View source for Q-Q plot

[[Image:Qqnormexp.png|350px|right|thumb|A normal Q-Q plot of randomly generated exponential(1) data.]] [[Image:Qqnormcorr.png|350px|thumb|right|A normal Q-Q plot of randomly generated normal(0,1) data.]]
In [[statistics]], a '''Q-Q plot''' ('''"Q"''' stands for ''[[quantile]]'') is a graphical method for diagnosing differences, between the [[probability distribution]] of a [[statistical population]] from which a [[random sample]] has been taken and a comparison distribution. An example of the kind of differences that can be tested for is [[normal distribution|non-normality]] of the population distribution.

For a sample of size ''n'', one plots ''n'' points, with the (''n''+1)-[[quantile]]s of the comparison distribution (e.g. the [[normal distribution]]) on the horizontal axis (for ''k'' = 1, ..., ''n''), and the [[order statistic]]s of the sample on the vertical axis. If the population distribution is the same as the comparison distribution this approximates a straight line, especially near the center. In the case of substantial deviations from [[linearity]], the statistician rejects the [[null hypothesis]] of sameness.

== Plotting positions ==
For the quantiles of the comparison distribution typically the formula ''k''/(''n''&nbsp;+&nbsp;1) is used.
Several different formulas have been used or proposed as [[symmetrical]] '''plotting positions'''. Such formulas have the form (''k''&nbsp;&minus;&nbsp;''a'')/(''n''&nbsp;+&nbsp;1&nbsp;&minus;&nbsp;2''a'') for some value of ''a'' in the range from 0 to 1/2. The above expression ''k''/(''n''&nbsp;+&nbsp;1) is one example of these, for ''a'' = 0.  Other expressions include:
* (''k''&nbsp;&minus;&nbsp;1/3)/(''n''&nbsp;+&nbsp;1/3) <ref>A simple (and easy to remember) formula for plotting positions.</ref>
* (''k''&nbsp;&minus;&nbsp;0.3175)/(''n''&nbsp;+&nbsp;0.365) <ref>[http://www.itl.nist.gov/div898/handbook/eda/section3/normprpl.htm Engineering Statistics Handbook: ''Normal Probability Plot''] &ndash; Note that this also uses a different expression for the first & last points.  [http://ase.tufts.edu/cee/faculty/vogel/probability1986.pdf] cites the original work by Filliben [[1975]].</ref>
* (''k''&nbsp;&minus;&nbsp;0.326)/(''n''&nbsp;+&nbsp;0.348) <ref>[http://cat.inist.fr/?aModele=afficheN&cpsidt=14151257 ''Distribution free plotting position'', Yu & Huang]</ref>
* (''k''&nbsp;&minus;&nbsp;0.375)/(''n''&nbsp;+&nbsp;0.25)<ref>This is Blom's earlier approximation [[1953]] and is the expression used in [[MINITAB]].</ref>
* (''k''&nbsp;&minus;&nbsp;0.44)/(''n''&nbsp;+&nbsp;0.12)<ref>This plotting position was used by Gringorten [[1963]] to plot points in tests for the [[Gumbel distribution]].</ref>
For large sample size, ''n'', there is little difference between these various expressions.

== Relation with rankit plots ==
Q-Q plots are similar to [[rankit]] plots, also called '''normal probability plots'''.  The difference is that in normal probability plots, instead of using the quantile of the normal distribution as the x-axis, one uses the [[expected value]] of the ''k''th order statistic from a normal distribution with expectation 0 and [[variance]] 1.  Only when ''n'' is small is there a substantial difference between a Q-Q plot and a normal probability plot.

== See also ==
* [[Probit]] analysis  is developed by [[Chester Ittner Bliss]] in [[1934]].

== References ==
{{reflist|2}}


[[Category:Statistical charts and diagrams]]

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