Science & Methodology
The mathematical models, empirical databases, and statistical techniques powering StrengthChecker.
1. Relative Strength & Log-Normal Curves
Human physical strength does not follow a pure Gaussian normal distribution. Due to physiological limits and training adaptation barriers, strength standards follow a right-skewed log-normal distribution.
We compute percentiles using the cumulative distribution function (CDF) of the log-normal distribution:
μ(bw) = a₀ + a₁ · ln(bw)
σ(bw) = b₀ + b₁ · ln(bw)
z = (ln(bwRatio) - μ(bw)) / σ(bw)
percentile = Φ(z)
Where μ is the location parameter, σ is the scale parameter, and erf is the standard Gauss error function.
2. Statistical Reference Coefficients
Standards adapt dynamically to athlete bodyweight through non-linear power regression:
| 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.
// Male:
BFP (%) = 495 / (1.0324 - 0.19077 · log₁₀(waist - neck) + 0.15456 · log₁₀(height)) - 450
// Female:
BFP (%) = 495 / (1.29579 - 0.35004 · log₁₀(waist + hip - neck) + 0.22100 · log₁₀(height)) - 450
5. VO₂ Max Estimation Formulas
To evaluate cardiorespiratory conditioning, we implement four scientifically validated field estimation methods, mapping results to 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
VO₂ Max = 132.853 - 0.0769 · W_lbs - 0.3877 · Age + 6.315 · Gender - 3.2649 · Time_min - 0.1565 · HR_bpm
// 3. 1.5-Mile Run Test
VO₂ Max = 88.02 - 0.1656 · W_kg - 0.85 · Time_min + 3.716 · Gender
// 4. Heart Rate Ratio Method
VO₂ Max = 15.3 · (HR_max / HR_rest)
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.