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.
; horizontal axis coordinates are rankits.

Example

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 and then sorted into increasing order, the expected values of the resulting order statistics are:
Suppose the numbers in a data set are
Then one may sort these and line them up with the corresponding rankits; in order they are
which yields the points:
data pointrankit
16−1.2672
22−0.6418
40−0.2016
430.2016
650.6418
751.2672

These points are then plotted as the vertical and horizontal coordinates of a scatter plot.

Alternative method

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:
the corresponding ranks are:
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:
data pointrankrankit
6550.6418
7561.2672
161−1.2672
222−0.6418
4340.2016
403−0.2016

Rankit 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.
A rankit plot is a kind of Q-Q plot – it plots the order statistics of the sample against certain quantiles of the assumed normal distribution. Q-Q plots may use other quantiles for the normal distribution, however.

History

The rankit plot and the word rankit was introduced by the biologist and statistician Chester Ittner Bliss.