HCS 493 Week 1 Basics of Statistics and Data Analysis Example

The Average Said 214 Minutes: Basic Statistics for an Emergency Department Manager, From Mean and Median to Spread and the National Benchmark

[Student Name]

University of Phoenix

HCS/493: Data Analytics for Health Care Managers

Week 1 Assignment

[Instructor Name]

[Date]

The hospital, its emergency department and its visit data are composites written for a model paper; the national benchmark comes from the public CMS data set listed.

The emergency department manager at a composite 210-bed community hospital forwarded the monthly dashboard to the analytics team with one line: our average stay for discharged patients was 214 minutes, and the chief operating officer wants it under three hours by spring. She asked whether the department was really that slow and what to do first. This paper is the analyst's answer. It uses the month's data to explain basic statistical ideas every health care manager needs and shows how they change the decision.

The Data

The data set contains every emergency visit in March that ended with the patient going home: 3,140 visits. Because it includes all such visits for the month, it is the population for that month, not a sample, although it can be treated as a sample when the question is about the department's typical performance over the year. The main variable is length of stay, measured in minutes from arrival to departure, which is continuous data. Other variables include triage level, an ordinal scale from one to five, and payer, a nominal category.

The Mean

The dashboard's figure of 214 minutes is the arithmetic mean: the total minutes of all visits divided by 3,140. The mean uses every value, which is its strength and its weakness. In March, 96 visits lasted more than 480 minutes, mostly patients waiting for a ride home after sedation, for social work to arrange a safe place to stay or for a delayed imaging result. Those long visits pulled the mean upward.

The Median

The median is the middle value when all visits are sorted from shortest to longest. Half of patients stayed less than the median and half stayed longer. In March the median was 171 minutes. The gap of 43 minutes between mean and median is the signature of right-skewed data, where a long tail of high values stretches the distribution. Altman and Bland (1995) noted that many clinical measurements are not normally distributed and that summaries that assume a symmetric bell shape can mislead when they are not. The typical patient in March stayed about two hours and fifty minutes, not three and a half hours; the department does not have a general speed problem so much as a long-stay problem.

The Mode and Range

The mode, the most frequent value when stays are grouped into 15-minute intervals, fell between 135 and 150 minutes. The range ran from 22 minutes to 1,310 minutes. The range takes seconds to compute, yet it is set by just two unusual visits, so it says little about the department as a whole.

Standard Deviation

Standard deviation summarizes, in the same minutes as the data, the usual distance between a single visit and the mean. In March it was 118 minutes. Altman and Bland (2005) distinguished the standard deviation, which describes variability among individuals, from the standard error, which describes how precisely a mean has been estimated. For a manager, the standard deviation answers a service question: do patients have similar experiences? A spread this wide says they do not.

Percentiles

Because the data are skewed, percentiles describe the spread more usefully than the standard deviation. The 25th percentile was 118 minutes and the 75th was 236. The 90th percentile was 352 minutes, meaning one patient in ten stayed nearly six hours or more. The long tail, not the middle, is where the department's performance is weakest.

Comparing With the Nation

The Medicare agency reports, for every hospital and for the nation, the middle value of the minutes that discharged emergency patients spend between walking in and leaving, excluding patients with mental health visits and those transferred. For discharges between October 2024 and September 2025, the national median was 162 minutes (Centers for Medicare & Medicaid Services, 2025). Before comparing, the analyst recalculated March using the same exclusions, which removed 71 visits and produced a median of 168 minutes. The department is six minutes slower than the national median, a small gap, and far from the 214 minutes the dashboard implied.

Breaking the Data Down

Splitting the visits by triage level shows where the long tail sits. Patients in levels four and five, the least urgent, had a median stay of 112 minutes and almost no visits over eight hours. Level three patients, who usually need tests or imaging, had a median of 204 minutes and accounted for 68 of the 96 visits over 480 minutes. Triage level is ordinal, so the analyst reports medians and counts for each level rather than averaging the level numbers themselves, which would treat the gap between levels one and two as equal to the gap between levels four and five.

Limits of the Analysis

One month may not represent the year; March had an influenza surge in its first two weeks. Recording errors also matter, since a few visits closed hours late in the electronic record would inflate the tail. The analyst checked the 20 longest visits by hand and corrected three.

Why the Choice of Statistic Matters

If leadership chased the mean, it might add staff across all shifts to speed every visit, an expensive response to a problem that lies mostly in fewer than 100 visits. Looking at the median and the tail points instead to the causes of very long stays: waiting for transportation, social work coverage at night and delayed imaging reads.

Recommendations

First, report the median, the 90th percentile and the number of visits over 480 minutes each month, alongside the mean, so leaders see both the typical visit and the tail. Second, use the federal measure definition for any comparison with national or state figures. Third, review the long-stay visits each week by cause, since the same three causes explained most of them in March.

Three Habits for Managers

When a number arrives, ask what population it describes and how the variable was measured. Ask whether the data are likely skewed and, if so, look for the median and percentiles. Ask whether any benchmark uses the same definition. These three questions would have turned a request to cut the average to under three hours into a targeted plan for the longest stays.

Conclusion

The March data show how basic statistics change a management decision. The mean of 214 minutes was accurate arithmetic but a poor summary of a skewed distribution. The median, percentiles and a like-for-like national comparison showed a department close to the national figure with a concentrated problem in its longest visits, which is where improvement should begin.

References

Altman, D. G., & Bland, J. M. (1995). Statistics notes: The normal distribution. BMJ, 310(6975), 298. https://doi.org/10.1136/bmj.310.6975.298

Altman, D. G., & Bland, J. M. (2005). Standard deviations and standard errors. BMJ, 331(7521), 903. https://doi.org/10.1136/bmj.331.7521.903

Centers for Medicare & Medicaid Services. (2025). Timely and effective care: National [Data set]. https://data.cms.gov/provider-data/dataset/isrn-hqyy