Estimating a one-rep max, and where the formulas diverge

A one-rep max estimate answers a narrow arithmetic question: given that somebody moved a certain weight for a certain number of repetitions, what single-repetition weight would be consistent with that? It is a projection from a set that happened to a set that did not.

The two formulas

Two are in common use, and both are one line.

Epley is 1RM = w × (1 + reps / 30). It is linear in the rep count: each extra repetition adds a thirtieth of the weight, forever.

Brzycki is 1RM = w × 36 / (37 − reps). It is a ratio with the rep count in the denominator, so it climbs slowly at first and then accelerates.

The free One-Rep Max tool uses Epley. Its worked default is 185 for 5 repetitions, giving 185 × (1 + 5 / 30) = 185 × 1.166667 = 215.833333.

They agree exactly at ten reps

This is the fact that makes the comparison worth doing, because it is not approximate. At 10 repetitions Epley gives a multiplier of 1 + 10 / 30 = 4/3. Brzycki gives 36 / (37 − 10) = 36 / 27, which is also 4/3. Identical, not close.

Below ten they disagree mildly. At one repetition Brzycki gives 36 / 36 = 1, which is the honest answer — a single is already a single. Epley gives 1 + 1 / 30 = 1.033333, so it projects a heavier max than the weight already moved, which is a known quirk of the linear form.

Above ten they come apart, and here are the figures

Take a round 100 kilograms and run both formulas at the same rep counts. Every figure below is the two formulas applied to that one weight:

• 5 reps: Epley 116.666667, Brzycki 100 × 36 / 32 = 112.5. Epley is higher by 4.166667, about 3.7 percent.
• 10 reps: Epley 133.333333, Brzycki 133.333333. The gap is exactly zero.
• 15 reps: Epley 150, Brzycki 100 × 36 / 22 = 163.636364. Now Brzycki is higher, by 13.636364, about 9.1 percent.
• 20 reps: Epley 166.666667, Brzycki 100 × 36 / 17 = 211.764706. Brzycki is higher by 45.098039, about 27 percent.
• 25 reps: Epley 183.333333, Brzycki 100 × 36 / 12 = 300. Brzycki is now more than 63 percent higher.

Read down that list and the shape is unmistakable. The two formulas cross at ten, and the gap then widens roughly threefold from fifteen reps to twenty and again from twenty to twenty-five. At thirty repetitions Brzycki returns 36 / 7 = 5.142857 times the weight while Epley returns 2 times it. At thirty-seven the denominator hits zero and Brzycki has no answer at all.

What the divergence tells you

The two formulas were fitted to broadly the same kind of data, and they are pinned together at the rep counts where that data was thick. Where they disagree, it is because neither of them is constrained by much — they are extrapolating, and extrapolations that start from the same place fan out.

So the disagreement is a usable signal. When two reasonable methods give answers within about four percent, the estimate is doing what an estimate should. When they are twenty-seven percent apart, the arithmetic is telling you that the question — what would a single repetition weigh, projected from twenty — does not have a well-determined answer, and any single figure printed to six decimal places is false precision.

There is a physical reason underneath the arithmetic. A set of three and a set of twenty are not the same kind of effort, and how far a person can extend a set is itself an individual trait. Two people with the same single-repetition strength will not necessarily manage the same number of repetitions with the same lighter weight, so a formula mapping one to the other is describing an average relationship, not a fixed one.

Reading the output honestly

The tool prints one number from the Epley line, and the tool's own text says it is most reliable for sets of about ten repetitions or fewer. The arithmetic above shows why that cut-off is exactly where it is: ten is the point where the two standard methods still agree.

An estimate is not a measurement. Nothing in 215.833333 records a weight anyone has lifted; it records what 185 × 1.166667 comes to. The only figure that is a measurement is the set you actually did, which is the input, and it stays the more informative number of the two.