One rep max: how the Epley and Brzycki formulas work

The Epley formula

Epley's equation dates from 1985. Lacio and colleagues cite it as the Boyd Epley workout chart from the University of Nebraska: a training chart, not a study.

The idea is linear. Each rep you complete adds a fixed one thirtieth of the lifted weight to the estimate.

The Brzycki formula

Matt Brzycki published his equation in 1993 in the Journal of Physical Education, Recreation and Dance, in an article titled Strength testing: Predicting a one-rep max from reps-to-fatigue. The same preprint describes it as a practitioner-oriented article rather than a formal empirical study.

Brzycki states the weight as a percentage of 1RM that falls by 2.78 points per rep: % 1RM = 102.78 − 2.78 × reps. Dividing the lifted weight by that percentage gives the estimate.

The worked example

Take a lift of 100 kg for 5 reps. In US units the same logic applies to pounds; 225 lb for 5 reps gives 262.5 lb by Epley and 253.1 lb by Brzycki.

Why the formulas diverge with reps

Epley grows in a straight line: every rep adds the same amount. Brzycki's divisor, 37 − reps, shrinks with each rep, so each extra rep adds a little more than the one before. A straight line and a curve that bends upward can cross, and these two cross at exactly 10 reps.

Check it: Epley at 10 reps is 1 + 10 ÷ 30 = 1.333. Brzycki is 36 ÷ 27 = 1.333. Both put the 1RM at one third above the lifted weight.

The percentage table under the result runs the other way. It multiplies an estimate by 50 to 95%. For example, 80% of 112.5 kg is 90 kg, and 80% of 116.7 kg is 93.3 kg. Small differences between formulas carry straight through into every row, so the table is only as good as the estimate it starts from.

• 1 rep at 100 kg: Epley 103.3 kg, Brzycki 100.0 kg.
• 3 reps: Epley 110.0 kg, Brzycki 105.9 kg.
• 5 reps: Epley 116.7 kg, Brzycki 112.5 kg.
• 8 reps: Epley 126.7 kg, Brzycki 124.1 kg.
• 10 reps: both 133.3 kg.
• 12 reps: Epley 140.0 kg, Brzycki 144.0 kg.

How reliable are the estimates?

Note the 1-rep row. Brzycki returns the lifted weight itself, as it should. Epley adds 3.3%, which is a quirk of the straight-line form. The calculator skips the quirk: for a single rep it shows the weight you lifted.

At low to moderate reps the formulas can track a measured 1RM closely. Lacio and colleagues (2010) tested gym users on the bench press with loads that allowed 4 to 10 reps. They found no significant difference between the predicted and measured values, with standard errors of 2.7 to 3.2 kg. They suggested testing other groups and conditions before generalizing.

Past about 10 reps there is little testing to lean on: that study used sets of 4 to 10 reps. The math shows why Brzycki in particular can't stretch far: as reps approach 37, the divisor approaches zero and the estimate runs off toward infinity. Epley never breaks that way, but its straight line has no reason to stay accurate either.