Riegel's formula predicts a race time by scaling a known one: T2 = T1 × (D2 / D1)^1.06, where T is time, D is distance and 1.06 is the exponent. On the World Athletics list as we read it on 15 October 2026, it takes Cheptegei's 10,000 m record to a marathon 57 seconds slower than Sawe's 1:59:30, and misses other record pairs by up to about 2% either way. For 289 recreational runners who reported both a 10 km and a marathon, it came out a median 12.1 minutes (5.5%) too fast, and about four in five were predicted too fast. Vickers and Vertosick, whose data these are, got there first, finding it gave times "at least 10 min too fast for half of runners".
Record times climb an almost even gradient from the mile to the marathon
The exponent sets how fast time grows with distance. At exactly 1, pace would never change; at 1.06, doubling the distance multiplies the time by 2.085 (our arithmetic). Naismith's rule for hill walking also turns distance into time at one fixed rate.
Seven men's records from 1500 m to the marathon fit an exponent of 1.063 in our calculation; the four men's road records from 5 km fit 1.045. World Athletics keeps separate women's road records for women-only and mixed races, where men also ran, and our seven-record women's fit gives 1.070 with the first and 1.057 with the second. George Ron (2017, on his blog) and Blythe and Király (2015) fitted records before us.
| Record pair | Formula at 1.06 | Record | Miss |
|---|---|---|---|
| Men, 10,000 m to marathon | 2:00:27 | 1:59:30 | 0.8% too slow |
| Men, 10 km road to marathon | 2:01:59 | 1:59:30 | 2.1% too slow |
| Women, 10,000 m to women-only marathon | 2:12:57 | 2:15:41 | 2.0% too fast |
| Women, 10,000 m to mixed-race marathon | 2:12:57 | 2:09:56 | 2.3% too slow |
Each pair is two athletes on different days, so the table tests the curve itself, and the 57 seconds is its friendliest pair.
Calculators flatten Riegel's reported exponents into a single 1.06
P. S. Riegel set out the power law in "Athletic records and human endurance", American Scientist, 1981 (PubMed record). We could not open the paper. Vickers and Vertosick write that "Riegel cited a k of 1.08 for elite runners and 1.05 to 1.06 for male recreational runners aged 40 to 70", and that Runner's World's online calculator uses 1.06. Blythe and Király's preprint, which says it has not been peer-reviewed, reports the same split. It also lists a 1977 Runner's World article by Riegel, which we have not read and credit with no exponent.
So 1.06 is one of several values Riegel reportedly gave, which calculators apply to every runner and distance. Today's records up to the marathon fit 1.045 to 1.070, below the 1.08 reported for elites; without the paper we cannot say which records Riegel used.
The marathon steepens the curve for most runners outside the record books
Vickers and Vertosick surveyed runners through Slate.com from 24 April to 16 June 2014, kept 2,303, and published the self-reported times with their 2016 paper, as public domain (CC0) data. We re-ran the formula at 1.06 on every runner who reported both races of a pair.
| Known race to predicted race | Runners | Median miss | Over 10 min too fast | Within 3% |
|---|---|---|---|---|
| 5 km to 10 km | 500 | 1.3% too fast | 3% | 42% |
| 10 km to half marathon | 529 | 1.1% too fast | 12% | 51% |
| Half marathon to marathon | 717 | 5.3% too fast | 54% | 26% |
| 10 km to marathon | 289 | 5.5% too fast | 54% | 24% |
Rows overlap, as one runner can sit in several pairs. Up to the half marathon the formula lands within about 1%, as in the authors' finding that it "was well-calibrated for races up to a half-marathon". The one-sided miss arrives with the marathon, though 17% of the 10 km to marathon group were still predicted too slow.
Each runner's two races also imply an exponent. The median was 1.08 from 5 km to 10 km and 1.07 from 10 km to the half, then jumped to 1.14 from the half to the marathon.
The times come from a self-selected, mostly US readership, with no heat or course data beyond a five-point difficulty rating. We found no comparable open data for Southeast Asian races, so none of this describes runners there.
Errors level out among runners who reported more weekly mileage
The survey asked for typical weekly mileage, in miles, and Vickers and Vertosick's marathon models "included only prior race time and typical weekly mileage". Our split of the 10 km to marathon pair shows the same association.
| Weekly mileage | Approx. km (our conversion) | Runners | Median miss at 1.06 |
|---|---|---|---|
| Under 20 miles | Under 32 km | 14 | 10.2% too fast |
| 20–30 miles | 32–48 km | 46 | 9.5% too fast |
| 30–40 miles | 48–64 km | 65 | 7.8% too fast |
| 40–50 miles | 64–80 km | 74 | 4.6% too fast |
| 50 miles or more | 80 km or more | 90 | 3.3% too fast |
Runners reporting more mileage were missed by less. The lowest group holds 14 people, and mileage travels with experience, which this split cannot separate or show as a cause. In the top group, 71% were still predicted too fast.
Marathon misses dip below four per cent on a speed-based curve
Dave Cameron posted an alternative to the t-and-f mailing list on 20 June 2001, by his account about eight years after developing it. He fitted speed against distance for the "top 10 people - not times" at each event, so his scaling depends on distance alone. From 10 km to the marathon it behaves like an exponent of 1.073 (our arithmetic).
On the same 289 runners Cameron's model missed by a median 3.8% (8.5 minutes) too fast. The price shows on the records: it puts the men's marathon record 2.7% (190 seconds) too slow from the 10,000 m, against Riegel's 0.8%.
Until 15 October 2026 our race time predictor suggested "1.05 for elites" and said predictions matched real races "within 3–5%". Rounding after splitting off minutes made about 1.5% of a sweep of outputs print times like "6h 33m 60s", and a blank, zero or negative exponent silently became 1.06, while 0.5 was accepted. A revision committed that day rounds the total first, refuses exponents outside 1.00–1.20 with a message, prints the exponent used and quotes the values Vickers and Vertosick report.
Ultra distances drop away from the straight line
Past the marathon the records leave the curve. From the men's marathon record at 1.06, a 100 km road race comes out at 4:58:15. Sorokin's record from 14 May 2023 is 6:05:35, so the formula runs 18.4%, or 67 minutes, too fast. Adding the 50 km and 100 km records to the four shorter men's road records moves the fitted exponent from 1.045 to 1.108. Emig and Peltonen argue that individual runners follow no single power law. In the record book, the straight line runs from 1500 m to 42.195 km and ends there.
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