Measuring volatility in per-capita school funding: why the rural–urban comparison depends on the metric

Per-capita funding formulas tie each school’s budget to its enrollment through a fixed rule, and a growing literature asks whether such formulas expose some schools to greater financial uncertainty than others. We study year-to-year volatility in per-pupil funding across rural and urban pre-university schools in Romania, using complete administrative records for 5,703 schools over 2017 to 2019. Our central message is methodological: the rural–urban volatility comparison depends on how volatility is measured. By the standard deviation of within-school annual changes, urban schools appear more volatile, with a rural-to-urban ratio of 0.87; but this gap is imprecise once a school-clustered bootstrap quantifies its uncertainty, it is confined to a single year-pair, and it is driven by a small group of urban schools whose large one-year declines reverse earlier one-year increases. By outlier-resistant statistics such as the interquartile range and the median absolute deviation, rural schools are more volatile, with pooled ratios near 1.08 whose intervals lie above one and that place rural above urban in both year-pairs; this excess is, however, modest (about 8 percent) and attenuates to about 3 percent, no longer distinguishable from equality, once common county-level shocks are removed. The reconciliation is that urban schools have heavier tails, much of which reflects transitory offsetting changes, while rural schools show somewhat greater central dispersion. A Bayesian variance-contrast model recovers the same picture: a normal likelihood reproduces the standard-deviation gap, while a heavy-tailed Student-t likelihood places the rural scale above the urban scale with posterior probability 0.99. A cross-sectional dispersion measure gives a ratio near 1.12, but this reflects persistent differences across schools, not temporal volatility. The results caution against summarizing funding stability with a single statistic and suggest attending to both ordinary variation and tail risk, each with its uncertainty.