The runs test for the small sample sizes

To demonstrate this test, we will consider series with even number of observations. Let us consider a series of 14 observations. The data are shown below

N	Value	N	Value

First of all we write data in ascending order and find the median.

The run test developed here separates the observations into a subgroup above the median and a subgroup below the median. Then letting a “ ”denote observations above the median and a “–“ denote observations below the median we find the following pattern over the sequence

This sequence consists of a run of one “ ”, followed by a one run of one “-“, a run of one “ +”, a run of one “-“, a run of one “-“, a run of three “-“, a run of three “+”, a run of two “-“, and one run of “+”. In total there are runs.

The null hypothesis is that the series is a set of random variables. The table 7 in the Appendix gives the smallest significance level against which this null hypothesis can be rejected against the alternative of positive association between adjacent observations, as a function of and n.

If the alternative hypothesis is two-sided hypothesis on randomness, the significance level must be doubled if it is less than 0.5. Alternatively, if the significance level, , read from table is greater than 0.5, the corresponding significance level for the test against the two sided alternative is .

In our case, , and . From table in the appendix we see that for observations, the probability under the null hypothesis of finding 9 or fewer runs is 0.791. Therefore, the null hypothesis of randomness can only be rejected against the alternative hypothesis of positive association between adjacent observations at the 79.1% significance level. We have not found strong evidence to reject the null hypothesis that series are randomness.

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