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 boxandwhisker 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 boxandwhisker
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 (Q_{1}) and 25%
fall above the upper end (Q_{3}). 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 (Q_{3}  Q_{1})
and one to the largest observation below the
median +1.5 (Q_{3}  Q_{1}).
Observations beyond these values are generally
denoted by a plotted dot. The boxandwhisker
plots may give some hint as to the shapes of
the various distributions, but neither the bar
charts nor the boxandwhisker plots tell whether
the differences between surgeons should be attributed
to specialcause variation or commoncause variation.
The control chart method addresses this critical
question directly.
Figure 1. Bar Chart of the Average Surgery Times
of Nine Surgeons
Figure 2. BoxandWhisker 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 commoncause
variation. However, the averages are not in
control, implying specialcause 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
boxandwhiskers 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 commoncause
variation or is specialcause variation also
present. This question is answered only by the
control chart, not by either the bar chart or
by the boxandwhiskers 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. 214.
For more information, contact Drs. Robert and
Marilyn Hart at hart@uwosh.edu
or (541) 4120425.
