Q-Q plot
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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 non-normality of the population distribution.
For a sample of size n, one plots n points, with the (n + 1)-quantiles of the comparison distribution (e.g. the normal distribution) on the horizontal axis (for k = 1, ..., n), and the order statistics 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.
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[edit] Plotting positions
For the quantiles of the comparison distribution typically the formula k/(n + 1) is used. Several different formulas have been used or proposed as symmetrical plotting positions. Such formulas have the form (k − a)/(n + 1 − 2a) for some value of a in the range from 0 to 1/2. The above expression k/(n + 1) is one example of these, for a = 0. Other expressions include:
- (k − 1/3)/(n + 1/3) [1]
- (k − 0.3175)/(n + 0.365) [2]
- (k − 0.326)/(n + 0.348) [3]
- (k − 0.375)/(n + 0.25)[4]
- (k − 0.44)/(n + 0.12)[5]
For large sample size, n, there is little difference between these various expressions.
[edit] Relation with probability plots
Q-Q plots are similar to probability plots (which for a normal distribution are called normal probability plots or rankit plots). The difference is that in a probability plot, instead of using the quantile of the distribution as the x-axis, one uses the expected value of the kth order statistic from the distribution. Only when n is small is there a substantial difference between a Q-Q plot and a probability plot.
[edit] See also
- Probit analysis was developed by Chester Ittner Bliss in 1934.
[edit] References
- ^ A simple (and easy to remember) formula for plotting positions.
- ^ Engineering Statistics Handbook: Normal Probability Plot – Note that this also uses a different expression for the first & last points. [1] cites the original work by Filliben 1975.
- ^ Distribution free plotting position, Yu & Huang
- ^ This is Blom's earlier approximation 1953 and is the expression used in MINITAB.
- ^ This plotting position was used by Gringorten 1963 to plot points in tests for the Gumbel distribution.
[edit] Links
- Alternate description of the QQ-Plot: http://www.stats.gla.ac.uk/steps/glossary/probability_distributions.html#qqplot
- Normal plots
