Box Plot

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. 

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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.

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References

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