Profiling: Comparisons Using Xbar and s Charts
Authors: Marilyn & Robert Hart
Comparisons
It is often of interest to compare two or more
items, such as procedure times of the various
surgeons. An excellent way to do this is by
the use of an Xbar and s chart. [1, 6] The Xbar
chart will demonstrate whether the differences
in the averages may be explained by commoncause
variation or does there appear to be evidence
of specialcause variation. Similarly, the s
chart will demonstrate whether the differences
in the variation (as calculated by the standard
deviation) may be explained by commoncause
variation or does there appear to be evidence
of specialcause variation.
Example
For scheduling purposes, three surgeons are
to be compared for the time to complete a procedure.
Over the past 3 months, 24 times were recorded
for each of the first two surgeons, but 100
times were recorded for the third. The control
limits are calculated based upon an underlying
normality assumption, but the assumption of
normality is quite robust, i.e., there can be
some departure from normality and the control
limits would still work quite well in separating
specialcause variation from commoncause variation.
So it is sufficient for the data to be “nearnormal”
[4, 6]. The easiest way to check that assumption
is through the use of a probability plot [7].
Figures 1 – 3 show the data for each surgeon
individually, and Figure 4 shows the data combined
for all three surgeons. Note that all the plots
do not depart much from a straight line, so
it can be assumed that the data are nearnormal
and that the control limits on the Xbar and
s chart will be valid.
Figure 1. Probability Plot for Surgeon 1
Figure 2. Probability Plot for Surgeon 2
Figure 3. Probability Plot for Surgeon 3
Figure 4. Probability Plot for All Surgeons
Combined
The Xbar chart and s chart are made using 2sigma
limits since there are only three subgroups
[2] and are shown in Figures 5 – 6. Figure
6 is examined first to see if the variability
differences between the surgeons may be attributed
to chance, which it may since the chart is in
control. But the Xbar chart shows that the differences
in average times are not due to chance. The
14 minute difference between Surgeon 1 (89 minutes)
and Surgeon 3 (103 minutes) must be accounted
for when scheduling.
If the data were found to be badly skewed,
the data would need to be transformed before
control limits could be calculated [3].
Figure 5. Xbar Chart by Surgeon, 2Sigma Limits
Figure 6. s Chart by Surgeon, 2Sigma Limits
Comparisons by Other Methods
Comparisons are often done by several other
methods, none of which is satisfactory: Histograms,
boxplots, overlapping confidence intervals,
comparison charts, and hypothesis testing/ANOVA.
Histograms and boxplots do not show if the differences
could be attributed to chancecause variation.
Overlapping confidence intervals do not give
the same results as hypothesis testing because
the standard error of the difference of two
means must be computed not by adding the individual
standard errors but by taking the square root
of a weighted sum of the variances [8]. Hypothesis
testing, comparison charts, and ANOVA assume
equal variances, which is typically never tested
(but that is what the s chart does), and are
difficult to understand.
References
[1] Hart, M., J. Robertson, R. Hart, S. Schmaltz.
“Xbar and s Charts for Health Care Comparisons,”
Quality Management in Health Care, 15 (1), January
– March 2006, pp. 2 – 14.
[2] Hart, M. and R. Hart. “Shewhart Control
Charts for Health Care with Other Than 3Sigma
Limits,” Statit, July, 2005.
[3] Hart, M. and R. Hart. “Transformations
of Skewed Data Distributions in Health Care,”
Statit, January, 2005
[4] Hart, M. and R. Hart. “Testing for
‘NearNormality’: The Probability
Plot,” Statit, September, 2004.
[5] Hart, M. and R. Hart. “Control Charts
for Skewed Data,” Statit, April, 2004.
[6] Hart M and Hart R. Statistical Process
Control for Health Care, Pacific Grove, CA:
Duxbury, 2002.
[7] Shapiro S, How to Test Normality and Other
Distribution Assumptions. Milwaukee, WI: American
Society for Quality, 1990.
[8] Wolfe R and Hanley J, “If We’re
So Different, Why Do We Keep Overlapping? When
1 Plus 1 Doesn’t Make 2,”Canadian
Medical Association Journal, 166, no. 1 (2002),
6566.
