Statistical Properties of Height of Japanese Schoolchildren

We study height distributions of Japanese schoolchildren based on the statictical data which are obtained from the school health survey by the ministry of education, culture, sports, science and technology, Japan . From our analysis, it has been clarified that the distribution of height changes from the lognormal distribution to the normal distribution in the periods of puberty.

💡 Research Summary

The paper investigates how the statistical distribution of height among Japanese schoolchildren changes with age, using a large‐scale dataset from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) school‑health survey. The survey, conducted annually, samples schools across Japan and records the heights of children aged 5 to 17. In 2006, for example, the dataset comprised 7755 schools and 695 600 children, with each age group’s heights binned into 55 one‑centimetre intervals.

The authors adopt a two‑model fitting approach. For each age i they fit (i) a normal (Gaussian) distribution and (ii) a log‑normal distribution to the observed frequency histogram. The fitting is performed with the GNUFIT routine (implemented in gnuplot) which uses the Marquardt‑Levenberg algorithm for non‑linear least squares. After fitting, the sum of squared residuals R²_i is computed for each model:

 R²_i = Σ_{j=1}^{m} (O_{ji} – E_{ji})²

where O_{ji} is the observed count in height class j and E_{ji} the model‑predicted count. To decide which model better describes the data, the authors define a ratio

 ε_i = R²(LN)_i / R²(N)_i

and examine its common logarithm (base 10). If log ε_i < 0 the log‑normal model yields smaller residuals (better fit); if log ε_i > 0 the normal model is superior.

Plotting log ε_i versus age for the 2006 dataset (and similarly for 1970) reveals a clear transition: for younger children the log‑normal model fits best, whereas around age 11 for girls and age 13 for boys the log ε_i crosses zero, indicating that the normal distribution becomes the better description. After about age 16 both models fit comparably well, reflecting a stabilization of growth variability. The same pattern is observed in a longitudinal cohort born in 1990, using data from 1995‑2007, confirming that the transition is not an artifact of a single year’s sample but a robust feature of Japanese growth.

The authors interpret this shift in statistical terms as a change in the underlying growth process. Prior to puberty, growth is assumed to be multiplicative: the size X_t at time t evolves as X_{t+1}=α_t X_t, where the growth factor α_t is independent of X_t. Under Gibrat’s law, repeated multiplicative updates generate a log‑normal distribution of sizes. During puberty, physiological changes (hormonal surges, rapid skeletal growth) cause the growth increments to become effectively additive rather than proportional, i.e., ΔX_t is added to the existing size. The sum of many independent additive increments converges to a normal distribution by the central limit theorem. Thus, the observed statistical transition mirrors a mechanistic transition from multiplicative to additive growth dynamics.

The paper also cites Limpert et al. (2001), who reported a similar mixture of log‑normal and normal characteristics in adult female height distributions, suggesting that remnants of the multiplicative regime may persist into adulthood.

Limitations acknowledged by the authors include: (1) the cross‑sectional nature of the survey—different individuals are sampled each year, preventing true longitudinal tracking; (2) the exclusive focus on height without simultaneous analysis of weight or BMI, which are closely linked to growth; and (3) the lack of comparative data from other countries to assess universality.

Future work is outlined as follows: (a) extending the analysis to international datasets to test whether the log‑normal‑to‑normal transition is a universal feature of human growth; (b) constructing a quantitative growth model (e.g., stochastic differential equations) that reproduces the observed distributional shift; and (c) integrating weight and BMI data to develop a multivariate growth framework.

In summary, the study provides robust empirical evidence that Japanese children’s height distribution evolves from log‑normal in early childhood to normal during puberty, reflecting a fundamental change in growth dynamics. This insight has practical implications for public‑health monitoring, early detection of growth disorders, and the formulation of evidence‑based educational and health policies.