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.

1 Example
2 Rankit plot
3 Relation with Q-Q plots
4 History
5 See also
6 External links

[edit] 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 (the standard normal distribution) 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:

$\begin{matrix} \mbox{data}\ \mbox{point} & & \mbox{rankit} \\ \\ 65 & & 0.64335 \\ 75 & & 1.2816 \\ 16 & & -1.2816 \\ 22 & & -0.64335 \\ 43 & & 0.20189 \\ 40 & & -0.20189\end{matrix}$

[edit] Rankit plot

A graph plotting the rankits on the horizontal axis and the data points on the vertical axis is called a rankit plot (sometimes called 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.

[edit] Relation with 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 quantiles of the normal distribution on the horizontal axis. The difference is tiny unless the sample is very small. Another difference is that for Q-Q plots, one may plot for the horizontal axis the quantiles of some distribution other than the normal distribution.

[edit] History

The word rankit was introduced by the biologist and statistician Chester Ittner Bliss (1899-1979), (not to be confused with the politician Chester Bliss Bowles).