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Commonly Used Graphs Versus Control Charts

Robert F. Hart, Ph. D.
Marilyn K. Hart, Ph.D.

(Note: You may want to print this article for best viewing of the charts.)

In order to look at data to make comparisons, two commonly used graphs are the bar chart and the box-and-whisker plot. The bar chart simply plots the frequencies of the various measurements. For instance, the example used here has surgery times for nine surgeons from a hospital.1 Figure 1 is a bar chart of the average surgery times of the nine surgeons. Figure 2 is a box-and-whisker plot of the same data. There are slight variations in the plotting procedure, but generally the boxes show the values for the middle 50% of the measurements, that is, 25% fall below the lower end of the box (Q1) and 25% fall above the upper end (Q3). The median is shown as a line through the box. A "whisker" is drawn from each end of the box, one to the smallest observation above the median -1.5 (Q3 - Q1) and one to the largest observation below the median +1.5 (Q3 - Q1). Observations beyond these values are generally denoted by a plotted dot. The box-and-whisker plots may give some hint as to the shapes of the various distributions, but neither the bar charts nor the box-and-whisker plots tell whether the differences between surgeons should be attributed to special-cause variation or common-cause variation. The control chart method addresses this critical question directly.



Figure 1. Bar Chart of the Average Surgery Times of Nine Surgeons



Figure 2. Box-and-Whisker Plots of Surgery Times of Nine Surgeons


A better way to show each individual surgeon's surgery times is by use of the probability plot and histogram. The times for Surgeon F are given in Figures 3 and 4. Note that the data are not normally distributed, so a transformation is needed to make the data normally distributed for the control chart to work. By trial and error, a reciprocal transformation brings the data to close to normal, as seen in Figures 5 and 6. (For further details on this analysis, see Reference 2.) The control chart on this transformed data (Figure 7) shows that the s chart is in control, i.e., the variability amongst the surgeons may be attributed to common-cause variation. However, the averages are not in control, implying special-cause variation for certain surgeon averages. Surgeons B, E, and G are below the lower control limit; Surgeon H is above the upper control limit. This cannot be learned from either the bar chart or the box-and-whiskers plot.



Figure 3. Histogram of Surgeon F's Surgery Times



Figure 4. Probability Plot of Surgeon F's Surgery Times (Original Data)




Figure 5. Histogram of Transformed Data for Surgeon F



Figure 6. Probability Plot of Transformed Data for Surgeon F



Figure 7. Control Chart on Transformed Data


Conclusions

The only useful question which data analysis can address is whether or not the observed variation may reasonably be attributed to common-cause variation or is special-cause variation also present. This question is answered only by the control chart, not by either the bar chart or by the box-and-whiskers plot.

References

1. M. Hart and R. Hart. Statistical Process Control for Health Care. Pacific Grove, CA: Duxbury, 2002.
2. M. Hart, J. Robertson, R. Hart, and S. Schmaltz. " Xbar and s Charts for Healthcare Comparisons." Quality Management in Health Care, 15(1) 2006, pp. 2-14.

For more information, contact Drs. Robert and Marilyn Hart at hart@uwosh.edu or (541) 412-0425.

If you would like additional information, please send email to statit.support@acs-inc.com.


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