Math in Fitness: Steps, Calories, and Heart-Rate Zones

Two watches, one walk, two different zones

My wife and I took the same evening walk last autumn. Same path, same pace, same four thousand steps, four days a week for a month. Her watch kept calling it Zone 3. Mine kept calling it Zone 2. Neither watch was broken, and neither of us was exaggerating how hard it felt. The watches simply disagreed about what the percentage was a percentage of.

That is this entire article. Fitness trackers speak almost entirely in percentages, and every percentage needs a denominator. The number everyone can see is never the one that is in dispute. The steps, the calories and the zone label are readable on the screen in a second. The denominator underneath them is invisible, unmeasured, and usually guessed. If you are not tracking the guess, you are not tracking your effort. You are tracking someone's arithmetic about you.

Nobody has measured your maximum heart rate

Zones are sold as percentages of your maximum, and the maximum comes from a formula: 220 minus your age. It is a genuinely useful estimate and a bad measurement. Real maximum heart rate scatters around that line by roughly twelve beats per minute in either direction, which means your true ceiling is plausibly 168 or plausibly 192.

Now put those two facts next to each other. If Zone 2 is defined as 60 to 70 percent of maximum, then with a max of 180 the zone runs from 108 to 126 beats per minute. The whole zone is 18 beats wide. But the bottom edge is 60 percent of 168 at one end of the plausible range and 60 percent of 192 at the other, so that single edge can sit anywhere between 101 and 115.

The uncertainty on one edge is 14 beats and the zone is 18 wide. The error bar on the boundary is nearly as wide as the band. A watch that says "Zone 2" is reporting a category whose boundary it cannot locate. That is not a flaw in the watch. It is a consequence of dividing by a number nobody has.

Zone 2 is not 60 percent of your maximum

There is a second method, and it is the one that changed how I read the readout. Take the percentage of your range instead of your maximum, then add your resting heart rate back in. Take a 40-year-old with a resting rate of 60 and a maximum of 180. The usable range is 180 minus 60, or 120 beats.

Sixty percent of the maximum is 108. Sixty percent of the range is 60 plus 0.6 times 120, which is 132. At the top of the zone the gap is just as wide: 70 percent of maximum is 126, while 70 percent of the range is 144. The two Zone 2s do not overlap at all. There is a stretch from 126 to 132 that is easy by one definition and Zone 2 by the other, and no amount of staring at the watch resolves it.

This is the part worth teaching. Both methods are internally consistent, and both are used by real laboratories. They disagree because one asks how hard you are pushing toward your ceiling, and the other asks how far you are from sitting still. Different questions, different answers, and only one of them has a resting rate in it. Our percentages guide covers the arithmetic both of them rely on.

A step count is a distance with an opinion

Your tracker counts steps, but what you want is distance, so it multiplies by an assumed stride length. For a typical adult that assumption lands near 0.7 metres per step, which is how ten thousand steps becomes roughly seven kilometres.

Here is the catch. Stride length is not a constant, it is an average over your own walk, and it moves with your height, your shoes, the slope, and how tired you are by the fourth lap. A ten percent swing is seven hundred metres on a ten thousand step day, about twelve city blocks of pure assumption. Our when to divide and when to multiply article is about exactly that habit.

Which is also why ten thousand is worth treating as a round number rather than a derived one. Nothing in the arithmetic produces it, and a target you chose for a reason you can say out loud beats a round number you inherited.

Calories are honest only as a range

Calorie estimates come from a table of MET values, where one MET is about one kilocalorie per kilogram per hour. Walking briskly is roughly 3.5 METs. Multiply 3.5 by 80 kilograms by one hour and you get about 280 kilocalories for an hour of walking. Clean formula, and it is worth understanding, because the next question is where the number came from.

Resting is about 1 MET, so that same hour of walking included roughly 80 kilocalories you would have burned lying on the sofa. Gross effort is 280. Net effort is 200. Those are the same hour, the same walk, and the difference is 40 percent. This is why two watches on the same evening can report the same workout as meaningfully different. One of them is counting everything, the other is counting the extra.

Then there is the table itself. MET values are averages taken across groups of people, and any individual's real value sits somewhere around 20 to 30 percent either side. So 280 is not a number. It is 280 plus or minus a band, somewhere around 200 to 360. This is not pessimism, it is what averaging does to you, and our statistics guide is where the spread-versus-the-average distinction lives.

Which gives you the practical rule. Treat a calorie figure as a range, and treat a difference between two trackers as noise until it exceeds the band. One hundred calories is not a real difference between two devices. It is the same answer arriving twice with different rounding.

What to try this week

Take one walk and write down three things: your step count, your average heart rate, and your resting heart rate from that morning. Now compute your range, subtract, and find 60 percent of it. Compare that with 60 percent of 220 minus your age. Then check which one your watch reported, and whether it matched either. You will have spent ten minutes and found out exactly which number you have been trusting. If you also work out your walking pace for the same walk, our distance equals rate times time walkthrough shows the same habit applied to a route.

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