# Q-Q plot

File:Qqnormexp.png
A normal Q-Q plot of randomly generated exponential(1) data.
File:Qqnormcorr.png
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 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.

## 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.

## 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 kth 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.

## References

1. A simple (and easy to remember) formula for plotting positions.
2. 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.
3. Distribution free plotting position, Yu & Huang
4. This is Blom's earlier approximation 1953 and is the expression used in MINITAB.
5. This plotting position was used by Gringorten 1963 to plot points in tests for the Gumbel distribution.
de:Quantile-Quantile-Plotit:Plotting position