Plot the empirical cdf of a sample data set and compare it to the theoretical cdf of the underlying distribution of the sample data set. In practice, a theoretical cdf can be unknown.

Generate a random sample data set from the extreme value distribution with a location parameter of 0 and a scale parameter of 3.

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

Plot the empirical cdf of the sample data set and the theoretical cdf on the same figure.

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

The plot shows the similarity between the empirical cdf and the theoretical cdf.

Alternatively, you can use the ecdf function. The ecdf function also plots the 95% confidence intervals estimated by using Greenwood's Formula. For details, see Greenwood’s Formula.

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

Perform the one-sample Kolmogorov-Smirnov test by using kstest. Confirm the test decision by visually comparing the empirical cumulative distribution function (cdf) to the standard normal cdf.

Load the examgrades data set. Create a vector containing the first column of the exam grade data.

load examgrades
test1 = grades(:,1);

Test the null hypothesis that the data comes from a normal distribution with a mean of 75 and a standard deviation of 10. Use these parameters to center and scale each element of the data vector, because kstest tests for a standard normal distribution by default.

x = (test1-75)/10;
h = kstest(x)

h = logical
   0

The returned value of h = 0 indicates that kstest fails to reject the null hypothesis at the default 5% significance level.

Plot the empirical cdf and the standard normal cdf for a visual comparison.

cdfplot(x)
hold on
x_values = linspace(min(x),max(x));
plot(x_values,normcdf(x_values,0,1),'r-')
legend('Empirical CDF','Standard Normal CDF','Location','best')

The figure shows the similarity between the empirical cdf of the centered and scaled data vector and the cdf of the standard normal distribution.