The variance of the discrepancy distribution of rounding procedures, and sums of uniform random variables

When ℓ probabilities are rounded to integer multiples of a given accuracy n , the sum of the numerators may deviate from n by a nonzero discrepancy. It is proved that, for large accuracies n → ∞ , the limiting discrepancy distribution has variance ℓ / 12 . The relation to the uniform distribution over the interval [ - 1 / 2 , 1 / 2 ] , whose variance is 1 / 12, is explored in detail.