women.

These come from a survey, so they show associations rather than proof. The authors argue that the training links are likely causal, pointing to the biology of interval training and the dose-response pattern for mileage.

Strength work is the other lever most plans add. Our review of running economy research covers how strength and plyometric training lower the energy cost of each stride, with a dumbbell plan. Mileage only adds up if you keep showing up, and if that is the hard part, our ranking of the best fitness apps for staying consistent compares which apps keep people training past the first few weeks.

When to ignore this calculator

The formula and the marathon model both assume a healthy adult who raced all out and trains for the target. In the cases below, use the prediction loosely or not at all.

• Your recent race was not all out. A training run, a race run with a friend or a race you paced for someone else understates what you can do. The study asked runners to report only races in which they tried as hard as they could.
• You have not trained for the target distance. A 5K on a few miles a week does not make you ready for a marathon. Read the marathon prediction with your mileage, and build the long run first.
• The course or the weather was unusual. Hills, heat, wind, a downhill course or a tailwind move your time in ways the formula cannot see. The study adjusted for difficult and fast courses and left out very difficult and very fast ones.
• The target is an ultramarathon or a trail race. The calculator stops at the marathon, the longest race in the studies behind it.
• You are coming back from injury, illness or a long break. Use a race from your current training, not an old personal best.
• You are over 60 and predicting a marathon. Only 21 of the study's marathon runners were over 60, so the mileage model is least tested there.
• You have heart symptoms or known heart, metabolic or kidney disease, or you are pregnant. Racing is vigorous exercise, and the American College of Sports Medicine's screening model calls for medical clearance before vigorous exercise in people with symptoms or known disease (Riebe et al. 2015). In pregnancy, ask your obstetric care team what suits you.

Related reading

References

1. Riegel PS. "Athletic records and human endurance." Am Sci. 1981;69(3):285-290. PubMed 7235349. The paper is not open access; its exponent values are given here as summarized by Vickers and Vertosick 2016.
2. Vickers AJ, Vertosick EA. "An empirical study of race times in recreational endurance runners." BMC Sports Sci Med Rehabil. 2016;8:26. doi:10.1186/s13102-016-0052-y, PMC5000509. Model formulas in Additional file 1; data in Additional file 2.
3. Blythe DAJ, Király FJ. "Prediction and quantification of individual athletic performance of runners." PLoS One. 2016;11(6):e0157257. doi:10.1371/journal.pone.0157257, PMC4919094
4. Drake JP, Finke A, Ferguson RA. "Modelling human endurance: power laws vs critical power." Eur J Appl Physiol. 2024;124(2):507-526. doi:10.1007/s00421-023-05274-5, PMC10858092
5. Riebe D, Franklin BA, Thompson PD, et al. "Updating ACSM's recommendations for exercise preparticipation health screening." Med Sci Sports Exerc. 2015;47(11):2473-2479. doi:10.1249/MSS.0000000000000664

How this calculator differs from typical race time predictors

Most online race time predictors run Riegel's formula with a fixed exponent and print one number for every distance, marathon included, as if it were equally reliable everywhere. Few say where the formula was tested, and fewer warn that it runs fast at the marathon for recreational runners.

This one gives Riegel's prediction for any distance from the mile to the marathon and the full ladder of standard distances with paces per mile or per kilometer. It also says how far each prediction can be trusted, from published calibration data. For the marathon it adds the Vickers and Vertosick (2016) model with your weekly mileage, checked against the authors' own published predictions. It does not fit an exponent from two of your races, which predicted worse in the largest test, and no number appears until the time, distances and mileage sit inside the ranges the research covered.