Medicine ball throw distance weighted with height-to-mass ratio aligns with other physical fitness tasks

Abstract

The traditional Medicine Ball Throw (MBT_T) for assessing upper-extremity explosive strength inherently favours heavier and taller participants. We designed three alternative indices to adjust for this anthropometric bias: (1) multiplying MBT_T by the height-to-mass ratio (MBT_A1), (2) dividing MBT_T by the product of mass and height (MBT_A2), and (3) dividing MBT_T by the body mass index (MBT_A3). Using age- and sex-adjusted LMS-based z-scores from two independent cohorts (821 children, aged 8.3 ± 0.7 years; 354 adolescents, aged 14.4 ± 2.2 years), we compared the variants’ alignment with the concave BMI-profiles of five standard fitness tests (6-minute run, standing long jump, jump sideways, 4 × 10 m shuttle run, push-ups). The three anthropometrically enriched variants yielded significantly better goodness of fit than MBT_T with MBT_A1 performing best overall. Furthermore, when deployed as independent covariates, the enriched indices demonstrated systematically higher predictive power for standard fitness tasks than the unadjusted distance. We recommend adjusting MBT_T via the height-to-mass ratio for a fairer assessment, though its structural overlap with BMI requires caution in predictive models.

Supplementary Information

The online version contains supplementary material available at https://doi.org/10.1038/s41598-026-58353-w.

Introduction

Globally, the prevalence of obesity and extreme obesity has increased alarmingly in recent years¹. Elevated Body Mass Index (BMI) values, waist circumference, and body fat percentage positively correlate with reduced fitness and a higher risk of metabolic syndrome later in life². Physical fitness in youth is a critical health indicator³; conversely, paediatric obesity strongly associates with adult cardiovascular risk factors⁴and coronary heart disease⁵. Consequently, school-based physical activity interventions are deployed worldwide to counter the obesity pandemic^6,7. Monitoring the efficacy of these strategies requires national and international fitness screening systems⁸using test batteries that are reliable, valid, cost-effective, and easy to administer.⁹

Beyond basic anthropometrics, comprehensive fitness monitoring in children and adolescents typically targets a latent construct of motor tasks comprising endurance, coordination, flexibility, speed, and muscle power^10,11. The parameter “muscle power” distinguishes between explosive strength and strength endurance, static and dynamic strength, and lower and upper body muscular strength¹². There are a variety of field tests that can be used to assess upper body muscular strength^12,13. While field tests for lower-body power (e.g., jumping) are easily implemented, many traditional upper-body assessments are poorly suited for large-scale school screenings due to prolonged administration times (e.g., timed hangs), equipment costs (e.g., handgrip dynamometers costing > €250), or structural barriers (e.g., pull-up bars).¹⁴

The medicine ball throw (MBT) circumvents these limitations as a very easy-to-administer and low-cost field test, yet its traditional assessment via throwing distance (MBT_T) fundamentally misaligns with other standard motor tasks, specifically tasks taxing cardiorespiratory endurance, lower body muscular strength, body coordination, action speed, and upper body muscular endurance¹⁰. While tasks like running and jumping display concave relationships where high body mass acts as a severe biomechanical penalty, MBT_Tcorrelates positively with height, mass, and BMI^15,16. This divergence occurs because MBT_Tinherently favours overweight children and adolescents due to their typically higher absolute fat-free mass and longer structural lever arms. Furthermore, it is the only common power task that does not require an individual to accelerate their own body mass against gravity¹⁵. Additionally, a reliable correlation exists between a participant’s body mass and their strength performance¹⁷. Failing to adjust for these growth metrics results in a systematic misjudgement of true explosive strength¹⁸. Because upper-body power remains a vital indicator of overall physical fitness, a method to preserve the MBT’s operational simplicity while neutralizing its structural size bias is highly desirable.

To address this, we propose enriching the traditional throwing distance (MBT_T) with anthropometric data, yielding three alternative indices. We evaluate whether these adjusted variants align more accurately with the biological BMI profiles of standard fitness tasks and assess their efficiency as performance predictors. The generalizability of these modelling approaches is validated with cross-sectional cohorts from primary, secondary, and high schools.

Methods

Study participants

The participants of this secondary analysis were selected from two different study populations. Study Population 1 (SP1)¹⁹ comprised a primary school cohort of 821 children (419 girls, 402 boys) in grades 2 and 3, with a mean age of 8.3 ± 0.7 years. Study Population 2 (SP2)²⁰ consisted of 354 adolescents (157 boys, 197 girls) from a secondary school (SS, n = 178) and a high school (HS, n = 176), with a mean age of 14.4 ± 2.2 years (see Supplement Figure S1). Due to strict time constraints the push-up test was only conducted in the secondary school cohort (n = 178). Both studies were approved by the Research Ethics Committee of the University of Graz, Styria, Austria. Active informed consent was obtained from the legal guardians of all participants or, from the adolescents themselves if they were older than 14 years. Demographic data including age, sex, and school type were recorded at enrolment.

Procedure

Anthropometric measurements and physical-fitness testing were conducted by trained research team members at the participating schools during regular physical education classes. All assessments took place in indoor sport halls. Participants wore standard sports attire and were tested barefoot, except for the 6-minute run (6MR), which required athletic sneakers.

Anthropometrics

Body height was measured to the nearest 0.1 cm using a mobile stadiometer (SECA 213, Hamburg, Germany), and body mass was measured to the nearest 0.1 kg using an electronic floor scale (BOSCH PPW4202/01, Nuremberg, Germany). BMI was calculated as body mass divided by height in meters squared (kg/m²).

Physical fitness assessments (SP1)

Cardiorespiratory endurance was assessed via the 6-minute run (6MR), lower body strength via standing long jump (SLJ), body coordination via jumping sideways test (JS), and action speed via a 4 × 10-meter shuttle run (4 × 10SHR). The action speed (m/s) was scored by dividing the total distance run (i.e., 40 m) by the elapsed running time (4 × 10SHR) in seconds. Upper body strength was measured using the traditional standing 1-kg medicine ball chest throw (MBT_T). All fitness tests followed the established field methodology described by Jarnig et al. (2022).¹⁹

Physical fitness assessments (SP2)

In the adolescent cohort (SP2), secondary school students (grades: 5 to 8) and high school students (grades: 9 to 12; maximum age: 18 years) completed the medicine ball overhead throw (MBT_SOT−2 kg). From a stationary standing position at the starting line, participants held a 2 kg medicine with both hands behind their heads and threw it forward overhead as far as possible. The shortest distance from the starting line to the initial impact point was recorded measured to the nearest centimetre using a tape measure. The maximum distance from two valid attempts was used for analysis.

All SP2 participants completed the SLJ following identical protocols to SP1. Additionally, the secondary school subgroup (n= 178) completed the push-up test (PU) according to the German Motor Test guidelines²¹. From a prone starting position, participants touched their hands together behind their lower back, placed their hands adjacent to their shoulders, and pushed up into a fully extended plank position. At the peak of extension, they lifted one hand to touch the back of the opposing hand before returning to the starting position in a controlled manner. The score reflected the number of correctly executed repetitions completed within 40 s.

Anthropometrically enriched measures of MBT

Traditional MBT_T performance assessment relies on the absolute throwing distance, which is conventionally adjusted for age and sex. However, raw performance is heavily confounded by H_B and M_Bdue to biomechanical advantages.^16,22 To mitigate this systemic bias, we designed three alternative indices that explicitly integrate individual growth metrics to isolate true upper-body explosive capacity:

MBT_A1. This index multiplies the absolute throwing distance by the height-to-mass ratio (HMR):

1

The resulting physical unit is square centimetres per kilogram cm²/kg (or cm²kg^− 1).

MBT_A2. This index divides the absolute throwing distance by the product of body height and body mass:

2

Because length units (cm) cancel out within the fraction, the resulting physical unit is inverse kilograms (1/kg or kg^− 1).

MBT_A3. This index scales the absolute throwing distance relative to the Body Mass Index (BMI).

3

The resulting physical unit is cubic meters per kilogram: m³/kg (or: m³kg^− 1).

Standardization and classification

To avoid structural distortions when mixing external reference populations, all anthropometric, fitness, and MBT_A variables were standardized using internal cohort metrics. Age- and sex-adjusted z-scores were calculated independently within each cohort (n = 821 for SP1; n = 354 for SP2; PU: n= 178) utilizing the generalized additive models for location, scale, and shape (GAMLSS) framework to execute the LMS method.²³ Smooth reference centiles were generated modelling the skewness (L), median (M), and coefficient of variation (S) using the Box-Cox Cole-Green (BCCG) family distribution.²⁴ For descriptive baseline purposes, raw BMI parameters were also mapped onto the international 5-level International Obesity Taskforce (IOTF) weight classification system (thinness, normal weight, overweight, obesity, and morbid obesity).²⁵

Statistical inference with linear mixed models (LMMs)

Primary analyses targeted the primary school cohort (SP1), while secondary and high school datasets (SP2) evaluated the generalizability of the modelling structures to adolescents.

The first set of LMMs assessed the alignment of the three anthropometrically enriched MBT_A variants compared with the traditional MBT_T across standard performance profiles. We specified four independent models where a five-level repeated-measures factor (task) represented the dependent variables: the four physical-fitness tasks plus either MBT_T or one of the 3 MBT_A variants. Fixed effects evaluated quadratic BMI and school grade. Random factors accounted for child- and school-level nesting, providing variance components (VCs) and correlation parameters (CPs) across the five repeated motor metrics.

The second set of LMMs evaluated the comparative predictive efficiency of the traditional and alternative metrics. Rather than acting as a dependent level of the task factor, each MBT index was repositioned as a fixed-effect covariate (independent predictor) for the remaining physical-fitness tasks. Control covariates and random-effect nesting structures were identical to the first set of LMMs.

Because all four MBT indices and standard fitness metrics were converted into unitless, internally standardized z-scores prior to LMM estimation, the dependent variables occupied an identical metric scale. This statistical standardization isolates model comparisons from the arbitrary physical dimensions of the raw scores, enabling Likelihood Ratio Tests (LRT), Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC) to directly reflect true relative goodness-of-fit. Since competitive model alternatives shared identical parameter dimensions and observations, delta values across LRT, AIC, and BIC converged identically.

Model optimization and selection proceeded by comparing LMM configurations with varying complexities in their fixed-effects structures. Final model selection was adjudicated using Likelihood Ratio Tests (LRT) and the Akaike Information Criterion (AIC), adopting the strict parsimony criterion where a more complex model alternative required an AIC reduction of more than five units to be selected over a simpler variant.²⁶

For the adolescent cohort (SP2), an identical analysis strategy was deployed using the standing long jump and push-up as comparison fitness tasks. Given that these data were collected within only two schools, the random factor school was omitted from the variance components. Consequently, the fixed effect of grade was replaced with a binary fixed-effect factor representing school type (secondary vs. high school).

Statistical computations and visualization were implemented using the tidyverse (version 2.0.0)²⁷ and easystats (version 0.7.5)²⁸ ecologies in the R statistical environment (version 4.5.2).²⁹ Cohort-specific LMS z-scores were computed via the gamlss package (version 5.5-0).²⁴ LMM fitting and estimation were executed utilizing MixedModels.jl (version 5.1.0)³⁰ within the Julia language framework (version 1.12.5)³¹ to optimize computational efficiency.

Use of artificial intelligence

AI assistants (Gemini Pro, Claude Sonnet 4.6) were consulted for brainstorming and technical language polishing. The authors cross-examined all outputs and retain complete ownership and responsibility for the integrity of the published content.

Results

Descriptive statistics

Descriptive statistics for the primary school (SP1) and adolescent (SP2) cohorts are reported in the Supplement (Table S1). All subsequent inferential models are evaluated utilizing internal, cohort-specific z-scores adjusted for age and sex via the GAMLSS LMS methodology.

Alignment of MBT_T and MBT_A variants with physical fitness

Bivariate correlation analyses of z-scores across the six physical-fitness tasks (i.e., MBT, 6MR, 4 × 10SHR, JS, SLJ, and PU) revealed that all three anthropometrically enriched MBT_A variants correlated substantially higher with the six physical-fitness tasks than MBT_T (see Supplement Table S2 below diagonal; correlations of raw scores are provided above the diagonal). Among the MBT_A options, MBT_A1 demonstrated the strongest association with other physical-fitness tasks, but, except for PU (A1: 0.69, A2: 0.52, A3: 0.43), differences between the alternative indices remained below 0.10.

Zero-order relationships for BMI revealed the expected concave alignment of anthropometrically enriched MBT_A1, MBT_A2, and MBT_A3 with 6MR, 4 × 10 SHR, JS, and SLJ for SP1 (see Fig. 1A) and with SLJ and PU for SP2 (see Fig. 1B). In stark contrast to MBT_T (left panel in the second row of Fig. 1A; left panel in the first row of Fig. 1B), —where throwing performance continues to rise linearly alongside elevated BMI due to an unadjusted inertial mass advantage—the anthropometrically enriched variants showed a pronounced performance decline at high BMI levels, matching the constraints observed in running and jumping.

Fig. 1.

(A). SP1: Zero-order relations for physical fitness z-scores by BMI z-scores, adjusted for age and sex with sample-based. LMS parameters. Curves are quadratic fits to children’s scores with 95% CIs; points are binned means for five BMI_IOTF groups; (B) same as (A) for SP2.

The goodness-of-fit statistics for LMMs using one of the four MBT indices in turn as the dependent variable and quadratic BMI and four physical-fitness tasks in SP1, respectively two physical-fitness tasks in SP2, as covariates are shown in the top two blocks of Table 1 for SP1 and SP2. The bottom two blocks of Table 1 contain corresponding statistics when the MBT indices were used in turn as covariates to predict performance in the four physical fitness tasks of SP1, respectively two physical-fitness tasks of SP2; quadratic BMI was always included as well.

Table 1.

Goodness-of-fit statistics for alignment LMMs (using MBT as the dependent variable) and predictive LMMs (using MBT as a covariate for predicting PF) for SP1 and SP2.

MBT_ as…	Data	Index	df	deviance	AIC	BIC
DV	SP1	A1	47	9643	9737	10,034
		A2	47	9615	9709	10,006
		A3	47	9745	9839	10,136
		T	47	9953	10,047	10,344
	SP2	A1	17	2091	2125	2206
		A2	17	2118	2152	2234
		A3	17	2127	2161	2242
		T	17	2193	2227	2308
Covariate	SP1	A1	38	7660	7736	7968
		A2	38	7732	7808	8040
		A3	38	7675	7751	7982
		T	38	7688	7764	7996
	SP2	A1	13	1223	1249	1305
		A2	13	1265	1291	1346
		A3	13	1235	1261	1316
		T	13	1248	1274	1330

Note. df: degrees of freedom (i.e., number of model parameters). DV: dependent variable. The best fitting models in each set of four are set in bold. Differences in goodness-of-fit for different DVs are identical for deviance, AIC, and BIC because of identical number of observations and model parameters.

Goodness-of-fit parameters for the first set of LMMs, which treated the respective MBT variants as a dependent level within the repeated-measures task factor, are detailed in the two upper tiers of Table 1 for SP1 and SP2, respectively. Corresponding statistics for the second set of LMMs, which treated the MBT variants as an independent covariate predicting the remaining performance markers, occupy the lower tiers of Table 1. Both modelling blocks integrated quadratic BMI for strict statistical control.

Information criterion comparisons strongly favoured the MBT_A1 variant across the majority of evaluations. It provided the most parsimonious data fit (ΔAIC > 5) in three of the four specified LMM sets, performing marginally second to MBT_A2 only within the primary school alignment framework (SP1 dependent variable tier). Overall, the drop in AIC and BIC values confirmed that MBT_A1 establishes a significantly superior fit compared to traditional baseline scoring.

The divergence between the unadjusted and enriched frameworks was verified by the LMM fixed-effect estimates. Table 2 contrasts the fixed effects of MBT_A1 and MBT_T; fixed effects for all four models are documented in Supplement Table S3. The structural pivot is illustrated by a complete reversal of the linear BMI coefficient. While a rising BMI predicted superior performance for MBT_T (BMI = 0.235), it acted as a severe performance penalty for the enriched MBT_A1 (BMI = − 0.547). Moreover, the quadratic component was significant only for MBT_A1 (BMI² = − 0.133, p <.001; MBT_T: BMI² = − 0.016, p=.618). Similar directional transformations were observed for MBT_A2 and MBT_A3 (Supplement Table S3). Finally, within the primary school cohorts, third-grade children achieved significantly higher overall margins across all models than second-grade children, even after accounting for the pre-standardized age and sex distributions.

Table 2.

LMM Fixed-effect estimates BMI, BMI², and grade on physical fitness in MBT_A1 (left part) or MBT_T (right part) and 6-minute run (6MR), jumping sideways (JS), shuttle run (4 × 10SHR), and standing long jump (SLJ). The estimates are from SP1.

Task: parameter	LMM MBTA1				LMM MBTT
	Coef	SE	z	p	Coef	SE	z	p
MBT: BMI	−0.547	0.028	−19.280	< 0.001	0.235	0.034	6.984	< 0.001
BMI 2	−0.133	0.027	−4.884	< 0.001	−0.016	0.032	−0.499	0.618
6MR: BMI	−0.365	0.030	−12.167	< 0.001	−0.365	0.030	−12.170	< 0.001
: BMI2	−0.185	0.029	−6.437	< 0.001	−0.185	0.029	−6.436	< 0.001
4 × 10SHR: BMI	−0.216	0.031	−6.899	< 0.001	−0.216	0.031	−6.896	< 0.001
: BMI2	−0.148	0.030	−4.935	< 0.001	−0.148	0.030	−4.936	< 0.001
JS: BMI	−0.133	0.031	−4.268	< 0.001	−0.133	0.031	−4.265	< 0.001
: BMI2	−0.056	0.030	−1.863	0.063	−0.055	0.030	−1.852	0.064
SLJ: BMI	−0.234	0.033	−7.039	< 0.001	−0.234	0.033	−7.039	< 0.001
: BMI2	−0.135	0.032	−4.245	< 0.001	−0.135	0.032	−4.245	< 0.001
Grade	0.085	0.022	3.933	< 0.001	0.087	0.022	3.916	< 0.001

Note. The dependent variables are MBT, 6MR, JS, 4 × 10SHR, and SLJ; the independent variables are BMI, BMI², and grade. The bold values highlight the alignment of MBT_A1 and the misalignment of MBT_T with other fitness tasks. Estimates for MBT_A2 and MBT_A3 are in Supplement Table S3.

This alignment was successfully replicated within the adolescent cohort (SP2; see Table 3). Linear BMI again operated in diametrically opposite directions (MBT_A1: BMI = − 0.537, p <.001; MBT_T: BMI = 0.388, p <.001) and the negative quadratic trend remained significant for MBT_A1 (BMI² = − 0.175, p <.001). Parallel modelling patterns for MBT_A2 and MBT_A3 are documented in Supplement Table S4. The fixed effect for school type (secondary vs. high school) was not statistically significant in any of the four LMMs. Thus, anthropometrically enriching MBT aligns the measure with the biomechanical constraints of running and jumping, keeping in mind that this reversal also mathematically results from how the adjusted indices are calculated.

Table 3.

LMM Fixed-effect estimates of BMI, BMI², and school on physical fitness in MBT_A1 (left part) or MBT_T (right part) and push-up (PU) and standing long jump (SLJ). The estimates are from SP2.

Task: parameter	LMM MBTA1				LMM MBTT
	Coef	SE	z	p	Coef	SE	z	p
MBT: BMI	−0.537	0.044	−12.251	< 0.001	0.388	0.049	7.950	< 0.001
BMI 2	−0.175	0.044	−3.977	< 0.001	−0.066	0.049	−1.352	0.176
SLJ: BMI	−0.310	0.049	−6.387	< 0.001	−0.310	0.049	−6.384	< 0.001
BMI2	−0.248	0.049	−5.087	< 0.001	−0.248	0.049	−5.085	< 0.001
PU: BMI	−0.135	0.071	−1.897	0.058	−0.155	0.072	−2.172	0.030
BMI2	−0.194	0.072	−2.681	0.007	−0.189	0.073	−2.593	0.010
School	−0.105	0.083	−1.265	0.206	−0.115	0.087	−1.315	0.188

Note. The dependent variables are MBT, SLJ, and PU; the independent variables are BMI, BMI², and school. The bold values highlight the alignment of MBT_A1 and the misalignment of MBT_T with other fitness tasks. The estimates of MBT_A2 and MBT_A3 are in Supplement Table S4.

Predictive power of MBT variants for physical fitness tasks

The anthropometrically enriched indices mirrored the BMI curvature profiles of the standard physical-fitness tasks, diverging sharply from the traditional MBT_T baseline. Because height and mass can be collected with minimal operational overhead, we evaluated the comparative efficiency of these metrics as frontline predictors of broader motor performance. Figure 2A (SP1) and Fig. 2B (SP2) plot the zero-order effects of the four MBT indices (lines) for each of the four (SP1), respectively, two (SP2) physical-fitness outcomes. While the predictive slopes of MBT_A1, MBT_A2, and MBT_A3 tracked tightly together, they were also visibly steeper than the flat pathway generated by traditional MBT_T.

Fig. 2.

(A) SP1: Zero-order relations for four sex- and age-adjusted six-minute run (6MR), jumping sideways (JS), shuttle run (4 × 10SHR), and standing long jump (SLJ) by covariates MBT_T, MBT_A1, MBT_A2, and MBT_A3. Curves are quadratic fits to children’s scores with 95% CIs. ((B) SP2: same for sex- and age-adjusted push-up (PU) and SLJ. Only the difference between MBT_T (red lines) and the three MBT_As is clearly visible.

To statistically test these predictive dimensions, each MBT variant was nested sequentially under the respective fitness-task levels within the secondary LMM design, utilizing quadratic BMI and grade (SP1) or school type (SP2) for control. The LMM was estimated separately for each of the four MBT indices for SP1 and SP2. Table 4 contains fixed-effect statistics for the four MBT indices as covariates for SP1 (top blocks) and for SP2 (bottom blocks). In agreement with Fig. 2 for all four physical-fitness tasks of SP1 and for the two physical-fitness tasks of SP2, all MBT-covariate slopes were significantly positive. The predictive magnitude was always lowest for MBT_T. The three enriched variants produced comparable, highly elevated coefficients, with minor numerical advantages for MBT_A1; exceptions were MBT_A2 for SLJ in SP1 (MBT_A1 = 0.423, MBT_A2 = 0.436) and PU in SP2 (MBT_A1 = 0.615, MBT_A2 = 0.746). Complete fixed-effect parameters are detailed in the Supplement Table S5 and Table S6.

Table 4.

LMM fixed effect estimates for eight LMMs with varying MBT covariates for SP1 (top four LMMs) and SP2 (bottom four LMMs).

Data	Covariate	Task	Coef	SE	z
SP1	A1	6MR	0.327	0.035	9.362
		4 × 10SHR	0.437	0.035	12.494
		JS	0.423	0.035	12.006
		SLJ	0.561	0.036	15.730
	A2	6MR	0.314	0.038	8.373
		4 × 10SHR	0.402	0.038	10.543
		JS	0.436	0.038	11.559
		SLJ	0.500	0.040	12.608
	A3	6MR	0.298	0.033	8.971
		4 × 10SHR	0.411	0.033	12.354
		JS	0.374	0.034	11.075
		SLJ	0.530	0.034	15.636
	T	6MR	0.254	0.030	8.553
		4 × 10SHR	0.367	0.030	12.415
		JS	0.327	0.030	10.890
		SLJ	0.458	0.030	15.096
SP2	A1	SLJ	0.697	0.046	15.170
		PU	0.615	0.085	7.200
	A2	SLJ	0.592	0.051	11.552
		PU	0.746	0.082	9.127
	A3	SLJ	0.673	0.044	15.187
		PU	0.469	0.085	5.497
	T	SLJ	0.607	0.042	14.464
		PU	0.418	0.077	5.402

Note. Other covariates were BMI, BMI², grade (SP1) and school (SP2); see Supplement Tables S5 and S6 for a complete list of all fixed effects; all p-values < 0.001; largest covariate coefficient for each PF task is set in bold.

Variance components and correlation parameters

Variance components (VCs) and correlation parameters (CPs) of the 16 LMMs are documented in Supplement Table S7. Within the primary school sample (SP1), all child-related CPs were significantly positive. On average, they were numerically slightly larger with MBT_A1 (r =.60) than with MBT_A2 (r =.57) or MBT_A3 (r =.58) for LMM1 and numerically slightly smaller for LMM2 (MBT_A1: r =.49, MBT_A2: r =.51, MBT_A3: r =.53). Thus, when included as an indicator, MBT_A1 yielded a slightly better-defined latent construct of physical fitness than did MBT_A2 or MBT_A3. Reciprocally, these individual variances contracted and shifted into the fixed-effect expressions when the enriched metrics were deployed directly as predictive covariates. Of course, these small differences are not significant. Residual correlation parameters across school-level nesting structures tracked with less systematic variance, presumably because there were only twelve schools for children and only three CPs for adolescents.

Discussion

Traditional and anthropometrically enriched MBT_A

The medicine ball throw (MBT) is a straightforward, low-cost field test that enables rapid group testing. It has been used since 1930 to assess upper-body explosive strength³². Despite its practical advantages, it is rarely incorporated into international health-related fitness batteries³³. A key reason for its omission is that throwing distance is directly confounded by body mass, reducing its validity for health-related screenings and systematically biasing performance rankings and talent identification protocols.^16,]34

Three fundamental differences separate MBT from weight-bearing physical-fitness tasks. First, high body mass confers an inertial advantage in projectile tasks (i.e., force = mass × acceleration), whereas excess mass acts as a significant ballast performance cost in the other five physical-fitness tasks (6MR, SLJ, JS, 4 × 10 SHR, PU). Second, the MBT does not require participants to move their own body weight, whereas the other physical-fitness tasks require displacing body mass over a target time or distance.

Third, heavier children and adolescents often exhibit functional adaptations to habitual weight-bearing—effectively ‘training’ absolute muscular strength through daily activities—which benefits throwing distance but hinders performance in weight-dependent motor tasks.^35[,36 Consequently, the observed divergence between MBT_T and the rest of the fitness battery is biomechanically expected.

All three MBT_A variants transform the assessment by penalizing excess body mass, neutralizing the structural advantage heavier children and adolescents possess in projectile tasks. This correction effectively shifts the raw metric from a proxy of absolute force into a measure of relative work efficiency, aligning the throw with running and jumping tasks. However, the indices differ in their biomechanical treatment of stature. The first formula (MBT_A1) treats height as a functional asset. Conversely, in the second formula (MBT_A2), applies height as a secondary performance penalty, while the third formula (MBT_A3) weights the height asset even more heavily. Scaling positively for height in the first and third indices is physically justified: taller participants benefit from a higher release point and longer skeletal levers (arm span), which extend the force-application path and increase release velocity.

While both MBT_A1 (HMR multiplication) and MBT_A3 (BMI division) treat height as an asset, the theoretical preference for MBT_A1 rests on the classical principles of allometric scaling. Biologically, muscle strength scales with muscle cross-sectional area (L²), whereas body mass scales with volume (L³)³⁷. MBT_A1 accounts for this dimensional discrepancy by scaling the absolute throw (a proxy for force, L²) by the height-to-mass ratio (L/L³ = L^− 2), resulting in a scale-invariant metric (L² L^− 2 = 1). Although this assumption of isometric geometric similarity may not capture nuanced allometric shifts in body proportions during puberty, it provides a mathematically orthogonal measure of relative strength that isolates performance from absolute size. In contrast, MBT_A3 normalizes by BMI (M/L²), which scales linearly with height (L). Consequently, MBT_A3 fails to fully eliminate the size advantage, leaving a residual bias that artificially favours taller participants (L²/L = L). This theoretical preference for the height-to-mass ratio is supported by independent evidence showing that HMR-based models yield a superior fit compared to BMI models when predicting physical fitness³⁸.

Empirically, all three MBT_A variants achieved the intended qualitative alignment with weight-bearing physical-fitness tasks and outperformed the traditional MBT_T. MBT_A1 provided the best goodness of fit in three of the four LMM comparisons, though the differences in goodness-of-fit between the variants were minor. The single exception occurred in the primary-school cohort (6–9 years) where MBT_A2 performed marginally better. This shift may reflect a developmental “coordination lag” in younger children, for whom greater height does not yet confer a biomechanical advantage because they cannot yet control longer limbs effectively. For young children, size acts primarily as “bulk”; during adolescence, it matures into functional “leverage.” This matches evidence indicating that anthropometric growth and motor-coordination development contribute independently to throwing mechanics across childhood.^39–41

The empirical performance of MBT_A1 aligns with the theoretical preference for a scale-invariant metric derived from the square-cube law (see above). The minor exception in the child cohort may simply represent a statistical miss for MBT_A1 or a statistical false positive for MBT_A2. Furthermore, because MBT_A1 and MBT_A2 correlate exceptionally high (r=.92 across both cohorts; see Supplement Table S2), the choice between them will rarely alter practical classifications. For reasons of theoretical consistency and parsimony, we recommend MBT_A1 (throwing distance scaled by the height-to-mass ratio) as the standard method for youth fitness batteries, hereafter refer to as hmrMBT (height-to-mass ratio medicine ball throw).

Implications for participants physical-fitness assessment and sports training in general

It is standard practice to take participants’ sex and age into account when assessing their physical fitness. This increases the validity because performance is interpreted relative to developmental baselines, facilitating the tracking of training gains or developmental change. Trivially, all else being equal, taller participants run faster and jump further than smaller peers. For the same reason that performance is evaluated relative to age and sex, accounting for anthropometrics provides a more accurate, differentiated motor profile. For example, hmrMBT mitigates the confounding influence of rapid growth spurts or delayed physical maturation on physical-fitness tracking.

In athletic training, macro-, meso-, and micro-cycles focus on distinct physical objectives. The hmrMBTmay prove valuable in strength-and-conditioning programs where a secondary goal is reducing body mass over a targeted micro cycle⁴². Increasing assessment precision via indices like hmrMBT allows practitioners to track specific goals more effectively. Weight loss or gain due to illness, injury or training adaptations can be directly accounted for upon returning to training, making is easier to isolate true performance gains.

Compared with competing with peers, transparency and objective performance assessments have been shown to increase athletes’ personal motivation and self-assessment accuracy.^43[,44 Furthermore, external and intrinsic motivation levels are directly tied to sports retention; as motivation rises, dropout rates decrease⁴⁵.

In summary, more precise assessments lead to clearer communication of physical performance, ultimately increasing participant acceptance and motivation. Utilizing the hmrMBTmay (1) reduce anthropometric bias in youth competitions, (2) increase athletes’ understanding of coaching and educational decisions, and (3) improve the acceptance of performance feedback among athletes and their legal guardians⁴⁶.

Limitations

As with any novel performance index, arguments for anthropometrically enriched carry the risk of misapplication. First, our modelling is restricted to body mass and height. In principle, other parameters could also be integrated; for instance, sitting height is relevant for calculating peak-height velocity (PHV) as a proxy of biological maturation during puberty. While hmrMBT effectively mitigates the impact of growth spurts on performance scoring, future research should incorporate maturation offsets to further refine these metrics, as omitting biological growth stages limits evaluation accuracy in paediatric populations.

Second, a mathematical limitation of the current modelling approach is the structural overlap between the MBT_A variants and the covariates. Because MBT_A1 (i.e., hmrMBT) incorporates the height-to-mass ratio (HMR) directly within the dependent variable, it is mathematically confounded with BMI (= m/h²) when the latter is used as a covariate. This structural overlap likely inflates the negative linear component of the relationship compared to other fitness tasks. However, the primary goal of this modelling was to achieve qualitative alignment with the established youth fitness profiles. All MBT_A variants successfully replicated the significant inverted U-shape (negative quadratic) relationship with BMI observed in running and jumping tasks, successfully correcting the linear inertial bias found in traditional assessments. While the absolute magnitude of the BMI coefficients should not be compared directly across different task types due to these structural differences, the resulting quadratic profile provides a more realistic assessment of physical fitness.

Third, a single index may not suit all physical-fitness tasks. For example, a high body mass is an asset in American football, whereas a low body mass is required in endurance-specific disciplines. Some sports may benefit while others are negatively affected if specific demands are ignored. Moreover, for certain motor tests, anthropometric adjustments are unnecessary; for instance, anthropometrics can be safely ignored in reaction-time tasks, where nerve conduction speeds render minor variations in body size irrelevant.

Fourth, the most appropriate index may change across developmental stages. In our study, the small advantage of MBT_A2 over hmrMBT in children may relate to a temporary lag in motor coordination development. Complementarily, during later development, adolescent growth spurts involve non-isometric changes in lever-arm mechanics and rigid-body proportions, which may introduce residual variance not captured by the scale-invariant metric. While we prefer hmrMBT for reasons of theoretical und practical parsimony, designing an index sensitive to developmental change remains a valuable topic for future research.

Fifth, measurement protocols for body height and body mass must be executed with high precision to minimize errors. Because these anthropometric values scale or divide the raw throwing distance, measurement errors can have compounding negative effects, substantially distorting individual assessment scores across multiple tests.

Sixth, increased data complexity often leads to unfamiliar data formats. The alternative MBT_A variants introduce unusual physical units such as kg^− 1, cm²kg^− 1 and m³kg^− 1. Clear explanations and targeted training will likely be necessary to achieve practical acceptance among physical education teachers, coaches, and administrators responsible for fitness testing.

Finally, a further limitation of this study is that these alternative indices have not yet been validated over longer periods within a longitudinal design. Additionally, field tests such as push-ups, jumping, and running tasks cannot provide the same level of kinetic precision as laboratory measurements.

Conclusion

The anthropometrically enriched hmrMBT index has the potential to optimize youth physical-fitness tracking by providing a fairer, closer monitoring of performance. This methodology could be extended to other tests within fitness batteries, such as the 6-minute run or the 4 × 10-meter shuttle run. Enhancing measurement validity in this manner can improve test acceptance and motivation among students and athletes, as well as the instructors and coaches evaluating them. Ultimately, this approach may support the implementation of large-scale or nationwide physical-fitness monitoring systems, an evaluation strategy that is urgently needed in the era of rising paediatric obesity.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

Conceptualization, G.J. and M.N.M.v.P.; methodology, G.J.; formal analysis, G.J. and R.Kl.; investigation, G.J.; resources, G.J.; data curation, G.J.; writing—original draft preparation, G.J. and R.Kl.; writing—review and editing, G.J., R.Ke., R.Kl. and M.N.M.v.P.; visualization, G.J. and R.Kl.; supervision, M.N.M.v.P.; project administration, G.J.; funding acquisition, G.J. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Austrian Federal Ministry of Education, Science and Research, grant number GZ:2024 − 0.780.614, and the Austrian Federal Ministry for Arts, Culture, Civil Service and Sport, grant number GZ:2025 − 0.021.608. Open Access was funded by the University of Graz. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Data availability

The data presented in this study are available upon request from the corresponding author. The data are not publicly available due to privacy/ethical restrictions.

Declarations

Competing interests

The authors declare no competing interests.

Institutional review board

The study was conducted according to the guidelines of the Declaration of Helsinki and approved by the Research Ethics Committee of the University of Graz, Styria, Austria (GZ. 39/68/63 ex 2021/22).

Informed consent

Informed consent was obtained from all the subjects involved in the study.