Run Test in statistics: Step by Step procedure

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In many situation, we are interested to whether the data collected for statistical analysis is random or not. In most all statistical inference we assume that the data is in a random sample. If the randomness of a sample is suspected, we test it’s randomness before, going to statistical analysis. In detail, There are many situations where we may investigate the assumptions of randomness. The data of statistical quality control analysis and regression analysis are the most important situations in while we must be sure about randomness. Run test is an non-parametric test.

Procedure for investigating the randomness are based on the number and nature of runs present in a data of interest.

 Length of run: The run which contains maximum number of item or symbols is called  length of run.

Wald- Wolfowitz Run Test for two variable


The data given for statistical analysis consists of a sequence of observations recorded in the order of their  occurrence can be classified into two mutually exclusive types.   

Let us consider,

n = the sample size

n1= number of observations in one type.

n2= number of observations in anathoter type.

Hypothesis of run test

H0 : The pattern of  accuracy of the two types of observation is determined by a random process.

For two tail : H1 : The pattern….. is not random.

For one tail : H1 : The pattern….. is not random.

(because there are too few runs to chance)

H1: The pattern…….. is not random

(because there are too may runs to be attributed to chance)

Test statistic

Test statistic is r.

r = the number of run

ConclusionIf r is less than or equal to lower critical value or greater than or equal to upper critical value then we reject H0.

 Run test for large sample (n1 or n2 > 20) 

The mean of r from the distribution is,

Run test for large sample

Comment: If the calculated value falls in critical region then we may reject H0 ; other wise we may accept it.

See other non-parametric Test:


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