Rankit

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In statistics, the rankits of the data points in a data set consisting simply of a list of scalars are expected values of order statistics of the standard normal distribution corresponding to data points in a manner determined by the order in which the data points appear.

Contents

1 Example
2 Rankit plot
3 Relationship between rankit plots and Q-Q plots
4 History
5 External links

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:

[-1.2816,\ \ -0.64335,\ \ -0.20189,\ \ 0.20189,\ \ 0.64335,\ \ 1.2816]

Suppose the numbers in a data set are

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:

[\begin\mbox\ \mbox & & \mbox \\ \\65 & & 0.64335 \\75 & & 1.2816 \\16 & & -1.2816 \\22 & & -0.64335 \\43 & & 0.20189 \\40 & & -0.20189\end]

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.

Relationship between rankit plots and Q-Q plots

One difference between a rankit plot and a Q-Q plot (short for quantile-quantile plot) is that in a rankit plot, one plots expected values of normal order statistics on the horizontal axis, whereas in a Q-Q plot, one plots the k/(n + 1)-quantiles of the normal distribution on the horizontal axis (for k = 1, ..., n). The difference is tiny unless the sample is very small.

Another difference is that Q-Q plots included plots in which, and the horizontal axis, one plots the k/(n + 1)-quantiles of some distribution other than the normal distribution.

History

The word rankit was introduced by the statistician Chester Bliss (not to be confused with the politician Chester Bliss Bowles).

External links

[Engineering Statistics Handbook]

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