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## normal probability plot

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

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Categorized Plots, 2D - Normal Probability Plots
This type of probability plot is constructed as follows. First, within each category, the values (observations) are rank ordered. From these ranks one can compute z values (i.e., standardized values of the normal distribution) based on the assumption that the data come from a normal distribution (see Computation Note ). These z values are plotted on the Y-axis in the plot. If the observed values (plotted on the X-axis) are normally distributed, then all values should fall onto a straight line. If the values are not normally distributed, then they will deviate from the line. Outliers may also become evident in this plot. If there is a general lack of fit, and the data seem to form a clear pattern (e.g., an S shape) around the line, then the variable may have to be transformed in some way (e.g., a log transformation to "pull-in" the tail of the distribution, etc.) before some statistical techniques that are affected by non-normality can be used.
For a detailed discussion of Categorized Graphs, see Categorized Graphs in Selected Topics in Graphical Analytic Techniques.

Normal Probability Plots
This type of graph is used to evaluate the normality of the distribution of a variable, that is, whether and to what extent the distribution of the variable follows the normal distribution. The selected variable will be plotted in a scatterplot against the values "expected from the normal distribution."

The standard normal probability plot is constructed as follows. First, the deviations from the mean (residuals) are rank ordered. From these ranks the program computes z values (i.e., standardized values of the normal distribution) based on the assumption that the data come from a normal distribution (see Computation Note ). These z values are plotted on the Y-axis in the plot. If the observed residuals (plotted on the X-axis) are normally distributed, then all values should fall onto a straight line. If the residuals are not normally distributed, then they will deviate from the line. Outliers may also become evident in this plot. If there is a general lack of fit, and the data seem to form a clear pattern (e.g., an S shape) around the line, then the variable may have to be transformed in some way .
See also, Normal Probability Plots (Computation Note)

Normal Probability Plots (Computation Note)
The following formulas are used to convert the ranks into expected normal probability values, that is, the respective normal z values.
Normal probability plot. The normal probability value zj for the jth value (rank) in a variable with N observations is computed as:
zj = -1 [(3*j-1)/(3*N+1)]

where -1 is the inverse normal cumulative distribution function (converting the normal probability p into the normal value z).
Half-normal probability plot. Here, the half-normal probability value zj for the jth value (rank) in a variable with N observations is computed as:
zj = -1 [3*N+3*j-1)/(6*N+1)]
where -1 is again the inverse normal cumulative distribution function.
Detrended normal probability plot. In this plot each value (xj) is standardized by subtracting the mean and dividing by the respective standard deviation (s). The detrended normal probability value zj for the jth value (rank) in a variable with N observations is computed as:
zj = -1 [(3*j-1)/(3*N+1)] - (xj-mean)/s
where -1 is again the inverse normal cumulative distribution function.

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