Distribution Plots

Use normal probability plots to assess whether data comes from a normal distribution. Many statistical procedures make the assumption that an underlying distribution is normal. Normal probability plots can provide some assurance to justify this assumption or provide a warning of problems with the assumption. An analysis of normality typically combines normal probability plots with hypothesis tests for normality.

This example generates a data sample of 25 random numbers from a normal distribution with mean 10 and standard deviation 1, and creates a normal probability plot of the data.

rng('default');  % For reproducibility
x = normrnd(10,1,[25,1]);
normplot(x)

The plus signs plot the empirical probability versus the data value for each point in the data. A solid line connects the 25th and 75th percentiles in the data, and a dashed line extends it to the ends of the data. The y-axis values are probabilities from zero to one, but the scale is not linear. The distance between tick marks on the y-axis matches the distance between the quantiles of a normal distribution. The quantiles are close together near the median (50th percentile) and stretch out symmetrically as you move away from the median.

In a normal probability plot, if all the data points fall near the line, an assumption of normality is reasonable. Otherwise, an assumption of normality is not justified. For example, the following generates a data sample of 100 random numbers from an exponential distribution with mean 10, and creates a normal probability plot of the data.

x = exprnd(10,100,1);
normplot(x)

The plot is strong evidence that the underlying distribution is not normal.

Probability Plots

A probability plot, like the normal probability plot, is just an empirical cdf plot scaled to a particular distribution. The y-axis values are probabilities from zero to one, but the scale is not linear. The distance between tick marks is the distance between quantiles of the distribution. In the plot, a line is drawn between the first and third quartiles in the data. If the data falls near the line, it is reasonable to choose the distribution as a model for the data. A distribution analysis typically combines probability plots with hypothesis tests for a particular distribution.

Create Weibull Probability Plot

rng('default');  % For reproducibility
x1 = wblrnd(3,3,[500,1]);
x2 = raylrnd(3,[500,1]);

figure
probplot('weibull',[x1 x2])
legend('Weibull Sample','Rayleigh Sample','Location','best')

x = poissrnd(10,[50,1]);
y = poissrnd(5,[100,1]);
qqplot(x,y)

x = normrnd(5,1,[100,1]);
y = wblrnd(2,0.5,[100,1]);
qqplot(x,y)

rng('default')  % For reproducibility
y = evrnd(0,3,100,1);

cdfplot(y)
hold on
x = linspace(min(y),max(y));
plot(x,evcdf(x,0,3))
legend('Empirical CDF','Theoretical CDF','Location','best')
hold off

ecdf(y,'Bounds','on')
hold on
plot(x,evcdf(x,0,3))
grid on
title('Empirical CDF')
legend('Empirical CDF','Lower Confidence Bound','Upper Confidence Bound','Theoretical CDF','Location','best')
hold off