World Library  
Flag as Inappropriate
Email this Article

Normal probability plot

Article Id: WHEBN0000543119
Reproduction Date:

Title: Normal probability plot  
Author: World Heritage Encyclopedia
Language: English
Subject: Rankit, Distribution fitting, Probability plot, Normal distribution, Q–Q plot
Collection: Normal Distribution, Normality Tests, Statistical Charts and Diagrams
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Normal probability plot

The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw data, residuals from model fits, and estimated parameters.

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 probability plot for a normal distribution. The theoretical quantiles are generally chosen to approximate either the mean or the median of the corresponding order statistics.

Contents

  • Definition 1
  • Other distributions 2
  • Plot types 3
  • See also 4
  • References 5
  • Further reading 6
  • External links 7

Definition

The normal probability plot is formed by plotting the sorted data vs. an approximation to the means or medians of the corresponding order statistics; see rankit. Some users plot the data on the vertical axis;[1] others plot the data on the horizontal axis.[2][3]

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

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

for i = 1, 2, ..., n, where

a = 3/8 if n ≤ 10 and
0.5 for n > 10,

and Φ−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 2-level fractional factorial experiment. With fewer points, it becomes harder to distinguish between random variability and a substantive deviation from normality.

Other distributions

Probability plots for distributions other than the normal are computed in exactly the same way. The normal quantile function Φ−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 of distributions, the location and scale parameters of the distribution can be estimated from the 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.

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

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

See also

References

 This article incorporates public domain material from websites or documents of the National Institute of Standards and Technology.

  1. ^ e.g., Chambers et al. (1983, ch. 6. Assessing distributional assumptions about data, p. 194)
  2. ^  
  3. ^ Titterington, D. M.; Smith, A. F. M.; Makov, U. E. (1985), "4. Learning about the parameters of a mixture", Statistical Analysis of Finite Mixture Distributions, Wiley,  

Further reading

  • Chambers, John; William Cleveland; Beat Kleiner; Paul Tukey (1983). Graphical Methods for Data Analysis. Wadsworth. 

External links

  • Engineering Statistics Handbook: Normal Probability Plot
  • Statit Support: Testing for "Near-Normality": The Probability Plot


This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.