Rankit

In statistics, rankits of a set of data are the expected values of the order statistics of a sample from the standard normal distribution the same size as the data. They are primarily used in the normal probability plot, a graphical technique for normality testing. This is perhaps most readily understood by means of an example. If an i.i.d. sample of six items is taken from a normally distributed population with expected value 0 and variance 1 (the standard normal distribution) and then sorted into increasing order, the expected values of the resulting order statistics are: −1.2672, −0.6418, −0.2016, 0.2016, 0.6418, 1.2672. Suppose the numbers in a data set are 65, 75, 16, 22, 43, 40. Then one may sort these and line them up with the corresponding rankits; in order they are 16, 22, 40, 43, 65, 75, which yields the points: These points are then plotted as the vertical and horizontal coordinates of a scatter plot. Alternatively, rather than sort the data points, one may rank them, and rearrange the rankits accordingly. This yields the same pairs of numbers, but in a different order. For: 65, 75, 16, 22, 43, 40, the corresponding ranks are: 5, 6, 1, 2, 4, 3, i.e., the number appearing first is the 5th-smallest, the number appearing second is 6th-smallest, the number appearing third is smallest, the number appearing fourth is 2nd-smallest, etc. One rearranges the expected normal order statistics accordingly, getting the rankits of this data set: Normal probability plot A graph plotting the rankits on the horizontal axis and the data points on the vertical axis is called a rankit plot or a normal probability plot. Such a plot is necessarily nondecreasing. In large samples from a normally distributed population, such a plot will approximate a straight line. Substantial deviations from straightness are considered evidence against normality of the distribution. Rankit plots are usually used to visually demonstrate whether data are from a specified probability distribution.