Statistical Inference: A Short Course

Appendix 10.A Assessing The Randomness of A Sample

One of the key assumptions made during our discussion of estimation and testing was that we were engaging in a process of simple random sampling from a normal population. Appendix 7.A considered the issue of assessing the normality of a simple random sample. Now, we look to whether or not we can view a given data set “as if” it was drawn in a random fashion.

The type of test that we shall employ to assess the randomness of a set of observations is the runs test. As the name implies, this test is based on the notion of a run—an unbroken sequence of identical outcomes/elements (denoted by, say, letters) that are preceded or followed by different outcomes or no outcomes at all. For instance, suppose a particular piece of machinery produces the following ordered sequence of defective (d) and non-defective (n) items:

nnnn/dd/n/d/nnn/d/nnn/d

According to the definition of a run, this sequence has eight runs, where a slash is used to separate individual runs. Thus each sequence of n's and d's, uninterrupted by the other letter, identifies a run. It is important to note that: (1) the order in which the observations occur must be preserved so that the various subsequences of runs can be identified; and (2) the length of each run is irrelevant.

As will be evidenced below, a runs test is used to determine if the elements within an ordered sequence occur in a random fashion, where each element within the sequence takes on one of two possible ...