One-Rep Max Formulas Disagree, and the Disagreement Is the Point

Five curves through one cloud of data

There is one physiological fact underneath all of this, and it is not controversial: a load you can lift once is heavier than a load you can lift five times, which is heavier than a load you can lift fifteen times. Plot enough real sets — load on one axis, repetitions to failure on the other — and you get a cloud of points that slopes downward and curves. Every one-rep max formula is somebody's attempt to draw a line through that cloud.

The lines differ because the clouds differed. Different authors sampled different lifters, different exercises and different rep ranges, then picked different functional shapes to fit. Brzycki chose a hyperbola. Epley and O'Conner chose straight lines with different slopes. Lombardi chose a power law. Wathan chose a saturating exponential. Each fit its own data well. None of them was ever a law, and none of them was tested against the others on the region where they disagree most, which is exactly the region where most people use them.

The five formulas, written out exactly

Write w for the weight lifted and r for the repetitions completed to failure. Epley: 1RM = w × (1 + r ÷ 30). Brzycki: 1RM = w × 36 ÷ (37 − r). Lombardi: 1RM = w × r^0.10. O'Conner: 1RM = w × (1 + 0.025 × r). Wathan: 1RM = 100 × w ÷ (48.8 + 53.8 × e^(−0.075 × r)). That is the whole of it. Every one of them takes exactly two inputs and returns a multiplier applied to the load you actually lifted.

Two of them already misbehave at a single repetition, which is a useful thing to notice before trusting them anywhere else. A set of one rep to failure is a one-rep max by definition, so the multiplier ought to be exactly 1. Brzycki gives 36 ÷ 36 = 1 and Lombardi gives 1^0.10 = 1, both correct. Wathan gives 1.013, a 1.3% overshoot. O'Conner gives 1.025. Epley gives 1 + 1/30 = 1.033, so it adds 3.3% to a number that was already the answer. On a 225 lb bench that is 7.5 lb of invented weight; on a 100 kg bench it is 3.3 kg.

One lift, seven rep counts, five answers

The table below runs all five formulas over the same 225 lb bench press at 1, 3, 5, 8, 10, 12 and 15 repetitions. Read it as a fan opening. At one repetition the widest disagreement is 7.5 lb. At five it is 11.2 lb. At eight, 17.3. At ten, 21.9. At twelve, 35.5. At fifteen, 73.2 lb — which is more than the entire spread at the first four rep counts added together. As a share of the lowest estimate that is 3.3%, 4.4%, 6.4%, 7.8%, 12.3% and 24.8%. Because those shares are ratios, they are the same whether you work in pounds or kilograms: the 100 kg bench fans out from 3.3 kg to 32.5 kg over the same range.

One detail in the table is worth pausing on. Epley and Brzycki return exactly the same number at ten repetitions — 300.0 lb on the 225 lb bench — because 1 + 10/30 and 36/27 are both precisely 4/3. That is a coincidence of arithmetic, not agreement of substance: below ten reps Brzycki reads lower than Epley, above ten reps it reads higher, and the two curves simply cross at that point. Anyone who has been told the formulas agree at ten reps has been shown a crossing and sold it as a consensus.

Where the divergence comes from: four different shapes

The fan opens because the four functional shapes behave completely differently once you leave the range they were fitted on. Brzycki is a hyperbola with a pole at 37 repetitions: as r climbs towards 37 the denominator shrinks towards zero and the estimate runs away to infinity. At 30 repetitions Brzycki implies your load was 19.4% of your maximum, which no one believes. O'Conner and Epley are straight lines and simply keep climbing at a constant rate for ever. Lombardi is a power law with a tiny exponent, so it flattens: it implies you could do fifteen repetitions at 76.3% of your one-rep max, which is far too generous for a bench press. Wathan saturates and stays plausible longest, but it too is extrapolating.

This is why the divergence is not noise you can average away. The five formulas do not scatter randomly around a true value; they pull systematically in different directions, and which one reads highest depends on the rep count. At three repetitions Lombardi is the highest and Brzycki the lowest. At fifteen the order is exactly reversed. Averaging the five gives you a number — 146.6 kg for the 100 kg bench at fifteen reps, 329.9 lb for the 225 lb bench — but that average has no more authority than the disagreement it papers over.

The relationship is lift-dependent, and the formulas ignore that

Not one of the five formulas asks which exercise you did. They take a weight and a rep count and nothing else. Yet the reps-to-percentage relationship is demonstrably not the same across lifts: a squat sustains more repetitions at a given percentage of maximum than a bench press does. The reasons are mechanical rather than mysterious — more muscle mass sharing the work, a larger contribution from the trunk and hips, and sticking points that behave differently as fatigue accumulates. LeSuer and colleagues tested seven prediction equations against measured maxima in the bench press, squat and deadlift and found the errors were not the same across the three: the equations were systematically off on the deadlift in a way they were not on the bench.

Training history moves the curve too. Richens and Cleather compared endurance-trained and strength-trained athletes and found the number of repetitions achievable at a given percentage of maximum was not the same in the two groups. That is a serious problem for a two-input formula, because it means the correct curve for you depends on something the formula never sees. Practically it means two things. Do not carry a percentage chart derived from bench press data across to your squat. And do not compare your calculated max to someone else's unless you both tested the same lift at a similar rep count.

Every formula assumes the set went to genuine failure

The input r is not the number of repetitions you did. It is the number of repetitions the load allowed. If you racked the bar with two clean repetitions left in the tank, you have fed the formula a smaller r than the truth and it will underestimate your maximum. The size of that error is easy to compute: on Epley, each extra repetition adds w/30, which is 7.5 lb on a 225 lb load or 3.33 kg on a 100 kg load, regardless of where in the rep range you are. Stopping two reps short therefore costs about 15 lb or 6.7 kg of estimate.

This cuts the other way as well, and more dangerously. Repetitions that finish with a bounced bar, a lifted hip on the bench or a rounded back on the deadlift are not the same repetitions the formulas were fitted on. They inflate r, inflate the estimate, and then that inflated estimate becomes the basis for the loads you put on the bar for the next twelve weeks. If you are going to test at all, test with the same technical standard you would use in a competition, and treat the last honest repetition as the last one that counts.

The rule that falls out: five reps or fewer

Below six repetitions the five formulas never disagree by more than 5.4%, and at five reps specifically the whole fan is 11.2 lb wide on a 225 lb bench — about one 5 lb plate per side. At that width the choice of formula is irrelevant next to the ordinary day-to-day variation in how strong you are, and any of the five gives you something you can program from. From twelve repetitions upward the fan is 12.3% wide and climbing, and at fifteen it is 24.8%. A number with a quarter of itself in dispute is not an estimate of your strength; it is an estimate of which author you happened to pick.

So: if you want a usable estimate, pick a load you can move three to five times and stop there. Pick one formula and keep using it, because the value of a repeated estimate is in the trend, not the absolute figure, and switching formulas between tests injects a change that has nothing to do with you. And once you have the number, remember what it is for. It is a starting point for percentage-based programming and a way to compare yourself to yourself over months. It is not a lift you have performed, and it does not belong on a competition entry form.

Frequently asked questions

Which one-rep max formula is the most accurate?

The question has no single answer, because accuracy depends on the rep count, the lift and the lifter. What can be said is narrower and more useful. At five repetitions or fewer all five formulas land within 5.4% of each other, so accuracy is dominated by how well you executed the test rather than by which formula you chose. At twelve or more they disagree by 12% to 25%, and no evidence picks a winner in that range because it is where every one of them is extrapolating. Wathan and Epley tend to sit near the middle of the fan across the whole range, which makes them reasonable defaults. The genuinely accurate way to know your maximum is still to work up to a single heavy repetition under a spotter, with a proper warm-up and, if you have any doubt, medical clearance first.

Why does Epley give a bigger number than the weight I actually lifted once?

Because the Epley multiplier at one repetition is 1 + 1/30 = 1.0333, not 1. The formula was fitted on sets of several repetitions, and nobody constrained it to return the input unchanged at r = 1. On a 225 lb single it invents 7.5 lb; on a 100 kg single it invents 3.3 kg. O'Conner has the same defect at 1 + 0.025 = 1.025, and Wathan is out by 1.3%. Brzycki and Lombardi both return exactly the load, since 36 ÷ 36 = 1 and 1^0.10 = 1. The practical consequence is simple: if you have actually completed a single repetition to failure, that weight is your one-rep max and no formula should be applied to it at all.

Can I use the same formula for the squat and the bench press?

You can run the arithmetic, but you should expect a different quality of answer. None of the five formulas takes the exercise as an input, yet the number of repetitions a lifter can complete at a given percentage of maximum is not the same across lifts: lower-body multi-joint lifts sustain more repetitions at the same percentage than the bench press does. LeSuer and colleagues found exactly this pattern when they checked seven prediction equations against measured maxima in the bench press, squat and deadlift — the errors differed by lift, and the deadlift was the worst served. The workable approach is to keep the rep count low, where all the formulas are tight, and to build your own percentage table per lift over time rather than importing one.

What happens if I stopped one or two reps short of failure?

The estimate comes out low, and by a computable amount. On Epley the multiplier rises by exactly w ÷ 30 per repetition, so on a 225 lb load each missing repetition costs 7.5 lb of estimate and on a 100 kg load it costs 3.33 kg. Stopping two short therefore understates the maximum by roughly 15 lb or 6.7 kg. The other formulas behave similarly in magnitude at low rep counts. This is a bigger source of error than the choice between formulas at five repetitions, which is why the discipline of the test matters more than the mathematics. If you are not willing to take the set to genuine failure — and there are good reasons not to, particularly alone or without a spotter — say so and treat the result as a floor rather than an estimate.

Why does Brzycki produce absurd numbers at high repetitions?

Because of its shape. Brzycki divides by 37 − r, so as the rep count climbs towards 37 the denominator shrinks towards zero and the estimate grows without limit; at exactly 37 repetitions the formula is undefined. At 30 repetitions it already implies the load you lifted was only 19.4% of your maximum, which is nonsense. The failure is structural rather than a bug: the formula was fitted on low and moderate rep sets and the hyperbola simply keeps doing what a hyperbola does once you leave that range. Treat 37 as a hard ceiling for Brzycki and, in practice, treat anything above ten or twelve repetitions as outside its useful domain — which is where the other four are also extrapolating, just less dramatically.