Describing Thirty Waits: Descriptive Statistics and a Histogram for Emergency Department Door-to-Provider Times
[Student Name]
University of Phoenix
NSG/509: Research and Applied Statistics for Quality Improvement
Week 2 Descriptive Statistics Exercise
[Instructor Name]
[Date]
The emergency department and all data are a composite written for a model paper.
A composite community hospital emergency department set a goal that patients be seen by a provider within 30 minutes of arrival. Before testing any change, the quality team needed to describe current performance. The nurse leading the project drew a sample of 30 adult patients across one week, stratified to include day, evening and night arrivals, and recorded the minutes from arrival to first provider contact. Before anyone could say whether a change worked, the team needed an honest picture of what a typical wait looked like and how much the waits varied. This exercise reports descriptive statistics and a histogram for those 30 times and interprets them for the team.
The Data
The 30 door-to-provider times, in minutes and sorted from shortest to longest, were: 12, 18, 22, 25, 27, 28, 30, 31, 33, 34, 35, 36, 38, 39, 41, 42, 44, 45, 47, 49, 52, 55, 58, 61, 66, 72, 79, 88, 104 and 131. The variable is measured at the ratio level: minutes have equal intervals and a true zero, so means, medians and standard deviations are all permissible (Polit & Beck, 2021).
Measures of Central Tendency
The mean is the sum of the values divided by their number. The 30 values sum to 1,443 minutes, so the mean is 1,443 divided by 30, or 48.1 minutes.
The median is the middle value of the ordered data. With 30 values, the median is the average of the 15th and 16th values, 41 and 42, or 41.5 minutes.
There is no mode: every time occurs once. With continuous data such as minutes, a mode is rarely informative unless values are grouped.
Measures of Dispersion
The range is the maximum minus the minimum: 131 minus 12, or 119 minutes. The range is easy to understand but depends entirely on the two most extreme patients.
The standard deviation, calculated with the sample formula, is 26.1 minutes. It summarizes how far values typically fall from the mean, but like the mean it is inflated by the long waits at the top of the distribution.
The quartiles give a more robust picture. Using the inclusive method that most spreadsheet software applies, the first quartile is 31.5 minutes, the median is 41.5 minutes and the third quartile is 57.25 minutes. The interquartile range, the spread of the middle half of patients, is 57.25 minus 31.5, or 25.75 minutes.
Against the target, 7 of the 30 patients, or 23%, were seen within 30 minutes, and 7 patients waited more than an hour.
The Histogram
A histogram displays the frequency of values within equal-width bins. With 30 values ranging from 12 to 131 minutes, bins of 20 minutes produce seven bars, enough to show the shape without leaving many bins empty, and equal widths keep the bar heights comparable (Grove & Cipher, 2020). The frequency distribution that forms the histogram is as follows.
0 to 19 minutes: 2 patients
20 to 39 minutes: 12 patients
40 to 59 minutes: 9 patients
60 to 79 minutes: 4 patients
80 to 99 minutes: 1 patient
100 to 119 minutes: 1 patient
120 to 139 minutes: 1 patient
In the histogram built from this table, the horizontal axis shows minutes in 20-minute bins and the vertical axis shows the number of patients. The tallest bar is the 20 to 39 minute bin, and the bars fall away to the right with a long, thin tail of three patients who waited more than 80 minutes.
Interpreting the Shape
The distribution is positively skewed, meaning its tail extends toward longer waits. Three pieces of evidence show this: the histogram's long right tail, the mean of 48.1 minutes sitting well above the median of 41.5 and the gap between the third quartile and the maximum, 73.75 minutes, being much larger than the gap between the minimum and the first quartile, 19.5 minutes. Waiting times are often skewed in this way, because a few patients are delayed by surges, triage decisions or boarding, while most are seen within a narrower range (Provost & Murray, 2011).
Because of the skew, the median is the better summary of a typical patient's wait, and the interquartile range is a better summary of variation than the standard deviation. Reporting only the mean of 48 minutes would suggest that the typical patient waits longer than most actually do, while reporting only the median would hide the long waits that matter most for safety and satisfaction. The team should therefore report the median, the interquartile range and the percentage of patients meeting the 30-minute target, and it should look separately at the patients in the tail.
Limits of This Sample
Thirty patients from a single week is a small sample, and the summary statistics carry uncertainty that the exercise should acknowledge. A different week, especially one with an influenza surge or a holiday, could produce a noticeably different median and a longer tail. The sample was stratified by shift but not by day of the week or acuity, so weekend patterns and the sickest patients, who are seen immediately, may be underrepresented. These limits do not make the description useless; they mean it should be read as a baseline to be extended, not as a final verdict on the department's performance.
What the Team Should Do Next
Descriptive statistics describe; they do not explain. The next step is to examine the long waits individually. A quick review found that five of the seven patients waiting more than an hour arrived between 6 p.m. and 10 p.m., when the department had the most boarded patients, which suggests that evening boarding, not triage speed, drives the tail. The team should also track the median and the percentage within target weekly on a run chart, since a single week's sample cannot show whether performance is stable or changing (Provost & Murray, 2011).
Conclusion
Thirty door-to-provider times from one emergency department produced a mean of 48.1 minutes, a median of 41.5 minutes, a standard deviation of 26.1 minutes and an interquartile range of 25.75 minutes, with only 23% of patients seen within the 30-minute target. The histogram and the gap between mean and median show a positively skewed distribution. For a skewed measure like waiting time, the median, the interquartile range and the percentage meeting the target give the quality team a more accurate picture than the mean alone, and the tail points to where improvement work should begin.
References
Grove, S. K., & Cipher, D. J. (2020). Statistics for nursing research: A workbook for evidence-based practice (3rd ed.). Elsevier.
Polit, D. F., & Beck, C. T. (2021). Nursing research: Generating and assessing evidence for nursing practice (11th ed.). Wolters Kluwer.
Provost, L. P., & Murray, S. K. (2011). The health care data guide: Learning from data for improvement. Jossey-Bass.
How this NSG 509 Week 2 example is structured
Course materials for NSG/509 describe the Week 2 assignment as a Descriptive Statistics Exercise in which students generate descriptive statistics and a histogram for a data set. The paper presents the raw data first so every statistic can be checked, reports measures of central tendency and dispersion with their arithmetic, builds the frequency table that produces the histogram and then interprets the shape. The interpretation section is where the exercise earns its grade, because it turns numbers into a decision about how to report performance. Students search this week as NSG 509 Week 2, NSG509 Wk 2 or NSG/509 Wk 2; all three are the same assignment.
NSG/509 Week 2 questions, answered
What does NSG/509 Week 2 usually ask for?
Course materials describe the NSG/509 Week 2 assignment as a descriptive statistics exercise: students generate descriptive statistics and a histogram for a data set and describe what they show. Your own instructions and data set decide the exact variables.
When should I report the median instead of the mean?
When the distribution is skewed or has outliers, as waiting times and lengths of stay usually do. The mean is pulled toward the long tail, while the median shows the typical value. Reporting both, with the reason for preferring one, is the strongest approach.
How many bins should a histogram have?
Enough to show the shape without making each bin nearly empty. For 30 values, five to eight bins of equal width usually work. State the bin width and make sure bins do not overlap.
Write yours, or have the desk draft it
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