RankIt

Rankit is a term used in statistics to denote a set of reference quantiles derived from data ranks, most commonly used in normal probability plots and in normal-score transformations. A rankit value corresponds to an expected order statistic from a standard normal distribution for a given rank in a sample of size n. The idea is to map the i-th smallest observation to a reference value so that, if the data are normal, the plotted points form an approximately straight line.

Computing rankits: The simplest approach assigns to the i-th rank a value q_i = Φ^{-1}((i - 0.5)/n), where Φ^{-1} is the inverse CDF of the standard normal distribution. Other plotting-position formulas exist, such as i/(n+1) or Blom's, Tukey's, and Gringorten's formulas, which adjust the placement of ranks along the distribution. Rankits can also be described as the expected values of the i-th order statistic from a standard normal sample, or as approximate normal scores used in practice.

Applications: Rankits are used in QQ plots to visually assess normality, and in transforming data to normal scores for nonparametric procedures or for stabilizing variances. They are independent of the actual data values and depend only on the sample size. Ties are typically handled by assigning average ranks or by using rankit formulas that accommodate ties.

Overview: In statistical software, rankits are often implemented as normal scores or plotting positions used for reference lines and data transformations. The concept is central to comparing empirical data to a reference distribution and to nonparametric methods that rely on ranks.