The Role of Statistical Graphics in Six Sigma

With the advent of Six Sigma, the use of statistical methods has become widespread by decision-makers at all levels of an organization. In each stage of the Six Sigma DMAIC strategy (Define, Measure, Analyze, Improve, Control), collection and analysis of data are vital. Yet if statistical results are to be helpful to decision-makers, they must be presented in a manner that makes them easy to understand.

Recent developments in statistical graphics are the key to successful use of statistical methods in Six Sigma. When results are displayed graphically rather than in tables, the proper inferences are much more likely to be made. The discussion which follows illustrates some typical output created at each stage of the DMAIC strategy.

Stage 1: Define

Six Sigma is a systematic approach to problem-solving. The first stage consists of defining the problem to be solved, the crucial inputs and outputs, and the team that needs to be involved to find a solution. Important tools in this stage include process maps, cause-and-effect diagrams, QFD (quality function deployment) matrices, and Pareto charts. The latter is an important graphical statistic that quantifies how often important events occur. For example, the Pareto chart below illustrates the frequency of complaints received about patient care at a selected hospital:

The height of each bar shows the percentage of all complaints related to a specific item, while the line above the bars plots the cumulative percentages from left to right. Combining the two leftmost bars, it can be seen that 73% of all complaints concern either wait time or staff. Such a chart shows clearly those areas where the potential for improvement is greatest.

Stage 2: Measure

Informed decisions regarding quality improvement depend upon the ability to quantify and measure performance. Performance data may consist of variables (such as the fill weight of cereal boxes) or attributes (such as the acceptability or unacceptability of an item based on visual inspection). In either case, measuring performance is critical to the success of any Six Sigma project.

Unfortunately, all measurements are subject to error. In the case of a gage, errors may be inherent in the functioning of the gage itself or due to the manner in which the gage is used. In the case of visual inspection, variability occurs due to the subjective nature of the assessment. In judging the adequacy of a measurement process, it is common to separate measurement error into differences between measurements made by the same appraiser (“repeatability”) and differences between measurements made by different appraisers (“reproducibility”).

The plot below shows the results of a typical R&R study in which 3 appraisers evaluated the acceptability of 50 parts, 3 times each. The blue squares indicate the percentage of times in which an appraiser was consistent on his or her evaluation of a given part, while the red diamonds indicate how often the evaluations were correct:

If the goal is to be correct and consistent 90% of the time, then only appraiser B has satisfactory performance.

It can also be quite helpful to take measurements at different points throughout a process to determine where excess variability is being introduced. Such a study, called a variance components analysis, can help determine where to concentrate one’s efforts in later stages. The graph below illustrates three variance components for the moisture content of pigment paste: variability among measurements made on the same item (“tests”), variability among measurements made on different items from the same batch (“samples”), and variability among measurements made on items taken from different batches:

The numbers along the right margin show the percentage that each component represents of overall variability. Note that 78% of the variability comes from variation among samples from the same batch, perhaps due to inadequate mixing.

Stage 3: Analyze

The third stage of the DMAIC strategy involves an intensive analysis of any measurements that are available. If specifications exist for the process under study, comparison of observed data to the specification limits can highlight where improvements need to be made. Based upon a probability model, the number of defects per million (DPM) can also be estimated:

The bell-shaped normal curve serves as a statistical model for the measured strength of product samples. Extrapolation of the normal curve beyond the specification limits yields an estimate of 11.3 defects per million, almost all above the upper limit.

By mining available data, the Six Sigma practitioner may also discover relationships that were previously unknown. For example, the interaction plot below shows the yield of a process as a function of both temperature and type of catalyst:

It is easy to see that the proper catalyst to use depends upon the temperature at which the process is run.

Stage 4: Improve

Once current performance has been quantified, process improvement can commence. Usually, this involves experimenting with new techniques or different formulations. Unfortunately, all data are not created equal. It is all too easy to spend a lot of time and money collecting data, only to find that it contains little useful information.

The statistical design of experiments (DOE) is a proven methodology for being sure that the data collected give the analyst the most information possible for the time and effort expended. By carefully planning which data are collected, the analyst can insure that the effects of different factors can be separated and avoid being misled by biased or uncertain results. In the chart below, created during a screening experiment, the magnitude of the main effects of each factor (shown as single letters) and the interactions between pairs of factors (shown with two letters) are illustrated as a barchart. Those bars that extend to the right of the vertical line correspond to statistically significant effects:

The resulting analysis allows mathematical models to be created which can then be used to determine optimal operating conditions:

The estimated strength is highest in the vicinity of catalyst = 230 and temperature = 60.

Stage 5: Control

Successful application of the first four stages of the Six Sigma DMAIC strategy often leads to improvements in process quality. To be sure that the improvements are not short-lived, the fifth and final stage puts procedures in place to monitor and control the process going forward. Statistical process control (SPC) charts are important tools that plot performance over time:

As long as the process remains in the green zone, it can be safely left alone. Too many points in the yellow zone (2 out of 3 consecutive points is a typical rule) is an early warning of approaching problems. Any point in the red zone needs to be investigated immediately, since it indicates performance that is out of the ordinary.

Conclusion

One of the great contributions of Six Sigma is its emphasis on collecting, examining and analyzing data. Statistical methods, once the exclusive province of statisticians, are now mainstream. Thankfully, the results of most analyses can be displayed in graphical format, leaving the details of the mathematics to statistical software programs. Decision-makers can then take the statistical conclusions and combine it with knowledge gleaned from other sources to make informed decisions.

Note: All graphics were generated by STATGRAPHICS Centurion XV from StatPoint, Inc.