Six Sigma and Beyond: Statistical Process Control, Volume IV

STATISTICAL ANALYSIS OF RUNS

In the previous section, we discussed the signals for the traditional out-of-control condition. In the case of the points out of limits, we gave a probability discussion as to whether or not it is truly an out-of-control condition. For the other conditions, we depended on a pictorial description to do the job. A somewhat advanced discussion on the analysis of runs was given, but it included nothing that an average person could use in a daily routine. In this section, we hope to provide an easier method by which to define the rest of the out-of-control conditions.

A run, in terms of process control, is strictly defined as a succession of items of the same class. In the case of control charts , an ordered series of points above and/or below the central line is considered a run.

A run, of course, is considered an out-of-control condition for a process, whether it occurs above or below the central line. Although there are a number of rule-of thumb guidelines ”based on probability ”that may be used to determine whether a run exists (e.g., 6, 7, 8, or 10 of 12 points on one side of the central line), it is possible to statistically determine whether nonrandom influences are present across the output of a particular process.

As an example of this technique, let us use an example of output from a punch press section in an aircraft factory (Duncan, 1986, 1948). In this case, a p chart yielded the results seen in Figure 11.10.

Figure 11.10: A p chart in an aircraft factory.

Reviewing this chart, one might wonder whether nonrandom (assignable) variation is at work in this process, causing a run or runs. There is, for example, one section in which 8 of 10 points are below the central line. This procedure will allow us to ask whether that is significant as a run, given the nature of the entire process. The steps in determining whether it is reasonable to believe that assignable causes are at work causing a run follow.

• Step 1 : Count the number of runs above the central line. In this case, we are referring to the number of runs of any length, including runs of 1. Using our example from Figure 11.10, we would find that the numbers of runs above the line total as follows :

  Runs of 1	5
  Runs of 2	2
  Runs of 5	1
  Total	8

• Step 2 : Count the number of runs below the central line.

  Runs of 1	2
  Runs of 2	1
  Runs of 3	2
  Runs of 4	1
  Runs of 6	1
  Total	7

• Step 3 : Add the total number of runs above and below the central line.

  Runs Below = 7

  +

  Runs Above = 8

  Total # of Runs = 15

• Step 4 : Count the number of points above and below the line. In our example, the number of points above the central line is 14, and the number of points below the line is 20. These two values are assigned to the symbols r and s, with r always taken as the smaller of the two values. Therefore, in our example,

  r = 14 and s = 20

• Step 5 : Find the critical value for the values r and s based on a table (or the given probability in a computer software package). For our example, the limiting critical value is 12.

• Step 6: Compare the value for the total number of runs with the critical limiting value (see Appendix D). If the total number of runs calculated for a process is equal to or less than the critical limiting value, the probability is 5% or less that the process runs are due to random causes. In other words, if the total number of runs present is equal to or less than the critical limiting factor, the assignable causes are present, and the process is not in control.

In our example, the total number of runs was 15. The critical limiting factor for the r and s value is 12. Therefore, the analysis of this process, insofar as runs are concerned , tells us that we cannot conclude that the number of runs is less than would be expected on the assumption of randomness.

STATISTICAL ANALYSIS OF TRENDS

With some minor modifications, the same procedure used for runs can be used for the determination of assignable causes as manifested by trends. You should note that many authors referring to this procedure (Duncan included) will refer to trends as "runs up" or "runs down."

Using the same control chart as with the example for runs, the steps for this procedure are as follows:

• Step 1 : Count the number of upward trends. For our example, the total number of upward trends is

  Trends up to 1	2
  Trends up to 2	6
  Trends up to 3	1
  Total	9

• Step 2 : Count the number of downward trends. For our example, the total number of downward trends is

  Trends down of 1	5
  Trends down of 2	2
  Trends down of 3	2
  Total	9

• Step 3 : Add the total number of upward and downward trends.

  Total number of trends = 9 + 9 = 18

• Step 4 : Count the number of changes (increases or decreases) in the process series. For our example, there were 15 decreases, 17 increases , and 1 "no change." Ignoring the no-change value, we would assign these values to r and s, so that (remembering that r is always the smaller figure) the following hold:

  r = 15 and s = 17

• Step 5 : Find the critical limiting value. In this case, the value is 9 or 11, depending on the table you use.

• Step 6 : Interpret the assignable variation. In our example, because 18 is greater than 9 or 11, we conclude that on the basis of the runs up and down, there is no reason to reject the hypothesis of randomness.