Statistical Thinking to Improve Quality

This posting describes the Box Plot (Box-and-whiskers plot) which is a tool for use in the seventh step, Study Cause and Effect, of the Hoerl-Snee Process Improvement Strategy.  The posting defines the plot and illustrates its use.   The Box Plot shows certain aspects of the distribution of data.  By classifying the data into categories, one can construct a Box Plot for each category and observe distributional differences among the categories.   These differences may reveal categories or factors that are increasing (or reducing) variability.

To illustrate the Box Plot, we refer to an example given by Breyfogle (2003, page 389).  An injection molding process produced plastic cylindrical connectors.   Breyfogle presents data from a sample of two parts collected hourly from four mold cavities for three hours consisting of measurements at three locations on the parts.   The Box Plot for the aggregated data appears below. 

The plot portrays key distribution characteristics as shown in the figure.  Twenty-five percent of the data are less than or equal to Q₁, half of the data are less than or equal to the median, and seventy-five percent of the data are less than or equal to Q₃.  The vertical lines are whiskers.   Call Q₁ the 25^th percentile, Q₃ the 75^th percentile, and the median the 50^th percentile. The lower whisker extends to the lower limit which is Q₁ – 1.5(Q₃ - Q₁), and the upper whisker extends to the upper limit which is Q₃ + 1.5(Q₃ - Q₁).   Values beyond the upper and lower limits are outliers and shown as asterisks (*).
 
The following figure illustrates the use of Box Plots to identify categories increasing variability and degrading quality.   Mold cavity 1 produces diameters greater than cavities 2, 3 and 4.  The 25^th percentile for mold cavity 1 diameters is greater than the 75^th percentiles for mold cavities 2,3 and 4.

References

1. Breyfogle, F. W. (2003). Implementing Six Sigma. Hoboken, New Jersey, John Wiley & Sons, Inc.
   

Posted by Gordon M. Clark at 8:24 PM | Permalink | Comments (0)

This posting discusses the seventh step, Study Cause and Effect, of the Hoerl-Snee Process Improvement Strategy.   Refer to the figure in the April 4 posting for an overview of the process.  Use Britz et al (2000) and Hoerl and Snee (2002) as references.

The previous step analyzed common-cause variation to identify the source (s) of variation.   If the previous step did not identify the source or if knowing the source does not reveal the root cause, we proceed to study cause and effect.  

Some of the tools we might use in this step are:

• Scatter plot.   A plot of a quality characteristic versus a potential explanatory variable.   See the plot in the 3/28/2008 posting showing the effect of solvent feed ratio on output weight.
• Cause & Effect Diagram.  A diagram portraying the potential causes of an effect.  See the diagram in the 2/28/2008 posting showing the potential causes of rejections at the grinding operations.  Frequently, the Cause & Effect diagram summarizes the results of a brainstorming session.   However, some improvement efforts will use data to substantiate the cause and effect diagram.
• Box Plot.   Box Plots depict the relationship between a discrete variable, such as location on a part, and the distribution of continuous variable, such as a dimension.
• Multi-Vari Charts.   Multi-Vari charts display variations in categories that aid in identifying causes.
• Interrelationship Digraphs.   Teams construct cause and effect relationships from a list of issues.

The next posting will summarize additional tools for this step.   Subsequent postings will give examples of Box Plots, Multi-Vari Charts and Interrelationship Digraphs.

References

1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
2. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.
   

Posted by Gordon M. Clark at 2:31 PM | Permalink | Comments (0)