2D Quantile-Quantile Plots

You can visually check for the fit of a theoretical distribution to the observed data by examining the quantile-quantile (or Q-Q) plot. In this plot, the observed values of a variable are plotted against the theoretical quantiles. A good fit of the theoretical distribution to the observed values would be indicated by this plot if the plotted values fall onto a straight line. To produce a Q-Q plot, STATISTICA first sorts the N observed data points into ascending order, so that:

x1 x2 ... ≤ xn

These observed values are plotted against one axis of the graph; on the other axis the plot will show:

F-1((i-radj) / (n+nadj))

where i is the rank of the respective observation, radj and nadj are adjustment factors (≤ 0.5) and F -1 denotes the inverse of the probability integral for the respective standardized distribution. The resulting plot is a scatterplot of the observed values against the (standardized) expected values, given the respective distribution.

Note that, in addition to the inverse probability integral value, STATISTICA also shows the respective cumulative probability values on the opposite axis, that is, the plot shows not only the standardized values for the theoretical distribution, but also the respective p-values. Note also that the adjustment factors radj and nadj ensure that the p-value for the inverse probability integral will fall between 0 and 1, but not including 0 and 1 (see Chambers, Cleveland, Kleiner, and Tukey, 1983; in STATISTICA, the default value for both adjustment factors is 1/3=.333).

You can choose a variety of distributions to use in creating the Q-Q plot. For a list of the theoretical distributions available, see Quantile-Quantile Plots - Advanced Tab.

See also, Graphs - Quantile-Quantile Plots and the Conceptual Overview for Quantile-Quantile Plots.