Half Marathon to Marathon Time Prediction: Riegel Formula vs the 2016 Correction

A 1:40:00 half marathon predicts a 3:28:30 marathon under the Riegel formula at exponent 1.06, while the 2016 Vickers-Vertosick correction predicts 3:44:16 for a runner logging 30 miles a week, a gap of 15.8 minutes. In that study of recreational runners, Riegel (at exponent 1.07) was well calibrated for races up to the half marathon but too fast at the marathon, by 10 minutes or more for about half of runners. Their correction adds one input: typical weekly mileage.

Vickers and Vertosick collected 2,303 runners' race results through a Slate.com survey and published both the prediction formulas and the full data set. Only seven of those runners reported a time meeting an Olympic qualifying standard at any distance, so the results describe ordinary marathoners, not elites. Among the 633 who reported both a half and a marathon, using the study's difficulty-adjusted times, the median runner finished 10.2 minutes slower than Riegel at 1.06 predicted.

The formulas side by side

Riegel, as the 2016 paper states it (Riegel's 1981 article in American Scientist is paywalled, so the formula and his exponent values are quoted from the 2016 paper):

time for race y ÷ time for race x = (distance of race y ÷ distance of race x)^k

The exponent k is a fatigue factor. According to the 2016 paper, Riegel cited k = 1.08 for elite runners and 1.05 to 1.06 for male recreational runners aged 40 to 70, and the Runner's World online calculator uses 1.06. Vickers and Vertosick used 1.07 as their benchmark, with 1.06 and 1.08 as sensitivity checks. Raising k to 1.08 improved Riegel slightly, but it still overestimated marathon speed for about 75% of runners. Because the Runner's World calculator uses 1.06, the gaps in the tables here are measured from 1.06.

With k = 1.06, the multipliers to a marathon (42,195 m) are:

Vickers-Vertosick Model 1 (one prior race). The formulas are in the paper's Additional file 1. Distances go in as meters, times as seconds, and mileage as miles in a typical training week.

1. Riegel velocity at k = 1.07: v_R = 42,195 ÷ [t × (42,195 ÷ d)^1.07], in meters per second.
2. Corrected velocity: v = 0.16018617 + 0.83076202 × v_R + 0.06423826 × (weekly miles ÷ 10).
3. Marathon time in minutes = (42,195 ÷ 60) ÷ v.

Vickers-Vertosick Model 2 (two prior races, shorter r1 and longer r2):

1. Your own fatigue exponent between the two races: k_r = ln(t2 ÷ t1) ÷ ln(d2 ÷ d1).
2. Predicted marathon exponent: k_m = 1.4510756 − 0.23797948 × k_r − 0.01410023 × (weekly miles ÷ 10).
3. Marathon time = t2 × (42,195 ÷ d2)^k_m, converted from seconds to minutes.

Model 1 keeps Riegel inside it but shrinks the result: the 0.83 coefficient means every extra meter per second of Riegel speed earns only 0.83 m/s of predicted marathon speed, and mileage adds speed back. Model 2 throws out the fixed exponent and estimates a personal one.

The study's data file stores each runner's Model 1 and Model 2 predictions. The formulas above reproduce all 171 stored Model 2 values to within 0.05 minutes and match the stored Model 1 values in nearly every case.

Worked example: a 1:40:00 half at 30 miles a week

Inputs: half marathon in 6,000 seconds, distance 21,097.5 m, typical week 30 miles.

• Riegel at 1.06: 6,000 × 2.0849 = 12,509.6 s, or 3:28:30.
• Riegel at 1.07: 6,000 × 2.0994 = 12,596.6 s. Velocity v_R = 42,195 ÷ 12,596.6 = 3.3497 m/s.
• Model 1 velocity: 0.16019 + 0.83076 × 3.3497 + 0.06424 × 3 = 0.16019 + 2.78280 + 0.19271 = 3.1357 m/s.
• Model 1 time: 42,195 ÷ 3.1357 = 13,456 s, or 3:44:16.

The corrected prediction is 15.8 minutes slower than Riegel at 1.06. Expressed as an exponent, 3:44:16 from a 1:40:00 half is the same as running Riegel with k = 1.165, far above the 1.06 most calculators use.

Adjusting a hilly or fast-course result

Both models expect an average course. The study asked runners to rate each race from very difficult to very fast, dropped the extremes, and converted "difficult" and "fast" results to an average-course equivalent by adding a fixed amount to race speed in meters per second. The adjustment is applied to speed, then turned back into a time: adjusted time = distance ÷ (distance ÷ time + adjustment).

A 1:40:00 half on a course you would call fast becomes 1:42:52 on an average course, and Model 1 at 30 miles a week then predicts 3:49:57 instead of 3:44:16. The same half on a difficult course becomes 1:39:03 and predicts 3:42:23. The fast-course penalty is larger than the hilly-course credit at the half and 10K, so a half-marathon personal best set on a course you rate fast costs 5 minutes 41 seconds of predicted marathon time. At the 5K the pattern flips: a difficult 22:00 5K converts to 21:22, worth almost six minutes of predicted marathon time, because the 5K carries the largest difficulty adjustment in the study. Only the paper's own adjustment values are shown here; a course you rate yourself is a judgment call, and the study had no objective measure of course difficulty.

Half marathon to marathon: the prediction grid

Each row takes a half-marathon time and shows the marathon it predicts under Riegel (both exponents) and under Model 1 at four training volumes. The last column is Model 1 minus Riegel 1.06, in minutes, for 20, 30, 40 and 50 miles a week. A positive gap means the correction is slower.

At 20 miles a week, the correction adds 15 to 21 minutes almost regardless of speed. At 50 miles a week, a 1:30 half still gets 9.2 minutes added, but a 2:10 half gets a prediction 2.8 minutes faster than Riegel. And the two Riegel exponents never differ by more than about two minutes from a half, so arguing over 1.06 versus 1.07 matters far less than accounting for training at all.

The negative cells in the bottom right are where Model 1 is least trustworthy. Few runners in the study combined a slow half with high mileage, and the speed-and-mileage check against real results below shows the model was optimistic for exactly that group.

Predicting from a 10K

The same calculation from a 10K finish. Riegel's multiplier jumps to 4.6002, so a small error in the exponent costs more minutes.

A 45:00 10K and a 1:40:00 half land on almost the same Riegel 1.06 marathon (3:27:01 against 3:28:30), and Model 1 gives them nearly identical answers at every mileage. That is what you would expect if the two races came from the same runner at the same fitness. When your two recent races do not line up like that, the two-race model below uses the mismatch.

Predicting from a 5K

From a 5K, the exponent choice alone moves a marathon prediction by four to seven minutes, because the distance ratio is 8.4 to 1. The 5K is the input most exposed to a wrong fatigue factor.

Shorter targets: where plain Riegel holds up

The 2016 paper tested whether Riegel's predictions were biased for the 10K and half marathon and found no evidence of it (calibration test p = 0.9 for the 10K and p = 0.3 for the half), against p < 0.0001 for the marathon. So for a 10K or half-marathon goal, the plain formula is the right tool, and the correction models were never built for those distances.

What the study's own marathoners actually ran

The study's data file lists every runner's reported races, weekly mileage, and difficulty-adjusted times. Running both formulas against the 633 runners who reported a half and a marathon (using the adjusted times the authors used for model building) gives a direct check of the grid above. A positive miss means the runner was slower than predicted.

Riegel at 1.06 ran fast in every band, and the miss shrank steadily as mileage rose: from about 17 minutes at 20 to 29 miles a week to about 6 minutes at 50 or more. Model 1 centered near zero from 20 miles a week up. Below 20 miles it still ran 11 minutes fast, on only 18 runners, which suggests the formula does not fully capture how much an undertrained runner slows over the last 10 miles.

These medians include the runners used to fit Model 1, so they flatter it slightly. The paper's own out-of-sample test, on 156 validation runners who reported a marathon and two shorter races, is the cleaner figure. Mean squared errors were 227.6 for Model 1, 208.3 for Model 2 and 380.7 for Riegel at 1.07, which as root-mean-square errors are about 15.1, 14.4 and 19.5 minutes. The paper's distribution of residuals shows the shape of the problem:

In that table a negative number means the prediction was too fast. Riegel's median runner finished 10:09 slower than predicted, and its 75th centile was still too fast, so at least three in four runners missed on the slow side. Nearly a quarter were more than 20 minutes slower than Riegel. The corrected models center near zero and miss in both directions, which is what a calibrated prediction looks like.

Speed and mileage together

Splitting the same 633 runners by both half-marathon time and mileage separates two effects that the single-band table mixes, since higher-mileage runners also tend to be faster:

The last row is the warning. Slower half-marathoners who trained 40 miles or more still finished a median 13 minutes behind Riegel, and Model 1, which credits their mileage generously, was 7.9 minutes too fast. That is the same region where the grid shows Model 1 dipping below Riegel. For that group, Riegel plus a fixed allowance tracks real results better than the mileage credit does.

Which prior race to predict from

The data supports a short decision rule. It assumes your races were on courses you would call average (the study dropped "very difficult" and "very fast" races and adjusted "difficult" and "fast" ones), run within about six months at similar fitness, which is what the survey asked participants to supply.

1. Goal is a 10K or half marathon: use plain Riegel from your most recent race. The study found no calibration problem at those distances.
2. Goal is a marathon and you have a half: predict from the half with Model 1 and your typical weekly mileage. In the validation group, Model 1 from a half missed by a median of 7.7 minutes either way, against 12.4 minutes for Riegel 1.06.
3. You also have a recent 5K or 10K at the same fitness: use Model 2 with the half as the longer race. Across all 195 runners in the data file (including those used to fit the models) with a 5K, a half and a marathon, the median absolute miss was 6.1 minutes for Model 2, 6.8 for Model 1 from the half, 12.4 for Riegel from the half and 17.1 for Riegel from the 5K. The 130 runners with a 10K and a half showed the same order: 6.7, 7.3, 11.2 and 12.8.
4. Only a 5K or 10K: Model 1 still works from the shorter race; in the validation group its median absolute miss was 8.0 minutes from a 5K and 8.5 from a 10K. Riegel from a 5K was the weakest option tested, a median 14.9 minutes fast.
5. Half slower than 1:50 and 40+ miles a week, or under 20 miles a week: treat the Model 1 time as a best case and plan on the slower of the two numbers, Model 1 or Riegel 1.06 plus 13 minutes.

Rule 4 softens the usual advice to predict only from your longest race. Under Model 1, the median miss from a 10K was within a minute of the miss from a half once mileage was in the formula, so a recent 10K at current fitness is a reasonable input when your last half is old or was run in poor shape.

How the two-race model reads your races

A worked example first: a 45:00 10K (2,700 s over 10,000 m) and a 1:40:00 half (6,000 s over 21,097.5 m), with a typical week of 30 miles.

• Your exponent between the two: ln(6,000 ÷ 2,700) ÷ ln(21,097.5 ÷ 10,000) = 0.7985 ÷ 0.7466 = 1.0696.
• Predicted marathon exponent: 1.4510756 − 0.23797948 × 1.0696 − 0.01410023 × 3 = 1.4511 − 0.2545 − 0.0423 = 1.1542.
• Marathon: 6,000 × 2^1.1542 = 6,000 × 2.2257 = 13,354 s, or 3:42:34.

That is 1:42 faster than Model 1 from the half alone and 14:04 slower than Riegel at 1.06. Mileage enters Model 2 as a reduction in the exponent: each extra 10 miles a week takes 0.0141 off it, worth a little over two minutes at this pace.

Model 2 calculates your personal fade between two races and then, somewhat counterintuitively, gives a slightly smaller marathon exponent to runners whose fade between the shorter races was larger. With a 1:40:00 half at 30 miles a week:

The effect is modest. A four-minute swing in the 10K moves the marathon about 4.4 minutes, which amounts to Model 2 treating a fast 10K as a sign of extra fitness the half did not show. If your 10K was run on tired legs or in heat, leave it out and use Model 1.

How weekly mileage moves the prediction

In Model 1, each extra 10 miles a week adds 0.0642 m/s to predicted marathon speed. At a 1:40 half that is worth about 4.5 minutes of marathon time per 10 miles (3:48:58 at 20, 3:44:16 at 30, 3:39:46 at 40, 3:35:27 at 50). At a 2:00 half it is worth about 6 minutes per 10 miles, because the same speed increment is a bigger share of a slower pace. Mileage is miles, not kilometers: 30 miles a week is about 48 km, and 50 miles about 80 km.

The mileage effect in the prediction is much smaller than the raw association between mileage and marathon time in the same study. With age, sex and other training held equal, the paper estimated that marathon times for 50 versus 30 miles a week differed by 25:32 for men and 31:41 for women. Model 1's difference for a fixed 1:40 half is 8:49. The gap exists because higher mileage already shows up in a faster half, so once the half is known, mileage adds only what the half missed: the endurance to hold pace past 13.1 miles.

The paper also notes that interval training and tempo runs were associated with faster race times (tempo runners' predicted marathon times were 4.2% faster than non-tempo runners' in the authors' adjusted model), but neither improved the prediction once prior race times and weekly mileage were in the model. That is why the correction asks for only one training number.

The mileage where the correction disappears

Setting Model 1 equal to Riegel 1.06 and solving for mileage gives the training volume at which the two methods agree for each half-marathon time:

A 1:40 half-marathoner would need to train about 67 miles a week before Riegel's 1.06 prediction matched the corrected one. The marathon runners in the study reported a median of 40 miles a week (interquartile range 30 to 50), so for most of them the uncorrected formula sits well on the fast side. For slower runners the crossover falls inside ordinary training volumes, and that is where the speed-and-mileage table showed Model 1 overshooting.

Pacing from the corrected prediction

The authors scored the models on the view that starting too fast costs more than starting too slow, using a penalized error that counted an overestimate of speed twice as heavily as an underestimate. On that penalized measure Riegel scored 1,429.8 against 646.1 for Model 1 and 525.0 for Model 2. The practical point is that Riegel's pace is a reach target, not an opening pace.

Here is what the difference means per kilometer and per mile, from a half-marathon time:

A 1:40 half-marathoner at 30 miles a week who starts at Riegel's 7:57 per mile is running 36 seconds per mile faster than the corrected estimate. Over the first 10 miles that is six minutes banked against a prediction that, for most runners in the study, was already too fast. The pace calculator converts any target finish time into per-mile and per-kilometer pace; its Quick Splits multiply a single pace across distances, so they show the pace a time requires, not a fatigue-adjusted forecast like the ones above.

If you train by heart rate, the heart rate zone calculator gives zone boundaries that can cap early-race effort while the pace settles. Heat, hills and wind will move any of these numbers; the study adjusted for course difficulty in its data, but neither formula has a weather input.

Limits of the 2016 data

• Sample makeup. Respondents were recruited from a Slate.com article, so they were younger and faster on average than the 2013 New York Marathon field the authors compared them with, though more than 300 reported marathons slower than four hours. Only 21 marathoners were over 60 and 3 over 70, so the authors cautioned against confident predictions in those age groups.
• Self-reported inputs. Race times, course difficulty and weekly mileage were all self-reported. Typical and peak weekly mileage were correlated above 0.9 in the data, so either figure gives a similar answer, but the formula was fit on typical mileage.
• One-time training data. The mileage input reflects the build-up to the races entered. A runner whose current training differs from the training behind the prior race is outside what the model saw.
• No physiological inputs. The authors deliberately left out lab measures such as VO2 max, aiming for inputs any runner knows. If you have a VO2 max estimate from the VO2 max calculator, it describes aerobic capacity, but neither model uses it, and the paper found that variables strongly associated with speed did not improve prediction after accounting for prior race times and mileage.
• Age and sex not in the formula. The study found women's relative speed improved with distance (about 20% slower than men at 5K, about 10% at the marathon), but neither prediction model includes sex or age, because they did not add predictive value once race times and mileage were known.

The formulas and figures above are general fitness information, not medical or coaching advice; anyone with a heart condition or other health concern should get medical clearance before marathon training.

Using the grid on race day

Pick the row nearest your recent half, 10K or 5K, read across to your typical weekly mileage, and note the gap to Riegel 1.06. If you fall in the slow-and-high-mileage corner or below 20 miles a week, take the slower of the two estimates. Then convert that time to splits with the pace calculator, and start at or slightly behind the corrected pace rather than the Riegel one. More equation-versus-equation comparisons for training numbers, including the one rep max formulas compared and the BMR and TDEE equations compared, are collected on the fitness calculators hub.

If a 40- or 50-mile week is new for you, energy needs rise with it: the TDEE calculator estimates daily calories from activity level, which is worth checking before a marathon block rather than during it.

Sources

Figures from the paper, its supplement and its data file were read on October 31, 2026. Prediction grids, pace tables, crossover mileages and the checks against the study's runners were calculated by CalculatorVillage on October 31, 2026, using the published formulas and data.

• Vickers AJ, Vertosick EA. An empirical study of race times in recreational endurance runners. BMC Sports Science, Medicine and Rehabilitation. 2016;8:26. Full text via Europe PMC (PMC5000509): sample of 2,303 runners, Riegel formula and exponents as cited from Riegel 1981, calibration p-values, mean squared errors, residual centiles, mileage and training associations, limitations.
• Vickers AJ, Vertosick EA. Additional file 1, "Formulae for Prediction Models," and Additional file 2, study data set, supplementary files via Europe PMC: Model 1 and Model 2 coefficients, difficulty adjustments, per-runner races, mileage and stored model predictions. The article states that a Creative Commons public domain waiver applies to its data.
• Riegel PS. Athletic Records and Human Endurance. American Scientist. 1981;69:285-290. Paywalled and not read; the formula and exponent values above are as stated in Vickers and Vertosick 2016.