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John Flaig

Six Sigma

Making and Interpreting Run Charts

Use this helpful tool to assess process stability and discover patterns

Published: Thursday, August 11, 2011 - 13:14

A run chart is a graphical display of data over time. Run charts are used to visually analyze processes according to time or sequential order. They are useful in assessing process stability, discovering patterns in data, and facilitating process diagnosis and appropriate improvement actions.

Creating the run chart

To start a run chart, some type of product, service, or process must be available on which to take measurements for analysis. Measurements must be taken over a reasonable period of time using a calibrated measurement tool that is being monitored with a calibration control chart. A measurement error study must indicate that the measurement process is acceptable during the data collection process. The data must be collected and stored in chronological or sequential order. You may start at any point in the data set and end at any point. To get meaningful results, at least 25 or more samples must be taken over a long enough period of time so all the components of variation are included.

Once the data have been collected in chronological or sequential order, they must be divided into ordered pairs of x and y values. The values for x represent time or sequence number, and the values for y represent the measurements taken from the product, service, or process.

Plot the y values versus the x values by hand or by using a computer program. Select an appropriate scale that will make the points on the graph visible. Next, draw vertical lines for the x values to separate time intervals such as observation number or time unit (e.g., days, weeks, months). Then draw horizontal lines to help distinguish where nonrandom observations appear (e.g., trends, shifts, spikes, cycles) in the process or operation. Your chart should look like the example in figure 1.

Fig. 1: An example of a run chart


Interpreting the run chart

To visually analyze a run chart, you should do the following:

Use your computer to draw a linear regression trend line from the beginning to the end of the data on the run chart. If the line is approximately horizontal, then the mean of the process can be considered stationary over this time interval. If not, then the process mean is considered nonstationary, or unstable. Remember that drawing this inference requires sufficient data, usually 50 or more observations (i.e., two points are not sufficient). There is a statistical test to determine if you can reject the null hypothesis that the slope of the linear trend line is zero. 

Look at the run chart. Does it appear that the variation in the data is increasing or decreasing over time (i.e., does the overall pattern or data envelope appear funnel-shaped or like a snake that swallowed a pig)? If the answer is yes, then the process variance can be considered nonstationary. If the answer is no, then the process variance can be considered stationary. You can use your computer to generate the two-point moving ranges and plot them. Then plot the linear trend line for the two-point moving ranges. If the trend line is approximately horizontal, then the variance is considered stationary over the interval. 

Are the points on the run chart scattered evenly and randomly around the trend line? If the answer is yes, then the data are not significantly auto-correlated (i.e., they are reasonably independent). If the data values tend to follow each other like a snake, then the data are auto-correlated (i.e., they may not be statistically independent). So you should ask: Is there a logical reason why this would be the case?

Is there a pattern in the run chart’s data (e.g., cyclic, trend, shift, spike, funnel shape)? To help spot a change, draw a horizontal line from the beginning of the data to the end, dividing the data in half. This is called the center line (CL) or median of the data. If you find cases where eight or more consecutive points are above or below the median line, or if eight points are steadily increasing or decreasing, then the process is probably unstable. If this is the case, then you should look for the causes. If it is not the case, then proceed to the next visual analysis test.

Is there a point or points on the run chart that appear to be isolated from the rest of the data? If the answer is yes, then investigate this rare observation to determine if it is valid and if so, then try to determine the causes.

Analysis of the run chart in figure 1

The process mean shown in figure 1 may be slightly nonstationary (i.e., the trend line is going down), but the small number of observations and the large amount of variation in the data make it impossible to validate this trend with a high degree of confidence. We should continue to monitor the process to see if the trend continues and becomes significant.

The process variance appears to be stationary, as the envelope of variation seems to be fairly consistent.

There appears to be auto-correlation in the data because the points do not appear to be randomly distributed around the trend line.

There appears to be a cyclic pattern of period 5 in the data. There does not appear be a trend, shift, spikes, or any other unusual systematic patterns in the data.

There do not appear to be any rare event points (i.e., outliers) in the data.

The run chart can be a very effective tool in understanding process behavior. The smart engineer will make it one of his commonly used tools.


About The Author

John Flaig’s picture

John Flaig

John J. Flaig, Ph.D., is a fellow of the American Society for Quality and is managing director of Applied Technology at www.e-at-usa.com, a training and consulting company. Flaig has given lectures and seminars in Europe, Asia, and throughout the United States. His special interests are in statistical process control, process capability analysis, supplier management, design of experiments, and process optimization. He was formerly a member of the Editorial Board of Quality Engineering, a journal of the ASQ, and associate editor of Quality Technology and Quantitative Management, a journal of the International Chinese Association of Quantitative Management.


Tests for runs

I missed you including material on Average Run Length (ARL) and Runs Tests for Randomness.

I agree with Mr. Moore's comments up to the point of not doing runs tests. I routinely did them first to see what the unaltered data told me. If I saw any trends or patterns then i would do an Individuals Chart for more information. If not then I would not be "out" anything.

Steve, My answer

We could discuss a process behavior chart but the only difference would be
that it has upper and lower natural process limits. Please review the following
comments and see if you understand why it makes sense to talk about run charts.

The famous statistician Yogi Berra once said, "You can see a lot just by
looking". This is different than having the computer tell you that rule 4
was violated on point 25. The first approach builds observational skill,
thinking and understanding of the process. The second approach only requires
that the analyst have a pulse.

Dr. Deming would often asked his students what was the difference between a
point just outside the control limits and one just inside. The answer is
essentially nothing! So the blind application of an algorithmic approach does
not teach people what they need to know to be a skilled SPC practitioner. I
have personally observed situations where the plug and crank method failed and
management concluded that SPC does not work here, even though the problem was
staring the analyst in the face.

The four Western Electric rules are pretty good but Dr. Shewhart originally
used just rule 1 and Dr. Nelson suggested eight rules. In fact, there are
infinitely many ways a process may exhibit instability so I would not be so
confident that four is adequate? Perhaps looking is more reasonable than just
assuming four is adequate.

summary, even if you use good SPC software you always need to look at the data
just the way I described in the run chart article. The software can speed the
analysis and help you find things, but if you don't OBSERVE carefully and THINK
it may speed you to the wrong conclusion. 


Why Not Process Behavior Charts???

In the modern era of laptop and desktop computers with excellent software packages, why not plot the data on a Process Behavior Chart and gain the insight needed to improve the process that creates the results?  Run Charts were OK when we had to generate our charts with a slide rule, #2 pencil, and a pad of engineering paper, but these days, with virtually no extra effort on a computer, a Process Behavior Chart can be created.  The four primary Western Electric rules are quite adequate for the analysis and everyone can understand them as operational definitions for signals of lack of a reasonable degree of statistical control.  No need to ride a rusty old bicycle when Harley-Davidson is available!  Vroooooom!!! 

What is the difference?

What is the difference between a run chart and a process behavior chart?