Editing Rankit

In [[statistics]], '''rankits''' of a set of data are the expected values of the [[order statistic]]s 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 test]]ing.

[[File:Normal probability plot.svg|thumb|380px|Sample [[normal probability plot]]; horizontal axis coordinates are rankits.]]

==Example==

This is perhaps most readily understood by means of an example.  If an [[Independent identically-distributed random variables|i.i.d.]] sample of six items is taken from a [[normal distribution|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 statistic]]s are:

:&minus;1.2672, &nbsp;&nbsp;&minus;0.6418, &nbsp;&nbsp;&minus;0.2016, &nbsp;&nbsp;0.2016, &nbsp;&nbsp;0.6418, &nbsp;&nbsp;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:
{| class="wikitable" style="text-align:right"
! data point !! rankit
|-
| 16 || &minus;1.2672
|-
| 22 || &minus;0.6418
|-
| 40 || &minus;0.2016
|-
| 43 || 0.2016
|-
| 65 || 0.6418
|-
| 75 || 1.2672
|}

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:
: 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:

{| class="wikitable" style="text-align:right"
! data point !! rank !! rankit
|-
| 65 || 5 || 0.6418
|-
| 75 || 6 || 1.2672
|-
| 16 || 1 || &minus;1.2672
|-
| 22 || 2 || &minus;0.6418
|-
| 43 || 4 || 0.2016
|-
| 40 || 3 || &minus;0.2016
|}

==Rankit plot==
{{main|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]].

A rankit plot is a kind of [[Q–Q plot]] – it plots the order statistics (quantiles) of the sample against certain quantiles (the rankits) 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]] (1899&ndash;1979).

==See also==
* [[Probit]] analysis developed by C. I. Bliss in 1934.

==External links==

* [http://www.itl.nist.gov/div898/handbook/eda/section3/normprpl.htm Engineering Statistics Handbook]

[[Category:Statistical charts and diagrams]]
[[Category:Normal distribution]]