Example 4.9 gave 2002 cost-to-charge ratios for inpatient services at 31 hospitals in Oregon. The same data source also provided the cost-to-charge ratios for outpatient services at these same 31 hospitals:

a. Use the given data to compute the quartiles and the interquartile range for the outpatient cost-to-charge ratio.

b. How does the interquartile range compare to the interquartile range for the inpatient cost-to-charge ratio from Example 4.9? Is there more variability in the inpatient or the outpatient ratio?

Data from Example 4.9

The Oregon Department of Health Services publishes cost-to-charge ratios for hospitals in Oregon on its web site. The cost-to-charge ratio is computed as the ratio of the actual cost of care to what the hospital actually bills for care, and the ratio is usually expressed as a percentage. A cost-to-charge ratio of 60% means that the actual cost is 60% of what was billed. The ratios for inpatient services in 2002 at 31 hospitals in Oregon were:

The sample size n = 31 is an odd number, so the median is excluded from both halves of the sample when computing the quartiles.

Each half of the sample contains 15 observations. The lower quartile is just the median of the lower half of the sample (62 for this data set), and the upper quartile is the median of the upper half (76 for this data set). This gives

The sample mean and standard deviation for this data set are 70.65 and 14.11, respectively. If we were to change the two smallest values from 45 and 48 to 25 and 28 (so that they still remain the two smallest values), the median and interquartile range would not be affected, whereas the mean and the standard deviation would change to 69.35 and 16.99, respectively.

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75 54 50 75 58 56 56 62 66 69 100 57 45 51 53 45 84 45 48 39 51 51 65 100 63 96 54 53 67 52 71