What is a Runs Test

A runs-test is a statistical procedure that examines whether a string of data is occurring randomly from a specific distribution. The runs test analyzes the occurrence of similar events that are separated by events that are different.


For example, a list of truly random single-digit numbers should only have a few instances where a sequence of numbers is ascending numerically. However, in many cases, it is difficult to assert the randomness of data in which there are thousands of sequences in a string of data, so the runs test was created as an objective method of determining randomness.

Importance of a Runs Test

The runs test model is important in determining whether an outcome of a trial is truly random, especially in cases where random versus sequential data has implications for subsequent theories and analysis.

The runs test is a shortened version of the full name: the Wald–Wolfowitz runs test, so named after mathematicians Abraham Wald and Jacob Wolfowitz. More precisely, it can be used to test the hypothesis that the elements of the sequence are mutually independent.

The Kolmogorov–Smirnov test has been argued to be more powerful than the Wald-Wolfowitz test for detecting differences between distributions that differ solely in their location.

Two powerful applications remain, however:

  • Testing the randomness of a distribution, by taking the data in the given order and marking with + the data greater than the median, and with – the data less than the median (numbers equalling the median are omitted.)
  • Testing whether a function fits well to a data set, by marking the data exceeding the function value with + and the other data with −. For this use, the runs test, which takes into account the signs but not the distances, is complementary to the chi-square test, which takes into account the distances but not the signs.