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.

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[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:

-1.2672,\ \ -0.6418,\ \ -0.2016,\ \ 0.2016,\ \ 0.6418,\ \ 1.2672\,.

Suppose the numbers in a data set are

65, 75, 16, 22, 43, 40.

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{align} \mbox{data}\ \mbox{point} & & \mbox{rankit} \\ 65 & {} \quad {} & 0.6418 \\ 75 & {} \quad {} & 1.2672 \\ 16 & {} \quad {} & -1.2672 \\ 22 & {} \quad {} & -0.6418 \\ 43 & {} \quad {} & 0.2016 \\ 40 & {} \quad {} & -0.2016 \end{align}

[edit] Rankit plot

Main article: 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 (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.

[edit] History

The word rankit was introduced by the biologist and statistician Chester Ittner Bliss (1899–1979).

[edit] See also

  • Probit analysis developed by C. I. Bliss in 1934.

[edit] External links