Epley vs. Brzycki vs. Lombardi: Why One Rep Max Formulas Disagree
A hand-verified comparison of the Epley, Brzycki and Lombardi one rep max formulas, showing exactly where and why they agree, cross over, and pull apart at higher rep counts.
Open three different strength apps, plug in the same 100 kilogram set for 12 reps, and there is a real chance you will get three different one rep max numbers back. That is not a bug in any of the apps. It is a direct, mathematically predictable consequence of which one rep max formulas each app happens to use, and how differently those formulas behave once rep count climbs past the range they were actually built for.
This article works through exactly where the three most common one rep max formulas, Epley, Brzycki and Lombardi, agree, where they cross paths, and where they pull apart, using a single hand-verified worked example across a full range of rep counts. The goal is not to crown a winner. It is to show precisely why a "one rep max estimate" is not one fixed number, and why the size of the disagreement between formulas depends heavily on how many reps the set that produced it actually contained.
Three one rep max formulas, three different shapes
All three one rep max formulas take the same two inputs, a weight lifted (w) and the number of reps performed with it (r), and return an estimate of what that same lifter could manage for a single rep. Where they differ is the shape of the mathematical curve connecting reps to estimated max.
Epley's formula, published by Boyd Epley in 1985, is linear in reps:
Every additional rep adds exactly the same fixed amount, one thirtieth of the working weight, to the estimate. A 100 kg lift adds 3.33 kg of estimated max per rep, whether that rep is the 2nd or the 14th.
Brzycki's formula, published by Matt Brzycki in 1993, is hyperbolic:
The denominator shrinks as reps climb, which means each additional rep adds a larger amount than the one before it. The formula has a mathematical asymptote at 37 reps, where the denominator hits zero and the estimate shoots toward infinity; well before that point, in the low double-digit rep range, the curve is already accelerating noticeably.
Lombardi's formula, published under the name Lombardi in 1989 and unrelated to the football coach of the same name, is a power law:
Raising the rep count to a small exponent like 0.10 produces a curve that rises quickly at first and then flattens out. Early reps add a comparatively large jump; later reps add almost nothing.
Three different shapes, published four years apart, in 1985, 1989 and 1993. There was never a guarantee they would agree outside the narrow range they were each built around.
The worked comparison: 100 kg across a rising rep count
Here is the actual arithmetic, at a fixed 100 kg input weight, run through all three formulas at five rep counts, rounded to one decimal place the same way this site's own one rep max calculator rounds:
| Reps | Epley | Brzycki | Lombardi | Spread (highest − lowest) |
|---|---|---|---|---|
| 5 | 116.7 kg | 112.5 kg | 117.5 kg | 5.0 kg |
| 8 | 126.7 kg | 124.1 kg | 123.1 kg | 3.6 kg |
| 10 | 133.3 kg | 133.3 kg | 125.9 kg | 7.4 kg |
| 12 | 140.0 kg | 144.0 kg | 128.2 kg | 15.8 kg |
| 15 | 150.0 kg | 163.6 kg | 131.1 kg | 32.5 kg |
A few things in that table are worth sitting with, because they are more specific and more useful than a vague "the formulas disagree more at high reps" statement.
First, look at 5 reps and 8 reps. The spread actually shrinks slightly between those two rows, from 5.0 kg down to 3.6 kg, before it starts widening again. This is the range where all three one rep max formulas were originally built to perform, and the numbers back that up directly: none of them strays far from the other two.
Second, look at exactly 10 reps: Epley and Brzycki land on the identical number, 133.3 kg, to one decimal place. This is not a quirk of the 100 kg input. At 10 reps Epley multiplies the weight by 1 + 10/30 = 4/3 and Brzycki multiplies it by 36/27 = 4/3, so a straight line (Epley) and a hyperbola (Brzycki) cross at exactly 10 reps for every weight. Below 10 reps, Brzycki reads lower than Epley. Above 10 reps, Brzycki reads higher, and keeps climbing away from it.
Third, and this is the real headline: from 10 reps onward, the spread grows every single step, from 7.4 kg at 10 reps to 15.8 kg at 12 reps to 32.5 kg at 15 reps. That is not a gentle widening, it is an accelerating one. Brzycki's hyperbolic curve is approaching its 37-rep asymptote and climbing faster with every additional rep, while Lombardi's power-law curve has already flattened out and is barely moving. Epley's straight line sits in between, agreeing more closely with Brzycki at low rep counts and with neither formula particularly well by 15 reps.
The full curve, rep by rep
The five-row table above makes the point cleanly, but it skips several reps in between, and skipping them hides a genuinely useful detail: the spread between these one rep max formulas does not fall smoothly and then rise smoothly, it wobbles on the way down before it accelerates on the way up. Here is the same 100 kg example, filled in for every rep count from 3 through 15:
| Reps | Epley | Brzycki | Lombardi | Spread |
|---|---|---|---|---|
| 3 | 110.0 kg | 105.9 kg | 111.6 kg | 5.7 kg |
| 4 | 113.3 kg | 109.1 kg | 114.9 kg | 5.8 kg |
| 5 | 116.7 kg | 112.5 kg | 117.5 kg | 5.0 kg |
| 6 | 120.0 kg | 116.1 kg | 119.6 kg | 3.9 kg |
| 7 | 123.3 kg | 120.0 kg | 121.5 kg | 3.3 kg |
| 8 | 126.7 kg | 124.1 kg | 123.1 kg | 3.6 kg |
| 9 | 130.0 kg | 128.6 kg | 124.6 kg | 5.4 kg |
| 10 | 133.3 kg | 133.3 kg | 125.9 kg | 7.4 kg |
| 11 | 136.7 kg | 138.5 kg | 127.1 kg | 11.4 kg |
| 12 | 140.0 kg | 144.0 kg | 128.2 kg | 15.8 kg |
| 13 | 143.3 kg | 150.0 kg | 129.2 kg | 20.8 kg |
| 14 | 146.7 kg | 156.5 kg | 130.2 kg | 26.3 kg |
| 15 | 150.0 kg | 163.6 kg | 131.1 kg | 32.5 kg |
The narrowest point in the entire table is 7 reps, at a 3.3 kg spread, right in the middle of the classic strength-training rep range. From there the spread climbs at almost every step, roughly doubling between 10 and 12 reps (7.4 kg to 15.8 kg) and again by 15 reps (32.5 kg). That acceleration is the real story: this is not three formulas drifting apart at a constant rate, it is three formulas whose disagreement compounds the further a set strays from the low-to-moderate rep range they were all built around.
It is also worth noticing what stays true across the entire table: Lombardi starts as the single highest estimate at 3 and 4 reps, then is overtaken by both Epley and Brzycki by 8 reps, and ends as the single lowest estimate by 10 reps onward. A formula's rank among the three is not fixed. It flips depending on how many reps went into the set, which is one more reason a calculator that quietly hides which formula it used, and reports one number as though it were the only possible answer, is giving up real information for the sake of looking simple.
Where these three formulas actually came from
Part of why Epley, Brzycki and Lombardi disagree is that none of them was derived from the same dataset or even the same decade of research. Epley's formula, credited to strength coach Boyd Epley and dating to the mid-1980s, grew out of practical strength-and-conditioning coaching work rather than a single controlled laboratory study, and its simple, linear structure reflects that: it is easy to compute by hand on a whiteboard between sets. Brzycki's formula followed in 1993, published in a coaching and physical-education journal, and its round numbers, 36 and 37, make it easy to work through mentally. Lombardi's formula falls between them, from 1989, and takes the form of a power law, a curve with one exponent that rises quickly at first and then flattens.
None of the three was designed with the other two in mind, and none of the three authors was trying to produce a number that would match the others at 12 or 15 reps specifically. They were each solving the same practical problem, estimate a max from a submaximal set, with different tools and different priorities, mostly built and checked against sets in the lower rep ranges lifters actually use for max-testing purposes. Their agreement in that range is the point. Their disagreement outside it is simply what happens when three independently built approximations get extrapolated past the data that shaped them.
Four more one rep max equations at 5 and 10 reps
Epley, Brzycki and Lombardi are the three this site's calculator shows, but four more equations are in common circulation. Here they are at the same 100 kg input (Wathan is sometimes spelled Wathen).
| Formula | Equation | 5 reps | 10 reps |
|---|---|---|---|
| Epley formula | w(1 + r/30) | 116.7 kg | 133.3 kg |
| Brzycki formula | (36 w)/(37 - r) | 112.5 kg | 133.3 kg |
| Lombardi formula | w r^(0.10) | 117.5 kg | 125.9 kg |
| O'Conner formula | w (1 + 0.025 r) | 112.5 kg | 125.0 kg |
| Lander formula | w/(1.013 - 0.0267123 r) | 113.7 kg | 134.1 kg |
| Mayhew formula | w/(0.522 + 0.419 e^(-0.055 r)) | 119.0 kg | 130.9 kg |
| Wathan formula | w/(0.488 + 0.538 e^(-0.075 r)) | 116.6 kg | 134.7 kg |
Across all seven, the 5-rep estimates span 6.5 kg (112.5 to 119.0) and the 10-rep estimates span 9.7 kg (125.0 to 134.7). More formulas do not settle the question. They repeat the same pattern: close agreement at moderate reps and a wider range as the set gets longer.
What independent research says about the most accurate 1RM formula
The disagreement visible in the arithmetic above is not just a quirk of one particular calculator; it shows up in published exercise-science research too. A frequently cited study from the late 1990s compared several one rep max prediction equations, including Epley's and Brzycki's, against directly tested maxes in the squat, bench press and deadlift, and found that no single formula was consistently the most accurate across all three lifts or across every lifter tested. Some equations performed better for upper-body pressing movements, others for lower-body movements, and individual results varied enough that the researchers stopped short of recommending any one formula as a universal replacement for direct testing.
That finding lines up exactly with what the hand-verified numbers above already show from first principles: these are three different curves, each reasonably well matched to the specific range and lifts they were built around, none of them a universally superior estimator. Treating any single one rep max formula as definitively "the accurate one" overstates what the underlying research actually supports.
How the spread carries into a 1RM percentage chart
The disagreement between formulas does not stay abstract once it feeds into a training percentage chart. A one rep max chart printed on a gym wall, or any rep max chart in an app, is only as reliable as the max it starts from. Take the 15-rep row from the table above: Epley says 150.0 kg, Brzycki says 163.6 kg, a 13.6 kg gap on the headline number alone. Apply a standard percentage chart, where a set of 5 reps is programmed at roughly 87 percent of the estimated max, and that single 13.6 kg disagreement becomes an 11.8 kg disagreement in the actual training weight prescribed for that day's work (150.0 × 0.87 = 130.5 kg versus 163.6 × 0.87 = 142.3 kg). A lifter switching between two apps that quietly use different formulas, with no change in their actual strength, could be told to load two meaningfully different weights for the identical prescribed set.
This is why showing Epley, Brzycki and Lombardi side by side, rather than silently picking one, matters: it surfaces a real source of disagreement between tools.
Why the shapes cause this specific pattern
The pattern in that table is not random, it falls directly out of how fast each curve's growth rate itself is changing.
Epley's growth rate never changes. Every rep is worth the same fixed fraction of the weight, always 1/30th, for as long as the formula is used. It is the one formula in this comparison with a perfectly constant slope.
Brzycki's growth rate accelerates. Because the formula divides by a shrinking number, (37 − reps), each additional rep matters more than the last, not the same amount. Near 37 reps that acceleration becomes extreme, which is exactly why this site's own calculator, and most others, cap accepted input at 15 reps: past that point, Brzycki's estimate starts becoming detached from anything a real lifter's body is actually doing.
Lombardi's growth rate decelerates. A power law with an exponent under 1 is a textbook diminishing-returns curve: the first few reps buy a comparatively large jump in the exponent's effect, and every additional rep after that buys less and less. By 15 reps, Lombardi has essentially stopped rising, which is why it goes from the single highest estimate at 5 reps to the single lowest estimate at 15 reps in the table above, without the formula itself doing anything unusual. It is simply the nature of a flattening curve being overtaken by two curves that are still climbing, one steadily and one at an accelerating rate.
What this means for programming, not just for math class
None of this is purely academic if you actually use a one rep max estimate to set training weights. A percentage-of-1RM chart, the kind that says "work up to a set of 5 at 87 percent," is only as good as the 1RM number it is applied to, and that 1RM number is only as good as the set that produced it.
The practical implication is straightforward: the closer your top set is to the 3 to 8 rep range, the less it matters which of these one rep max formulas your app or spreadsheet happens to use, because that is exactly the range where the formulas agree most closely. Push the same estimate out to a 12 or 15 rep top set, and the choice of formula starts to materially change the training weights a percentage chart tells you to use, sometimes by a meaningful double-digit amount of weight at heavier absolute loads.
Two lifters entering the identical 100 kg for 15 reps into two different single-formula calculators could walk away with training numbers over 30 kg apart, neither one told that the other formula exists.
A practical way to use three disagreeing numbers
Rather than treating the disagreement between one rep max formulas as a flaw to be resolved, treat it as information. When Epley, Brzycki and Lombardi are close together, as they are through most of the 3 to 8 rep range, you can be reasonably confident in the estimate regardless of which single number you end up using. When they spread apart, as they increasingly do past 10 reps, that spread is itself telling you something real: the set you used to generate the estimate is far enough from a true max attempt that any formula-based number deserves less confidence, not more.
A few practical habits follow directly from that:
Prefer lower-rep top sets for estimating a max. A hard set of 3 to 6 reps, taken close to failure with good form, sits squarely in the range where these formulas were built and where they agree most closely with each other and with direct testing.
Re-test periodically with an actual near-max attempt. No formula, however well it agrees with its peers, substitutes for a real, properly warmed-up top single every so often. Use the formula-based estimate to program week to week, and use an occasional true test to keep that estimate honest.
It does not matter which app or tool you used to get there. Whether that top set went into a squat 1RM calculator, a dedicated bench press calculator, a bench press max calculator, a plain strength calculator, or this site's own one-rep max calculator, the underlying formulas are the same three described in this article, and the disagreement between them is a property of the rep count, not of any particular app.
Do not chase the highest number among several formulas. It is tempting to look at three estimates and quietly prefer whichever one is largest, since it flatters the lifter. The comparison above shows exactly why that instinct is backwards: the highest number in a spread is frequently the product of one formula's curve accelerating away from reality, not evidence of extra strength the other two formulas somehow missed.
Treat the average as a reasonable middle ground, not a magic fix. Averaging Epley, Brzycki and Lombardi, the approach this site's own calculator takes, smooths out some of the disagreement without pretending it does not exist.
The bigger picture
Strength coaches have used estimating formulas like these for decades precisely because testing a true one rep max every week is impractical and risky. That tradeoff is worth keeping in view whenever the numbers from different one rep max formulas do not match: the disagreement is the visible cost of a genuinely useful shortcut, not evidence that the shortcut is broken. Epley, Brzycki and Lombardi were each built from real lifters, each validated within a specific rep range, and each still does its job well inside that range. The mistake is asking any one of them to speak confidently about a rep range it was never built to describe, and then being surprised when a different formula, built differently on purpose, says something else.