The Cumulative Distribution Function (abbreviated cdf) describes the probability distribution of a quantitative random variable, X, completely. For every possible value, x, in the range, the cdf is given by
- F(x) = Pr[X<=x],
that is the probability that X is no greater than x.
If X is a discrete random variable, then the probability is concentrated on discrete points and F(x) can be described as a sequence of pairs <x,p(x)> where p(x) = Pr[X=x].
The Kolmogorov Smirnov Test is based on cumulative distributions and can be used to test to see whether two empirical distributions are different or whether an empirical distribution is different from an ideal distribution. The closely related Kuiper Test (pronounced in Dutch the way an Cowper might be pronounced in English) is useful whether the domain of the distribution is cyclic as in day of the week. For instance we might use Kuiper's test to see if the number of tornadoes varies during the year or if sales of a product vary by day of the week or day of the month.