One in every ten runners using the Riegel formula to calculate their race finish times will finish more than 30 minutes and 24 seconds slower than they should. That formula is the equation behind nearly every online marathon-time calculator, and the figure comes from a 2016 study of 2,303 recreational runners published in BMC Sports Science, Medicine and Rehabilitation. It is not an outlier case. The median runner in that study got a prediction 10 minutes and 9 seconds too fast. The formula takes a known race time, multiplies it by the ratio between two distances, and raises that ratio to an exponent, a shape published by research engineer Peter Riegel in Runner's World in 1977 and expanded four years later in American Scientist. For most calculators run that exponent is 1.06. When running a 10K or a half marathon, that 2016 study found the formula well calibrated. Upping the distance to a marathon something breaks.
An equation built for world records
Riegel was not a statistician handed a spreadsheet of recreational finish times. He was a research engineer at Battelle Memorial Institute in Columbus, Ohio, whose day job involved investigating airflow in coal mines and developing deep-sea diving equipment. He had run more than 20 marathons himself and built a pace computer for other long-distance runners, and he later chaired the USATF Road Running Technical Council and headed the team that designed marathon courses for the 1984 and 1996 Olympics.
The equation he published, though, was fitted on world records rather than on runners like him. Riegel gathered world-record times across running, swimming, cycling, speed skating and several other endurance sports and plotted each one on a log-log graph of time against distance. For running men, the fitted exponent, what he called the fatigue factor, came out to 1.07732; for running women, 1.08283. For men aged 40 to 70, it dropped to a range of 1.05352 to 1.06370. Riegel raised the obvious problem himself, in his own paper: "What about real people? We have dealt with a composite of world records as though they represented the performances of a single person." His answer for ordinary runners was an informal one, not a study: a survey of distance runners in Ohio, from which he judged that a runner performing at some fixed percentage of world-class speed at one distance tends to hold roughly that percentage at others.
Where the data disagrees. Several pace-calculator sites tell runners that elite marathoners get a gentler exponent, somewhere near 1.04 to 1.05, while newer runners get a harsher one, up around 1.07 to 1.08. Riegel's own fitted table runs the other way. World-class men and women came out at 1.08; masters men aged 40 to 70 came out lower, at 1.05 to 1.06. The 1.06 that recreational calculators borrow sits closest to the value he fit for older, slower record-holders, not the one he fit for Olympic finalists.
Riegel also set a boundary on his own equation, and it is one most calculator sites never mention. The straight-line relationship, he wrote, held only "in the range from about 3.5 to 230 minutes," and beyond that ceiling he said he had "avoided extrapolation... because of the obvious pitfalls." Two hundred thirty minutes is three hours and fifty minutes. His fitted table for running men only covers 3.5 to 129 minutes, a range that tops out around a 2:09 marathon. A four-hour marathoner is using an equation whose own author declined to stand behind it that far out, built from a table that was never fitted to a marathon that slow in the first place.
How often (and how badly) it misses
The clearest test of what that means for a real marathon comes from the 2016 study, run by researchers Vickers and Vertosick, who surveyed 2,303 runners with race results at two or more distances between April and June 2014. The cohort had a median weekly training volume of 30 miles, a median marathon time of 3:28:02 for men and 3:54:36 for women, and it skewed noticeably faster than the general marathon field: about 40 minutes faster at the median than New York City Marathon finishers, with only 11 percent of the men and 19 percent of the women running slower than four hours. If anything, that makes the errors below conservative for a typical recreational field, not inflated.
The formula's marathon prediction was tested against each runner's actual finish, and the errors were not a tight cluster around zero. A formal calibration test found the coefficient on Riegel's marathon predictions significantly different from a perfect match, at p < 0.0001, meaning the mismatch was not something that could be explained by chance in a sample that size. Nearly a quarter of runners got a prediction more than 20 minutes too fast. The same test run on half-marathon predictions found no such gap, with a p-value of 0.3, and the same held for 10K predictions, at p = 0.9. Same formula, same runners, same paper. The break is specific to the marathon.
The same formula that holds for a 10K breaks at 42.2 kilometers
The traditional explanation for that break is glycogen depletion, the point at which stored carbohydrate runs low and pace collapses, commonly called hitting the wall. Buman and colleagues, in a 2008 study, surveyed 315 marathoners at three US races and found that 43 percent reported hitting the wall during the race, describing generalized fatigue, an unintentional slowing of pace and a shift toward simply surviving to the finish. The same study found the length of a runner's longest training run associated with whether they reported hitting the wall at all, an early, independent signal that training volume and marathon-specific collapse are connected.
A larger, more objective dataset points to the same collapse from timing-chip data rather than memory. Smyth's 2021 analysis of 4.18 million race records across 270 marathons found that 28 percent of male runners and 17 percent of female runners showed a significant late-race slowdown, beginning around the 29-kilometer mark and costing an average of 31.5 minutes for men and 33.2 minutes for women by the finish. That average cost sits almost exactly on top of the 10th-percentile Riegel error of 30 minutes and 24 seconds from the Vickers and Vertosick data, two entirely separate studies landing on the same order of magnitude.
A more recent line of research suggests glycogen is not the whole explanation. Hunter and Muniz-Pumares tested 18 marathon finishers before and after a 90-minute run and found that speed at lactate threshold, a marker of sustainable pace, dropped by 5.5 percent on average. Fresh-state threshold speed correlated strongly with marathon speed (r = 0.937), which is expected. What stood out was that the size of the drop after 90 minutes also correlated with marathon speed on its own (r = 0.680), even after adjusting for how many comparisons were run. Researchers have started calling this quality durability, the ability to hold physiological values steady as fatigue accumulates, distinct from the fresh-state fitness a short race measures. A 10K or a half marathon captures how fast someone is before fatigue sets in. It cannot see how much that ability erodes over three or four hours, and that gap is a plausible reason a formula built from one short race keeps missing at 42.2 kilometers.
The runners it fails hardest
A separate, independent dataset makes the same point from a different angle. Oficial-Casado and colleagues followed 7,663 runners who completed both the Valencia Half Marathon and the Valencia Marathon in the same year, and fitted a regression predicting marathon time directly from half-marathon time. The 1.06 exponent the calculators use implies a half-to-marathon multiplier of 2.0849. The Valencia data, timed by chip across thousands of real double starters, fit a multiplier of 2.28, a steeper slope than the formula assumes.
That steeper slope means the gap is not constant. For a man running a fast recreational half marathon, around 1:20:00, Riegel's marathon prediction lands roughly 9 minutes ahead of what the Valencia model expects. For a man running a 2:30:00 half, closer to the pace of a first-time or back-of-pack marathoner, the gap opens past 23 minutes. The formula is least reliable for exactly the runners most likely to be searching for a marathon-time calculator in the first place: newer, slower athletes rather than the sub-3:00 crowd the equation was built to describe. A 2:30 half also predicts a marathon in the neighborhood of five hours, which crosses back over Riegel's own 230-minute boundary from the other direction. The Valencia researchers, testing their own model against the widely used Daniels VDOT method, put it plainly: Riegel's equation is good for distances up to the half marathon, but it can be improved at the marathon.
Weekly mileage predicts the size of the miss
None of the studies above publish an error table sorted by training volume directly, but Vickers and Vertosick published something more useful: the full equation behind their improved prediction model, coefficients included. That model starts with the Riegel prediction and adds a term for typical weekly training mileage, which means the size of the correction can be computed directly for any given mileage level, holding the half-marathon input fixed.
Starting from the same 1:45:00 half marathon, the correction changes by roughly 19 minutes across the mileage range shown. At 20 miles a week, the model says Riegel is more than 20 minutes too optimistic. At 60 miles a week, the same starting half-marathon time comes out only about a minute off. Nothing about the half-marathon time changed between those two rows; only the weekly mileage did. A second, separately fitted model in the same paper, based on two prior races rather than one, moves the same direction: more weekly mileage flattens the fatigue exponent the model predicts for the marathon. Two independently estimated corrections in one paper agree on which way training volume pushes the answer.
One caution belongs here. The same study found that in its multivariable model of marathon speed alone, the interaction between weekly mileage and race distance was small in size, even though statistically detectable. That is a different question from the one above. Mileage does not make marathon pace disproportionately more sensitive to training volume than 10K pace is; the claim here is narrower and better supported: holding a runner's shorter-race time fixed, weekly mileage predicts how far off the Riegel formula's marathon guess will land, and the published correction models size that miss directly. No published study breaks Riegel's actual prediction errors down by mileage band the way the calibration and half-marathon tests do; this is a computation built from the published model equations, not a reproduction of an existing table, and it is worth being clear about that distinction.
What a runner can do
None of this means the Riegel formula is useless. Up through the half marathon it holds up well, and even at the marathon it gives a usable starting point rather than a random guess. It takes a time and two distances and nothing else, so it knows nothing about how much training has gone into the legs doing the running, which is precisely the gap the weekly-mileage correction fills, and nothing about the road either, where a course profile can move a finish time on its own. A runner training 20 miles a week and one training 60 can post the same half-marathon time and still owe two very different marathon predictions, which is one reason the training plans compared here diverge so sharply in weekly volume.
42cal's race time predictor starts from that same idea of translating a known result into a projected one. Pairing that projection with an honest read on weekly volume, and checking goal pace against 42cal's training paces tool before committing to a number on race day, does more to close the gap than adding a flat percentage ever will. The full detail behind the numbers here, including Riegel's 1981 paper and the Vickers and Vertosick study, is publicly available: Riegel's paper in American Scientist, hosted by the USATF Road Running Technical Council he once chaired, and the 2016 study in BMC Sports Science, Medicine and Rehabilitation that produced the error figures above. The Valencia Marathon study that produced the 2.28 multiplier is there too, for anyone who wants to see the regression itself rather than take the summary on faith.
