A swimmer goes 1:02.40 for the 100. The coach wants a 500 goal to put on the board, and the arithmetic looks like it does itself: a 500 is five 100s, so five times 1:02.40 is 5:12.00.
She will chase that number until March and never touch it. The multiplication is fine. The problem is that the 5 is doing a job it was never measured for, and swapping one kind of coefficient for another is the quietest way a pace chart stops describing anything real.
The 500 is not five 100s
Peter Riegel was a research engineer at Battelle in Columbus who spent his spare time measuring road courses for what became USA Track & Field. While writing a pace-prediction piece for Runner's World in August 1977, he noticed that plotting world-record time against distance on log-log paper produces something close to a straight line. A straight line on log-log paper means time is a power of distance: t = ax^b. He expanded the observation into a full paper for American Scientist in 1981, volume 69, number 3, pages 285 to 290.
The exponent b carries the whole argument. If b were exactly 1, time would scale linearly with distance and a 500 really would be five 100s stapled together. It isn't 1. Riegel named b the fatigue factor, because its size is exactly how fast average speed bleeds away as the race gets longer.
Fitted to men's freestyle world records, b came out at 1.02977. Run our swimmer's 1:02.40 out to 500 three ways:
| Method | Predicted 500 | Against five flat 100s |
|---|---|---|
| Five times the 100 | 5:12.00 | reference |
| Riegel, swimming men, b = 1.02977 | 5:27.31 | 15.31 s slower |
| Riegel as usually quoted, b = 1.06 | 5:43.63 | 31.63 s slower |
Fifteen seconds on a five-minute swim is the gap between a target and a fantasy. Thirty-one seconds is a different event. And the two rows underneath the first one disagree with each other by more than half a minute, which is the part worth sitting with, because both of them claim the same author.
Riegel's own table has fourteen exponents in it
Every race predictor on the internet ships 1.06, and it is genuinely Riegel's number. It traces to the 1977 magazine piece, where one round exponent for runners was the entire point of the exercise. What it isn't is the value his own analysis produced for running four years later, which came out at 1.07732. The 1981 paper carries a table of fourteen activities, each with its own b, computed from records standing as of 1 November 1979.
| Activity | Fitted exponent b | Distance range | Time range |
|---|---|---|---|
| Nordic skiing, men | 1.01421 | 15 to 50 km | 44 to 149 min |
| Swimming, men | 1.02977 | 0.4 to 1.5 km | 3.9 to 15 min |
| Swimming, women | 1.03256 | 0.4 to 1.5 km | 4.1 to 16 min |
| Cycling, men | 1.04834 | 4 to 100 km | 4.4 to 128 min |
| Race walking, men | 1.05379 | 1.6 to 50 km | 5.9 to 222 min |
| Speed skating, men | 1.06017 | 3 to 10 km | 4.1 to 15 min |
| Running, men | 1.07732 | 1.5 to 42.2 km | 3.5 to 129 min |
| Running, women | 1.08283 | 1.5 to 42.2 km | 3.9 to 147 min |
| Roller skating, men | 1.13709 | 3 to 10 km | 5.6 to 22 min |
The only rows sitting near the famous 1.06 are speed skating and the masters running brackets, which Riegel fitted between 1.05352 and 1.06370 for men aged 40 to 70. The number the calculators inherited is a rounded convenience from a magazine article, and it has been quoted back at three significant figures ever since by people who have never opened the paper.
The spread matters more than the provenance. Take a 400 in 4:30.00 and predict the 1500, a ratio of 3.75. Flat multiplication says 16:52.50. The swimming exponent says 17:33.13. The borrowed 1.06 says 18:16.07. The running exponent says 18:41.45. One conversion, four answers, and 42.94 seconds between the two that both call themselves Riegel.
Every exponent carries the range it was fitted inside
Those last two columns aren't decoration. Riegel drew them because his own analysis has edges, and he was explicit about where they are.
Below roughly 3 to 4 minutes the log-log plot stops being straight. Riegel put sprints outside the study on the grounds that they involve transient body processes rather than steady-state endurance, and he said so in the paper rather than leaving it to be discovered. Above about 230 minutes the curve bends the other way as the multi-hour events start behaving differently again.
Which means the swimming exponent of 1.02977 was fitted to 400, 800 and 1500 metre world records: 0.4 to 1.5 km, 3.9 to 15 minutes. A 100 swum in 1:02.40 is about a minute. It sits below the floor of the fit. That first table is an extrapolation off the end of the data, and the honest thing is to label it that way rather than quote it to two decimals and let the precision do work the evidence cannot.
The 400 to 1500 conversion lands inside the fitted range at both ends. That one you can lean on.
Plus fifteen seconds and times 1.20 agree at exactly one swimmer
Distance scaling is only half of where training coefficients come from. The other half is intensity: same distance, swum easier. Two conventions are in wide use for that, and they are different mathematical objects wearing the same label.
Offset zones state effort as seconds added to a threshold pace per 100. Critical Swim Speed is the usual anchor. It traces to Wakayoshi and colleagues in the European Journal of Applied Physiology, March 1992, volume 64, pages 153 to 157, where critical velocity was defined as the speed a swimmer could hold indefinitely without exhaustion and recovered as the slope of a distance-against-time line. Coaches estimate it from two time trials: CSS pace per 100 is the 400 time minus the 200 time, divided by two. The zone prescriptions built on top of it are written as offsets, easy at CSS plus 15 to 20 seconds per 100, threshold within 2 or 3 seconds of CSS.
Percentage zones state the same effort as a multiplier. Easy at 1.20 times base, threshold at 1.10, race at 1.00. That is the dialect of Jack Daniels and Jimmy Gilbert's Oxygen Power tables, self-published in 1979 and later the spine of Daniels' Running Formula, and of most physiological zone models that followed.
A fixed offset and a fixed coefficient describe the same zone at exactly one pace. Everywhere else they separate, and the direction is systematic:
| CSS per 100 | Easy at CSS + 15 s | That offset as a coefficient | Easy at CSS × 1.20 | That coefficient as an offset |
|---|---|---|---|---|
| 1:05.00 | 1:20.00 | 1.231 | 1:18.00 | +13.00 s |
| 1:15.00 | 1:30.00 | 1.200 | 1:30.00 | +15.00 s |
| 1:25.00 | 1:40.00 | 1.176 | 1:42.00 | +17.00 s |
| 1:35.00 | 1:50.00 | 1.158 | 1:54.00 | +19.00 s |
They cross at 1:15.00, because 15 seconds is 20 percent of 75 seconds and nowhere else. Above the crossover the offset version runs relatively harder: the swimmer holding 1:35 gets 15.8 percent of cushion where the multiplier would have handed her 20. Below it the offset runs relatively softer, and the fastest swimmer on the team gets the easiest easy day.
At the slow end that is 4 seconds per 100. Over a set of 10 x 100 it accumulates to 40 seconds of difference between two charts that both say "easy" at the top of the column. On a squad where everyone finishes within a few seconds of each other the gap is invisible. On a high school team with half a minute between the fastest and the slowest swimmer, it is the workout.
Neither convention is wrong. They encode different beliefs about what should stay constant across a roster. A fixed offset assumes the absolute recovery matters and should be the same for everyone. A fixed coefficient assumes relative effort matters and should be the same for everyone. Both are arguable positions, and coaches hold them sincerely. Inheriting one by accident, because it happened to be what the calculator you found on a Tuesday emitted, isn't a position at all.
What a borrowed coefficient is actually a measurement of
Every published number above was fitted to somebody, and it wasn't your team.
Riegel's exponents came from world records standing in 1979. Daniels and Gilbert's tables came from oxygen-cost equations run on trained distance runners. Jan Olbrecht's Z1 to Z5 scheme in The Science of Winning is anchored to lactate testing, which is a real measurement of a real athlete and still not a measurement of the fifteen-year-old in lane four in the second week of September.
That doesn't make published models useless. A curve fitted to elite athletes is usually right about shape even where it's wrong about magnitude, which is why starting from 1.20 for an easy column beats starting from nothing. It does mean the published coefficient is a hypothesis rather than a constant, and the squad's own repeat times are the only evidence that can settle it.
Checking is cheap. Run a set at a stated zone, write down what the athletes actually held, and compare it to what the column predicted. If the whole team beats the easy column by four seconds every time, the coefficient is wrong for this team and the fix is to change the number, not to keep writing times nobody swims. Two or three seasons of that and the headings on the chart mean something local and testable, which is more than a fatigue factor fitted to Nordic skiers in 1979 can offer.
The practical obstacle is usually that the coefficients live somewhere you can't reach. A calculator built on a fixed model hands you Z1 through Z5 and no way to say the threshold column should be 1.08 this season because that's what the mid-season time trial showed. If you want the coefficient sitting on the column where you can edit it, with the whole roster underneath so the change shows up across every athlete at once, that's what Coeffpace is for.