We want a single scalar number to summarize the differences between n pairs of numbers (R1, S1), (R2, S2), …, (Rn, Sn), where Rk is either 0 or 1 and Sk can take any real value from 0 to 1 (inclusive), for k = 1, 2, …, n. Perfect calibration is when the expected value of Rk is equal to Sk, for k = 1, 2, …, n.
The Kuiper metric is the absolute value of the sum of (Rk – Sk) / n, summing over only those k for which Sk falls in an interval. The interval is chosen such that the absolute value of the sum is greatest.
The reason for restricting to this worst-case interval is to minimize possible cancellation between positive and negative differences. The Kuiper statistic can take values ranging from 0 to 1 (inclusive).