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220 minus age: the formula nobody derived
In August 2000, an exercise physiologist at the University of New Mexico named Robert Robergs got in touch with Martti Karvonen, the Finnish physiologist then in his early eighties, to ask a simple question. Where did the formula come from?
The formula was maximum heart rate = 220 − age. It sat in nearly every exercise textbook and certification exam, usually printed next to Karvonen’s name, and it quietly set the training zones of millions of people. Karvonen’s answer was that he had never published it. He suggested Robergs ask Per-Olof Åstrand, the Swedish giant of the field.
A month later Åstrand was in Albuquerque to collect a lifetime achievement award from the American Society of Exercise Physiologists, and Robergs asked him in person. Åstrand said he had not published it either. He had, he recalled, said in lectures that such a formula looked close to the data and would be convenient to use.
So the two men most often credited with the most famous number in fitness both disowned it. Robergs and his colleague Roberto Landwehr kept digging, and in 2002 they published what they found under a title that undersold it: “The surprising history of the ‘HRmax = 220 − age’ equation.” The surprise was that there was no history. There was a remark.
A ceiling everyone could feel and nobody could predict
The idea underneath the formula is real and old. Push the intensity of exercise up in steps and heart rate climbs with it, roughly in line with the amount of blood the heart is moving. Then, near exhaustion, it stops climbing. More effort, more training, more will: the number does not go higher. By the early twentieth century physiologists treated that ceiling as a fact of the body.
Heart rate had one huge advantage over everything else in the laboratory. You could count it. Measuring cardiac output or oxygen uptake meant masks, bags, catheters and hours of work; a pulse needed a finger and a watch. So if you knew where someone’s ceiling was, you could turn a cheap number into an expensive one. A heart rate of 150 would mean something different in a person whose top was 160 and in one whose top was 200.
The catch was that finding the ceiling meant driving someone to their limit. That is unpleasant in a healthy student and a genuine clinical risk in a middle-aged man with chest pain. A way to predict the maximum from something harmless, like age, would be enormously useful. People tried.
The earliest prediction equation Robergs could find came from Sid Robinson in 1938: data from 92 healthy men aged 6 to 76, and a line of 212 − 0.77 × age. Note the shape. Not 220, and not a slope of exactly one. Åstrand’s own doctoral data from 1952 covered 225 young people tested to exhaustion. When Robergs refitted the 196 of them older than ten, the line came out close to 217 − 0.84 × age — with a scatter around it of about 11 beats per minute.
Karvonen’s real idea was a range, not a ceiling
Karvonen’s name ended up on the formula for an understandable reason. In 1957 he and two colleagues, Kentala and Mustala, published a small longitudinal training study that asked how hard exercise had to be to change the heart. Their answer, as later summaries put it, was that the useful threshold sat around 60% of the distance between resting and maximal heart rate.
That distance is the heart rate reserve, and it was a genuinely good idea. It scales intensity to the individual at both ends: a fit person with a low resting pulse and an unfit person with a high one get targets that mean something comparable. It is still the backbone of how intensity is prescribed.
But the reserve needs a maximum to work, and Karvonen’s paper was not a study of maximal heart rate. Textbooks printed the reserve method and the age formula side by side, often without a citation for the second, and the association did the rest. Karvonen, who also helped bring Finland into Ancel Keys’s Seven Countries Study and pushed for the North Karelia Project, was credited for decades with a line he never drew.
1971: a review with a figure in it
The trail ended in Washington. Samuel M. Fox III was a cardiologist who ran the US Public Health Service’s Heart Disease Control Program, made physical activity one of its themes, and had earlier built an in-patient cardiac rehabilitation programme at the National Heart Institute. With John Naughton and an exercise physiologist named William Haskell, he wrote a long review on physical activity and coronary heart disease, published in the Annals of Clinical Research in 1971.
In a section on intensity there is a figure. It shows roughly 35 data points drawn from about 11 sources, some published papers and some unpublished compilations, plotting maximal heart rate against age. There is no regression. The legend, as Robergs quotes it, says in effect that no single line adequately represents the data — and then observes that the line 220 − age passes not far from many of the points.
That is the whole origin. Not an experiment, not a fitted model, not a claim that the line described any individual. Clinicians writing for clinicians pointed at a convenient approximation and said, in the same sentence, that it was only approximate.
Haskell said as much himself thirty years later. Speaking to Gina Kolata of the New York Times in April 2001, he explained that the data had come from people who were never meant to be a representative sample, that he had laughed about the formula’s career over the years, and that it was never supposed to be an absolute guide to anyone’s training.
The receipt: what the original points actually say
Robergs and Landwehr did the thing nobody had bothered to do. Several of the original references were cited too loosely to find, so they read the data points back off Fox’s figure and fitted a line to them.
The fitted line was 215.4 − 0.91 × age. Close enough to 220 − age to see why the eye settled there. But the correlation was modest (r = 0.51), and the standard error — roughly, how far a typical point sat from the line — was about 21 beats per minute. The very data the formula was read off does not support using it as a precise number for anybody.
This is the part of the story that usually gets told wrongly. The popular version is “the formula was made up on the back of an envelope.” That is a little unfair to Fox, who never pretended otherwise. The truer version is stranger: the caveat was printed in the original legend, and it was the caveat, not the line, that failed to travel.
Figure · four equations, one spread
The formulas argue about a few beats. People differ by twenty.
| 220 − age (Fox, 1971) | 190 |
| 208 − 0.7 × age (Tanaka, 2001) | 187 |
| 207 − 0.7 × age (Gellish, 2007) | 186 |
| 211 − 0.64 × age (Nes, 2013) | 192 |
Formulas disagree by 6 beats. A typical person sits about 11 beats from any of them, in either direction.
| 220 − age (Fox, 1971) | 180 |
| 208 − 0.7 × age (Tanaka, 2001) | 180 |
| 207 − 0.7 × age (Gellish, 2007) | 179 |
| 211 − 0.64 × age (Nes, 2013) | 185 |
At 40 the old and new formulas agree almost exactly, which is why the argument looks trivial in mid-life.
| 220 − age (Fox, 1971) | 170 |
| 208 − 0.7 × age (Tanaka, 2001) | 173 |
| 207 − 0.7 × age (Gellish, 2007) | 172 |
| 211 − 0.64 × age (Nes, 2013) | 179 |
Formulas disagree by 9 beats. The individual spread is still wider than the disagreement.
| 220 − age (Fox, 1971) | 160 |
| 208 − 0.7 × age (Tanaka, 2001) | 166 |
| 207 − 0.7 × age (Gellish, 2007) | 165 |
| 211 − 0.64 × age (Nes, 2013) | 173 |
The old formula now sits lowest. Several large datasets find it tends to underestimate maximal heart rate in older adults.
| 220 − age (Fox, 1971) | 150 |
| 208 − 0.7 × age (Tanaka, 2001) | 159 |
| 207 − 0.7 × age (Gellish, 2007) | 158 |
| 211 − 0.64 × age (Nes, 2013) | 166 |
Formulas disagree by 16 beats, and each still carries its own individual error on top.
How a remark became a law
It is easy to see why the line won. Subtraction is the friendliest operation in arithmetic. You can do 220 − age in your head on a bike, and you can print it on a poster above a treadmill. Combined with Karvonen’s reserve, it produced a full training prescription from two facts anyone knows about themselves.
Then the citations fell away. When Robergs and Landwehr surveyed the exercise physiology textbooks in print around 2000, most gave the formula with no reference at all; the rest pointed at secondary sources, or at Karvonen, or at Åstrand. Certification exams tested it. The number moved from a figure legend into the part of knowledge that no longer needs a footnote, which is the most powerful place a number can live.
From there it went into equipment. Heart-rate monitors, gym consoles and, later, wrist devices needed a default ceiling before they had seen you work hard, and the default was very often this one. Nobody had lied. A caution had simply been lost in copying, like a footnote dropped in a reprint.
2001: a better line, and the same problem
The same Kolata article carried the correction. Hirofumi Tanaka, Kevin Monahan and Douglas Seals at the University of Colorado had pooled group averages from 351 studies — 492 groups, 18,712 people — and found that 208 − 0.7 × age fitted far better. They then tested it in their own laboratory on 514 healthy adults and got 209 − 0.7 × age, practically the same. Men and women did not differ, and habitual activity did not shift the line.
Their headline point was about age. Because the old line falls a full beat per year, it drifts too low as people get older, and so it tends to underestimate the maximum in older adults. At 40 the two formulas agree; at 70 they are nine beats apart. Robergs, pooling 30 published equations the following year, landed on nearly the same line.
The later datasets broadly agreed. Gellish and colleagues at Oakland University followed 132 people through 908 maximal tests between 1978 and 2003 — the same people getting older, not different people of different ages — and got 207 − 0.7 × age. In Norway, the HUNT Fitness Study measured 3,320 healthy adults in 2007–2008, kept only tests where a true maximal effort could be verified, and got 211 − 0.64 × age. Previous formulas, they reported, underestimated measured maximum in people over 30, and healthy older adults and women ran higher than earlier work had suggested.
So the intercept moved. What did not move is in the fine print of every one of those papers.
The real problem was never the 220
The HUNT authors put a number on it: a standard error of 10.8 beats per minute. If the errors are roughly bell-shaped, that means about a third of healthy people sit more than 11 beats from the prediction, and about one in twenty sits more than 21 beats away. Robergs’ survey found the same thing across the literature: typical errors of 7 to 11 beats for almost every age-only equation.
A 2025 study in PLOS One by Martin and colleagues is a good illustration of where that leaves us. They compared seven equations against measured maximum in 230 adults who had completed graded exercise tests. The old formula was not the villain there: its average bias was about one beat. But for every equation, the limits of agreement for an individual ran to roughly ±18 to 24 beats. Average accuracy and individual accuracy are different questions, and only the second one is about you.
This is the revisionist point, and I think it is the useful one. The long argument about whether the number should be 220 or 208 or 211 is an argument about the average of a crowd. It is worth having, and the newer lines are better for older adults. But it is a smaller correction than the one nobody made: printing a range instead of a point. Fox did print the caveat. We kept the line.
There is a quieter source of error too. Robergs stressed that maximal heart rate is partly specific to the kind of exercise: the ceiling you reach running uphill is often not the ceiling you reach on a bike or in a rowing boat. A single number for all of them was always a simplification.
What a wrong ceiling does to your zones
Take a 60-year-old whose real maximum happens to be 175, which is well inside the normal spread. The old formula tells them 160. Seventy per cent of 160 is 112; seventy per cent of 175 is about 122. Every zone built on that ceiling is shifted by around ten beats, all in the same direction, and the error grows toward the top of the range where the hard sessions live.
In practice that person is told that easy running is harder than it feels, or that a hard interval is beyond them when it is not. A different person, whose true maximum sits below the formula, gets the opposite message. Neither of them can tell, from the formula alone, which kind of person they are. Robergs judged that an error of up to about 8 beats is tolerable for setting training ranges, and too large for estimating fitness from heart rate.
None of this makes heart-rate zones useless. It makes them a starting estimate, to be checked against how the effort actually feels and against what you see over weeks. That is a large part of why zone 2 is best anchored to how you breathe and talk, not only to a percentage of a guessed maximum.
How it reached your wrist
If you have ever set up a sports watch, you have met Samuel Fox’s figure legend, whether or not you saw his name. The device asks your age, subtracts, and draws coloured bands. Some newer devices learn a better ceiling from the hardest efforts they record; many still begin with an age-based guess and show it to you as though it were a measurement.
It is the same pattern this series keeps finding. Wunderlich published a range and we kept his 37.0. The risk factor was a statement about groups that we learned to read as a verdict about individuals. Here, a line drawn loosely through a cloud of dots became a personal number printed to the beat. Every time, the average survived the journey and the spread did not.
The number on the screen is not dishonest. It is just answering a different question from the one you are asking. It says what is typical for your age. You want to know what is true for your heart.
What the story is actually evidence for
Maximal heart rate falls with age. That part is robust: cross-sectional data, longitudinal data and very large cohorts all agree, and age alone explains a good share of the difference between groups.
It is not a fitness score. Tanaka found habitual activity did not shift the line, and HUNT found no interaction with fitness level. A high maximum is not a sign of a better heart, and a low one is not a sign of a worse one. It is closer to a personal constant, like shoe size, that happens to drift down slowly. If you want a number that does reflect fitness, VO₂max, resting heart rate and how fast your heart rate comes down after effort are better places to look.
And it is not something to chase. Nothing in this history suggests you should test your own maximum, and a true maximal test is not a casual thing. Medication, including some common heart and blood-pressure drugs, can change heart-rate responses a great deal. If you have a heart condition, symptoms during exercise, or take medication that affects heart rate, talk to your doctor before any very hard effort or before you rely on heart-rate targets at all.
The bottom line
Read any age-based maximum as a first guess with a margin of about ten beats either side. This week, look at the highest heart rate your watch has actually recorded in a hard session you already do, well warmed up, over the past month or two. That is not your maximum, but it is a real floor under it, and if it sits above the formula you now know which way the formula is wrong for you. Then set your easy days by breath and conversation, and let the numbers check you rather than rule you.
Fox wrote that no single line would do. He was right. It took thirty years, and one physiologist willing to ask where the number came from, for anyone to read the legend again.
