Editing Normal probability plot

{{Short description|Graphical technique in statistics}}
{{Use dmy dates|date=December 2023}}
The '''normal probability plot''' is a [[graphical technique]] to identify substantive departures from [[normal distribution|normality]].  This includes identifying [[outlier]]s, [[skewness]], [[kurtosis]], a need for transformations, and [[Mixture distribution|mixtures]].  Normal probability plots are made of raw data, [[Errors and residuals in statistics|residuals from model fits]], and estimated parameters.

[[Image:normprob.png|thumb|350px|A normal probability plot]]

In a normal probability plot (also called a "normal plot"), the sorted data are plotted vs. values selected to make the resulting image look close to a straight line if the data are approximately normally distributed.  Deviations from a straight line suggest departures from normality. The plotting can be manually performed by using a special [[graph paper]], called ''normal probability paper''.  With modern computers normal plots are commonly made with software.

The normal probability plot is a special case of the [[Q–Q plot|Q–Q]] probability plot for a normal distribution.  The theoretical [[quantile]]s are generally chosen to approximate either the mean or the median of the corresponding [[order statistic]]s.

==Definition==
The normal probability plot is formed by plotting the sorted data vs. an approximation to the means or medians of the corresponding [[order statistic]]s; see [[rankit]].  Some plot the data on the vertical axis;<ref>e.g., Chambers et al. (1983, ch. 6.  Assessing distributional assumptions about data, p. 194)</ref> others plot the data on the horizontal axis.<ref>{{Citation | last1 = Box | first1 = George E. P. 
| last2 = Draper | first2 = Norman | author-link = George E. P. Box 
| year = 2007 
| title = Response Surfaces, Mixtures, and Ridge Analysis 
| edition = 2nd 
| publisher = Wiley 
| isbn = 978-0-470-05357-7}}</ref><ref>
{{Citation
| last1 =  Titterington
| first1 = D. M. 
| last2 = Smith
| first2  = A. F. M. 
| last3 = Makov 
| first3 = U. E. 
| year = 1985 
| title = Statistical Analysis of Finite Mixture Distributions 
| chapter = 4. Learning about the parameters of a mixture 
| publisher = Wiley
| isbn = 0-471-90763-4
}}</ref>

Different sources use slightly different approximations for [[rankits]].  The formula used by the "qqnorm" function in the basic "stats" package in [[R (programming language)]] is as follows:

: <math> z_i = \Phi^{-1}\left( \frac{i-a}{n+1-2a} \right), </math>

for {{math|''i'' {{=}} 1, 2, ..., ''n''}}, 
where

:{{math|''a'' {{=}} 3/8}} if {{math|''n''&nbsp;≤&nbsp;10}} and 
::0.5 for ''n''&nbsp;>&nbsp;10,

and {{math|&Phi;{{sup|−1}}}} is the standard normal [[quantile function]].

If the data are consistent with a sample from a normal distribution, the points should lie close to a straight line. As a reference, a straight line can be fit to the points. The further the points vary from this line, the greater the indication of departure from normality. If the sample has mean 0, standard deviation 1 then a line through 0 with slope 1 could be used.

With more points, random deviations from a line will be less pronounced.  Normal plots are often used with as few as 7 points, e.g., with plotting the effects in a saturated model from a [[Fractional factorial design|2-level fractional factorial experiment]].  With fewer points, it becomes harder to distinguish between random variability and a substantive deviation from normality.

==Other distributions==
{{main|Q–Q plot}}
Probability plots for distributions other than the normal are computed in exactly the same way. The normal quantile function {{math|&Phi;{{sup|−1}}}} is simply replaced by the quantile function of the desired distribution. In this way, a probability plot can easily be generated for any distribution for which one has the quantile function.

With a [[Location-scale family|location-scale family of distributions]], the [[location parameter|location]] and [[scale parameter]]s of the distribution can be estimated from the [[Y-intercept|intercept]] and the [[slope]] of the line.  For other distributions the parameters must first be estimated before a probability plot can be made.

==Plot types==
This is a sample of size 50 from a normal distribution, plotted as both a histogram, and a normal probability plot.

<gallery widths="200" heights="180">
File:normprob.png|Normal probability plot of a sample from a normal distribution – it looks fairly straight, at least when the few large and small values are ignored.
File:normhist.png|Histogram of a sample from a normal distribution  – it looks fairly symmetric and unimodal
</gallery>

This is a sample of size 50 from a right-skewed distribution, plotted as both a histogram, and a normal probability plot.

<gallery widths="200" heights="180">
File:normexpprob.png|Normal probability plot of a sample from a right-skewed distribution – it has an inverted C shape.
File:normexphist.png|Histogram of a sample from a right-skewed distribution – it looks unimodal and skewed right.
</gallery>

This is a sample of size 50 from a uniform distribution, plotted as both a histogram, and a normal probability plot.

<gallery widths="200" heights="180">
File:normunifprob.png|Normal probability plot of a sample from a uniform distribution – it has an S shape.
File:normunifhist.png|Histogram of a sample from a uniform distribution  – it looks multimodal and supposedly roughly symmetric.
</gallery>

==See also==
* [[P–P plot]]
* [[Q–Q plot]]
* [[Rankit]]

==References==
{{NIST-PD}}
{{more footnotes|date=July 2011}}
{{reflist}}

== Further reading ==
*{{cite book|last = Chambers|first= John|author2=William Cleveland |author3=Beat Kleiner |author4=Paul Tukey |year = 1983|title = Graphical Methods for Data Analysis|publisher = Wadsworth}}

==External links==
{{Commons category|Normal probability plots}}
* [http://www.itl.nist.gov/div898/handbook/eda/section3/normprpl.htm Engineering Statistics Handbook: Normal Probability Plot]
* [http://www.statit.com/support/quality_practice_tips/testingfornearnormality.shtml Statit Support: Testing for "Near-Normality": The Probability Plot]

{{Distribution fitting}}

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