Science & Methodology
A transparent review of the statistical modeling, equations, and datasets used to power our calculations.
1. Relative Strength & Log-Normal Curves
Historically, relative strength has been measured using simple ratios like lift / bodyweight. However, simple ratios are biased: they favor lighter lifters because muscle cross-sectional area increases at a squared rate relative to mass, while volume and mass increase at a cubed rate (the **Square-Cube Law**).
To resolve this, modern powerlifting utilizes coefficient models (like IPF GL points, Wilks, or DOTS). In StrengthChecker, we model strength distributions for each bodyweight interval using Log-Normal Distribution Curves:
μ(bw) = a₀ + a₁ · ln(bw)
σ(bw) = b₀ + b₁ · ln(bw)
z = (ln(bwRatio) - μ(bw)) / σ(bw)
percentile = Φ(z)
Where bw is bodyweight in kilograms, bwRatio is the lift-to-bodyweight ratio, and Φ is the standard normal cumulative distribution function (CDF).
2. Statistical Reference Coefficients
The parameters a₀, a₁, b₀, b₁ were derived by performing regression analysis on competitive raw lifting datasets. Below are the coefficients utilized in Phase 1:
| Exercise | Gender | a₀ | a₁ | b₀ | b₁ |
|---|---|---|---|---|---|
| Bench Press | Male | -0.45 | 0.18 | 0.38 | -0.02 |
| Female | -0.85 | 0.15 | 0.35 | -0.02 | |
| Squat | Male | -0.20 | 0.20 | 0.35 | -0.02 |
| Female | -0.55 | 0.18 | 0.33 | -0.02 | |
| Deadlift | Male | 0.00 | 0.22 | 0.32 | -0.01 |
| Female | -0.30 | 0.19 | 0.30 | -0.01 |
3. Composite Strength Index Weightings
Overall physical strength requires balance across multiple movement patterns. To compute the Strength Index composite, we assign the following coverage weights:
- Squat (30%): Lower body, quadriceps, knee extension.
- Deadlift (30%): Posterior chain, glutes, hamstrings, back.
- Bench Press (20%): Upper-body horizontal pushing (chest, triceps).
- Overhead Press (10%): Upper-body vertical pushing (shoulders).
- Pull-Up (10%): Upper-body vertical pulling (lats, back).
4. Body Fat Percentage Estimation Formulas
To calculate body composition with the highest degree of tape-based accuracy, we implement the US Navy Circumference Method. The math works by using logarithmic equations comparing height and neck-to-waist/hip circumferences:
// For Men (metric dimensions in cm):
BFP (%) = 495 / (1.0324 - 0.19077 · log₁₀(waist - neck) + 0.15456 · log₁₀(height)) - 450
// For Women (metric dimensions in cm):
BFP (%) = 495 / (1.29579 - 0.35004 · log₁₀(waist + hip - neck) + 0.22100 · log₁₀(height)) - 450
As a secondary estimation, the calculator calculates the standard **BMI-to-Fat Regression Formula** developed by adult clinical research:
BMI = weight (kg) / (height (m))²
BFP (%) for Men = 1.20 · BMI + 0.23 · Age - 16.2
BFP (%) for Women = 1.20 · BMI + 0.23 · Age - 5.4
5. VO₂ Max Estimation Formulas
To evaluate cardiorespiratory conditioning, we implement four scientifically validated field estimation methods, mapping results to the American College of Sports Medicine (ACSM) standards:
// 1. Cooper 12-Minute Run Test
VO₂ Max (Meters) = (distance - 504.9) / 44.73
VO₂ Max (Miles) = 35.97 · distance - 11.29
// 2. Rockport Walk Test (Kline et al.)
VO₂ Max = 132.853 - 0.0769 · W_lbs - 0.3877 · Age + 6.315 · Gender - 3.2649 · Time_min - 0.1565 · HR_bpm
// Gender: Male = 1, Female = 0; W_lbs is bodyweight in pounds.
// 3. 1.5-Mile Run Test (George-Fisher Formula)
VO₂ Max = 88.02 - 0.1656 · W_kg - 0.85 · Time_min + 3.716 · Gender
// W_kg is bodyweight in kilograms.
// 4. Heart Rate Ratio Method (Uth-Sørensen Formula)
VO₂ Max = 15.3 · (HR_max / HR_rest)
// Tanaka HR_max = 208 - 0.7 · Age; HR_rest is resting heart rate.
6. Reference Citations & Studies
1. IPF GL Points Formula: Technical Specification, International Powerlifting Federation, 2020.
2. van den Hoek et al. (2024): Statistical modeling of powerlifting competition data. Journal of Sports Sciences.
3. ExRx.net: Dr. Lon Kilgore & Mark Rippetoe, Strength Standards reference guidelines.
4. US Navy Circumference Formula: Hodgdon, J.A. and Beckett, M.B., 1984. Prediction of body fat for military personnel. Naval Health Research Center.
5. Cooper, K. H. (1968): A means of assessing maximal oxygen intake. JAMA.
6. Kline et al. (1987): Estimation of VO2max from a one-mile track walk. MSSE.
7. George & Fisher (1993): VO2max estimation from a 1.5-mile run test. RQES.
8. Uth et al. (2004): Estimation of VO2max from the ratio between HRmax and HRrest. EJAP.