Epley, Brzycki, Lombardi, Lander: A Methodology Deep-Dive on 1RM Math

Every classical 1RM formula is a one-parameter curve fit to a small dataset. The four most-cited — Epley, Brzycki, Lombardi, Lander — produce different predictions, and the differences matter at the practical edges of training (high reps for hypertrophy, low reps for peaking). This article walks through the math behind each, where the constants came from, and where each one fails.

The formulas and their constants

Epley:    1RM = weight × (1 + reps / 30)
Brzycki:  1RM = weight / (1.0278 − 0.0278 × reps)
Lombardi: 1RM = weight × reps^0.10
Lander:   1RM = weight / (1.013 − 0.0267123 × reps)
Mayhew:   1RM = (100 × weight) / (52.2 + 41.9 × e^(−0.055 × reps))

All five take a sub-maximal weight × reps performance and project a one-rep max. The differences are entirely in the rep-curve shape: how aggressively the formula extrapolates as reps increase.

Derivation: how each formula emerged

Epley (1985)

Epley's formula was derived from observational data on collegiate strength athletes performing bench press at varying rep counts. The linear (1 + reps/30) form was chosen for arithmetic simplicity, not theoretical optimum. The /30 divisor reflects the empirical observation that adding one rep at the same weight corresponds to roughly 3.3% more 1RM equivalent — a value that holds within ±5% for 1–8 reps.^[1]

Brzycki (1993)

Brzycki's reciprocal form was derived from bench press data at the United States Military Academy. The constants (1.0278 and 0.0278) were chosen to fit the cadet population's performance curve. The formula collapses to weight at 1 rep (correctly) and asymptotes toward a stable 1RM as reps grow large; for very high reps (15+) the formula's denominator approaches zero and the prediction explodes — a known weakness.

Lombardi (1989)

Lombardi's power-law form was derived from a broader cohort (track athletes, not just lifters). The 0.10 exponent makes the formula more conservative than Epley above 5 reps and less aggressive at the high end. The mathematical motivation was the published observation that strength-vs-reps curves tend toward power-law rather than linear in the moderate-rep range.

Lander (1985)

Lander's formula was calibrated specifically for the squat. The constants are very close to Brzycki's but shifted to fit the squat's slightly different fatigue curve. For lifters using one formula across all three competition lifts, Lander on squat / Brzycki on bench is a common pairing.

The empirical record

Three comparison studies anchor the modern read:

1. LeSuer et al. 1997 — compared Brzycki, Epley, Lander, Mayhew, O'Connor, and Wathan against tested 1RMs on bench, squat, and deadlift in 67 subjects. Brzycki and Epley emerged as the most accurate for moderate reps (3–8); Lander best on squat specifically.^[1]
2. Reynolds et al. 2006 — tested 70 adults (18–69 years) at 1, 5, 10, and 20RM on bench press and leg press and fitted prediction equations from each rep count. The 5RM data predicted 1RM best (R² = 0.993 bench, 0.974 leg press), and accuracy fell off progressively at 10RM and 20RM.^[2]
3. Grgic et al. 2020 — systematic review of 32 test-retest reliability studies of the 1RM test (pooled n = 1,595): median ICC 0.97, median coefficient of variation 4.2%. That ~4% test-retest noise in the reference measurement itself dwarfs the differences between formulas at typical training ranges.^[3]

Worked predictions

For a 100 kg × 5 reps performance, the four formulas predict:

Epley:     100 × (1 + 5/30)         = 116.7 kg
Brzycki:   100 / (1.0278 − 0.139)   = 112.5 kg
Lombardi:  100 × 5^0.10              = 117.5 kg
Lander:    100 / (1.013 − 0.134)    = 113.7 kg

Spread:    112.5 to 117.5 (5.0 kg, 4.4%)

At 5 reps, the formulas agree within ~4–5%, with Brzycki at the low end and Lombardi at the high end. The agreement narrows as reps decrease and widens as reps grow. At 10 reps from the same 100 kg performance:

Epley:     100 × (1 + 10/30)        = 133.3 kg
Brzycki:   100 / (1.0278 − 0.278)   = 133.4 kg
Lombardi:  100 × 10^0.10             = 125.9 kg
Lander:    100 / (1.013 − 0.267)    = 134.0 kg

Spread:    125.9 to 134.0 (8.1 kg, 6.4%)

At 10 reps, Lombardi diverges meaningfully from the others. The 6.4% spread crosses meaningful planning boundaries (e.g., picking the right next-percentage for a training block).^[1]

Where each formula fails

Above 10 reps

All four formulas degrade above 10 reps. The mechanism is straightforward: at high reps, fatigue dominates rather than strength, and the rep-vs-strength curve becomes non-linear in ways the simple linear forms can't capture. The multiple-RM validation work shows exactly that pattern — the load drops non-linearly as the rep count rises, and predictions built on 20RM data are markedly worse than those built on 5RM data.^[2]

Below 70% intensity

Below ~70% 1RM, sub-maximal sets are limited by anaerobic endurance rather than maximal strength. A lifter who fails a set at 60% × 25 reps is not testing maximal strength but rather the lactate-buffering capacity that has weak correlation with 1RM. The formulas mechanically over-predict in this range.

Untrained subjects

The Reynolds 2006 equations needed resistance-training volume, sex, age, and anthropometry as covariates to predict accurately^[2] — the tell that the rep-to-strength curve is not the same for every lifter. Beginners' curves are dominated by neural-adaptation noise, so treat any single high-rep estimate for a novice as a wide range rather than a number.

Different lifts

Deadlift behaves slightly differently from squat and bench across all four formulas. The hip-hinge pattern's mechanical disadvantage at lockout means deadlift rep tests fatigue faster than squat or bench, so the formulas systematically over-predict deadlift 1RM by 2–4% relative to squat. Lander is the calibration closest to squat; no widely-used formula is deadlift-calibrated.

The load-velocity alternative

Load-velocity profiling measures bar speed at known sub-maximal loads and uses the linear relationship between load and concentric velocity to predict 1RM. It is not automatically more accurate: the systematic review of 25 validation studies (842 participants, 16 exercises, 12 devices) found the validity of the prediction hinges on four things — how many loads you use, which exercise, which velocity metric, and which measuring device.^[4] Done well it is a measured estimate rather than a population formula; done casually it is not better than Epley.

The catch is hardware: load-velocity profiling requires a velocity-measuring device (linear position transducer, accelerometer, or vision-based system). For lifters with the equipment, the Velocity-Based 1RM tool replaces all four classical formulas with measured velocity input.

Bottom line: which formula for which use case

1. Quick gym estimate (3–8 reps): Epley. Simple math, accurate within the rep range it was fit to.
2. Bench-specific (3–6 reps): Brzycki. Marginally better fit on bench from the LeSuer cohort.
3. Squat-specific: Lander. Slightly better than Brzycki for the squat-fatigue curve.
4. Trained athletes with velocity tracking: Load-velocity profiling. Replaces all four formulas with measured slope.^[4]
5. High-rep tests (10+): None of the above. Use a 3–5 rep test instead and apply Epley.

Cross-checking against related tools

The One-Rep Max Calculator exposes Epley, Brzycki, Lombardi, and Lander side by side. The RPE to Percentage Converter provides the inverse direction (predicting RPE from a known percentage and rep count). The Velocity-Based 1RM tool replaces formula-based prediction with measured velocity.

Related reading: How To Calculate One-Rep Max Safely for the testing-protocol implementation, Velocity-Based Training: 1RM From Bar Speed for the measured-velocity replacement, and RPE-Based Programming: Math vs Coach for the auto-regulated alternative to formula-based load selection.

FAQ

Which formula is "the right one"?

None is strictly best at all ranges. Epley is the safest default for general gym use at 3–8 reps. Brzycki edges Epley on bench specifically. Lander on squat. At 10+ reps, all four degrade and load-velocity profiling outperforms them substantially.^[1]

Why does Lombardi diverge at high reps?

The power-law form (reps^0.10) grows slower than the linear or reciprocal forms as reps increase. Lombardi is therefore more conservative at high reps — predicting lower 1RMs than Epley or Brzycki for the same input. Whether that conservatism helps depends on the lifter and the rep count: the multiple-RM validation work shows the load-versus-reps curve is non-linear and that prediction accuracy falls away as the rep count rises, for every formula.^[2]

Does the formula choice matter for programming?

At 3–8 reps for a trained lifter, differences between formulas land within ±2% of tested 1RM — close to the noise floor of 1RM testing itself. The formula choice rarely changes the practical training prescription. Above 10 reps, the formulas disagree enough to matter; use lower-rep tests and Epley instead of high-rep tests and any formula.^[3]

What about my recent AMRAP set — should I trust the calculator?

Yes if the AMRAP went to true failure at 3–8 reps. Keep the benchmark in mind: a directly measured 1RM itself repeats with a median coefficient of variation of about 4%^[3], so a well-executed low-rep AMRAP estimate is not meaningfully less precise than re-testing a max. AMRAP at 12+ reps becomes a strength-endurance test rather than a strength test; the formulas over-predict in that range.

References

1. 1 The Accuracy of Prediction Equations for Estimating 1-RM Performance in the Bench Press, Squat, and Deadlift — Journal of Strength and Conditioning Research (LeSuer et al.) (1997)
2. 2 Prediction of one repetition maximum strength from multiple repetition maximum testing and anthropometry — Journal of Strength and Conditioning Research (Reynolds, Gordon, Robergs) (2006)
3. 3 Test-Retest Reliability of the One-Repetition Maximum (1RM) Strength Assessment: a Systematic Review — Sports Medicine - Open (Grgic et al.) (2020)
4. 4 Load-velocity relationships and predicted maximal strength: A systematic review of the validity and reliability of current methods — PLOS ONE (Marston et al.) (2022)