The average of these 199 successive differences is 2.42.*Average successive difference = 2.42*

One caveat is needed. Since the Ramirez-Runger test depends upon the time-order sequence of the data, it should always be used with data in their native ordering. Specifically, it can’t be used on data that have been rearranged into a ranking where the values are placed in ascending or descending order.

### So what happens next?

These data are not sufficiently well-behaved to be represented by a single probability model. However, that doesn’t keep your software from drawing a bell-shaped curve over the histogram as in Figure 8.

### Summary

This test will quantify the chances that you can successfully fit *any* probability model to your data. By using this simple test to examine the assumptions behind all probability models, you can avoid making serious mistakes. This column will illustrate this test and explain why it works.

### Example 1

A probability model is a limiting property for an infinite sequence of random variables that are independent and identically distributed. And a sequence of independent and identically distributed random variables will display the same amount of variation regardless of whether the computation is carried out globally or sequentially.

So the Ramirez-Runger test compares a global estimate of dispersion with a sequential estimate. The global estimate is the usual standard deviation statistic. The sequential estimate is the average of the successive differences divided by its bias correction factor, 1.128. The ratio of these two estimates, when squared, will be approximately distributed according to an *F*-distribution with (*n*–1) and (0.62*(*n*–1)) d.f. The probability of exceedance, or *p*-value, for this test statistic quantifies how reasonable it is to consider the two estimates as being equivalent.

When the *p*-value for the Ramirez-Runger test is small, you’ll know that you can’t fit a probability model to your data. Neither can you estimate process parameters nor compute confidence intervals, test hypotheses, or use any other statistical analysis techniques. Rather, you’ll need to use a more fine-grained approach, looking for the assignable causes of exceptional variation within the data themselves. And, of course, this will lead to the use of process behavior charts.

So, while the process behavior chart remains the final arbiter of when a process is operated unpredictably, the Ramirez-Runger test provides a computation-based alternative that can keep you from making serious mistakes. If you’re not already starting your analysis with a process behavior chart, then the Ramirez-Runger test is the test to use before all other tests.