Heart-rate training: lies damn lies and formulae

Sports Performance Bulletin looks at new research on the accuracy of maximum heart rate prediction formulae, and what this means for athletes who use heart rate monitoring to guide their training intensity

How hard? How long? How often? These are the three key factors when it comes to designing and executing a successful training program. And of these, determining how hard – ie the intensity of a training session – is the one most likely to cause confusion. We’re not going to look at ideal training intensities in this article since this depends on numerous factors such as the athlete’s sport, the desired training goal, athlete experience and background and current fitness levels. (For a fuller discussion of optimum training intensities, readers are directed to these articles.) But regardless of the training intensity set by the athlete or coach, there’s a common factor that needs to be considered – intensity measurement and monitoring.

Measuring intensity

There are several different ways to measure training intensity including power output (using a power meter), speed and perceived effort levels. However, of all the methods to monitor and control training intensity, heart rate is the most universally applicable across all sports. This is because of the fairly linear relationship between heart rate and power output – you can think of heart rate as a kind of ‘built-in power meter’. Since it’s very easy to track heart rate in real time with a heart rate monitor, as long as an athlete knows his or her maximum heart rate (MHR), training intensity can easily be set using a percentage of the MHR.

From the above, we can see that establishing an athlete’s MHR is a prerequisite for setting a training intensity or ‘zone’ based on heart rate. The complication however is that an athlete’s actual maximum heart rate can only be properly determined by performing a graded test to exhaustion – ie by working harder and harder until they can no longer continue and seeing what the maximum heart rate at that point is. This procedure presents a considerable challenge; it is quite unpleasant for even the fittest athlete and demands very high levels of motivation in order to be successful. Also, heart rate during exercise can be affected by other factors – for example poor recovery after previous training sessions, or if an athlete is fighting off a virus, both of which will elevate heart rate for any given workload.

A bigger potential problem however is that a graded test to exhaustion is NOT recommended for some individuals (especially those who are older) who are not already extremely fit because of the (very small) risk of a sudden cardiac arrest. MHR testing can be carried out on older, less fit individuals, but it needs to be done under medical supervision with adequate prior screening. A full discussion of how this can be achieved safely can be found in this excellent paper by published the American College of Sports Medicine(1).

The use of formulae

Because an accurate measure of MHR using maximal testing is not easy to obtain, a number of MHR estimations have been developed by scientists over the years to give a close approximation. A commonly-used formula to calculate MHR is the one developed by Fox and his colleagues in 1971(2). This simply states that maximum heart rate is calculated using ‘220 minus age in years’. The equation, however, has been reported to have a standard deviation of between 10 and 12 beats per minute. In short, although around two thirds of people will have an actual MHR within 10-12 beats per minute of that predicted by the Fox equation, one third will have MHRs that are even less accurately predicted. In fact, studies have shown that the ‘220 – age’ calculation for MHR significantly over and underestimates MHR in both younger and older adults(3). Despite its shortcomings, the Fox formula is still used in clinical settings and published in resources by well-established organizations in many settings(4).

In more recent years, a number of other MHR formulae have been developed, which claim to offer a greater degree of accuracy than the Fox calculation. These include (authors’ names alongside):

  • Gellish – MHR = 207 – (age x 0.7)
  • Tanaka – MHR = 208 – (age x 0.7)
  • Arena – MHR = 209.3 – (age x 0.7)
  • Astrand – MHR = 216.6 – (age x 0.84)
  • Nes – MHR = 211 – (age x 0.64)
  • Nes (males only) MHR = 208 – (age x 0.8)
  • Gullati (females only) MHR = 206 – (age x 0.88)
  • Fairbarn (females only) MHR = 201 – (age x 0.63)

The obvious question at this point is which of these formulae (and others) is most likely to be accurate when estimating MHR? Another question is how widely applicable these formulae are? Are they better at estimating MHR in certain types of individuals than others (as is assumed from the last three formulae above)?

New research

In order to try and find the answer to this question, a new study by US scientists has investigated how accurately these age-related MHR formulae were when compared with the actual data derived from a maximal exercise test on the treadmill(5). To do this, data was collected from 99 participants who completed maximal exercise tests from May 2017, to October 2019 at a sports performance clinic. All participants were screened to ensure they were healthy and free of any medical condition that would prevent them from completing the test.

Each participant performed an incremental test on a treadmill lasting 8–12 minutes. Following a brief warm up, the athletes started running at their normal 5km running pace, starting at a 0% incline (ie flat). The gradient was then increased by 2% every two minutes until the athletes could no longer continue. Measurements of maximal heart rates and maximal oxygen uptake (VO2max) were determined through the testing procedure. These were then compared for each athlete to the MHR predicted by the various formulae described above.

What did they find?

The key findings were as follows:

  • The Gulati (206–0.88*age), Astrand (216.6-0.84*age), Nes (211-0.64*age), and Fairbarn male (208-0.8*age) equations all produced results that were significantly different from the actual MHRs measured during treadmill testing. Each of these showed a large bias and wide ranges between limits of agreement suggesting poor agreement with actual MHRs.
  • The other formulas, Fox (220-age), Gellish (207–0.7*age), Tanaka (208 – 0.7*age), Arena (209.3–0.72*age), and Fairbarn female (201-0.63*age) showed less bias and had similar limits of agreement with MHR measured during the treadmill testing. However, the wide scatter of results away from the actual MHRs measured during testing demonstrated that these formulae too could not be considered as good predictors of MHR.
  • Given the above, the researchers concluded that these equations all produced poor predictions of MHR, with unsatisfactory limits of agreement.
  • All equation plots showed non-zero slopes, suggesting each proportional bias is present in each; in general, these formulae tend to overestimate MHR in younger people and underestimate it in older people. However, the Fox equation (with the flattest slope – see figure 1) is less likely to under or overestimate MHR as a function of age, and therefore may be the best formula to use for a wide range of both young and old athletes.

Figure 1: Plots showing agreement between measured and predicted MHR

Scatter plots showing agreement between measured and predicted MHRs. The vertical axes represent the difference in heart rate (beats per minute) between the actual MHR obtained during testing on the treadmill and that predicted by each of the equations. Notice that a good number of subjects had their MHR under or overestimated by very significant amount – 15-25 beats per minute! A perfectly accurate MHR prediction formula would have all the points lying on the ‘0 line’, along the middle of the graph. The wide scatter above and below this 0 line demonstrates the high levels of error in the prediction results. Off all the formulae, the Fox equation has the flattest trendline (dark blue), which suggests that while prone to a large degree of inaccuracy (like the other formulae), it’s the least likely to systematically produce over or underestimates of MHR according to a subject’s age.

Practical implications for athletes

What do these findings mean for athletes who use heart rate monitoring to guide training intensities? The main take-home message is that none of these formulae should be considered a reliable way of estimating your MHR. Indeed, for some people, the over and underestimations are so large as to render the training-zone guidelines almost meaningless. For example, an athlete performing a critical power workout (around 85-90% of VO2max and using heart rate monitoring plus MHR prediction as a guide could be working well under the required level or over it, resulting in excessive lactate accumulation, leading to an impossibly hard session.

Although the use of the Fox formula is less likely to result in age-related systematic errors, the fact remains that if you use heart rate monitoring to guide intensity, a maximal graded test should be considered as an essential requirement to determine your MHR. While this is not particularly easy or pleasant, and may require the supervision of a qualified fitness professional or medic, a maximal test needs only to be performed once.

Athletes with a high level of fitness, an extended training history including maximal efforts (eg intervals) and no previous history of any cardiovascular condition should be clear to undergo maximal testing, especially if they are younger (under 40). Older athletes, novices, and any athlete with a prior history of any type of cardiovascular pathology should consult a physician or other suitably qualified professional before undertaking a maximal test.

Where maximal testing to assess MHR is not possible or desirable, athletes are encouraged not to rely exclusively on heart rate monitoring as a guide to intensity, but to use other measures too, including perceived exertion and ventilation rates. This article explains how these other ‘self-monitoring’ measures can be easily used by athletes to ensure training intensity is optimized!


  1. Med Sci Sports Exerc. 2015 Nov;47(11):2473-9
  2. Ann Clin Res. 1971;3(6):404–32
  3. Med Sci Sports Exerc. 2007 May; 39(5):822-9
  4. Circulation. 2013;128(8):873–934
  5. Int J Exerc Sci. 2020; 13(7): 1242–1250

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