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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 common-cause variation or does there appear to be evidence of special-cause variation. Similarly, the s chart will demonstrate whether the differences in the variation (as calculated by the standard deviation) may be explained by common-cause variation or does there appear to be evidence of special-cause 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 special-cause variation from common-cause variation. So it is sufficient for the data to be “near-normal” [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 near-normal 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 2-sigma 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, 2-Sigma Limits


Figure 6. s Chart by Surgeon, 2-Sigma 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 chance-cause 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 3-Sigma 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 ‘Near-Normality’: 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), 65-66.