Control charts were built for factories, but the problem they solve is not specific to manufacturing. Any process that produces a stream of measurements over time, where people need to know whether this month's number is a real change or ordinary fluctuation, is a control chart problem. Hospitals produce those measurements constantly, and over the past two decades control charts have become a core tool in healthcare quality improvement.
This article covers why they fit healthcare so well, how to chart the three most common hospital metrics, and the mistakes that are specific to the setting.
Why healthcare needs control charts
The typical hospital quality report shows this month's infection rate next to last month's, with an arrow. The rate went from 2.1 to 2.6 per 1,000 device-days; the arrow is red; a meeting is scheduled. Next month it's 1.9; the arrow is green; the intervention is declared a success.
This is reacting to noise. With small monthly counts, the rate will swing widely from month to month even if nothing changes. A control chart puts those swings in context by showing the range of values that a stable process produces. The question changes from "did the number go up?" to "did the number go outside what this process normally does?" That's a question that can be answered, and it stops both the false alarms and the false reassurance.
The second reason is that healthcare improvement is typically tested with small, sequential changes (plan-do-study-act cycles) rather than controlled trials. A control chart with annotations at each change is the natural way to see whether a change moved the process.
Metric 1: Infection rates
Hospital-acquired infections are usually reported as a rate: infections per 1,000 device-days (central line-associated bloodstream infections per 1,000 central line-days, catheter-associated urinary tract infections per 1,000 catheter-days) or per 1,000 patient-days.
This is count data with a varying exposure, which calls for a u chart. Each point is the rate for a period (usually a month), and the limits vary with the exposure in that period: months with more device-days have tighter limits.
For a u chart, the center line is the total infections divided by total exposure across the baseline period, and the limits for a period with exposure n (in thousands of device-days) are
ū ± 3 √(ū / n)
Example: over 12 baseline months, a unit records 26 CLABSIs across 14,400 central line-days, so ū = 26 / 14.4 = 1.81 per 1,000 line-days. A month with 1,200 line-days (n = 1.2) has limits 1.81 ± 3√(1.81/1.2) = 1.81 ± 3.68, so UCL = 5.49 and LCL below zero (treated as no lower limit). A month with 3 infections in 1,200 line-days is a rate of 2.5, inside the limits. A month with 7 is 5.83, outside them.
When infections become rare (fewer than one per month on average), the u chart stops working well because most points are zero. The standard fix is a g chart or t chart, which plots the number of days between infections. A run of long gaps signals improvement; a cluster of short gaps signals a problem. This is the right chart for events like surgical site infections in a low-volume service.
Metric 2: Wait times
Emergency department door-to-provider time, time from order to result, days from referral to appointment: these are continuous measurements, and the right chart depends on how they're collected.
If each patient's wait time is recorded, and you want to monitor at the daily or weekly level, subgroup the patients within each period and use an X̄-S chart (the S chart rather than R because subgroup sizes are typically larger than 10 and vary). The X̄ chart tracks the average wait; the S chart tracks how consistent waits are within the period.
If you only have a summary figure per period (the daily median, say), use an I-MR chart on that summary.
Wait time data are almost always right-skewed: most patients wait a moderate time, a few wait a very long time. For the X̄ chart with a subgroup of 20 or more patients, this doesn't matter. For an I-MR chart on daily medians it usually doesn't matter either, because medians are more stable than means. For an I-MR chart on individual patients' wait times, it does matter, and a log transformation or a switch to subgrouping is warranted.
A useful practice: chart the daily 90th percentile alongside the daily mean. A process can hold its average steady while the long tail gets longer, and the tail is what generates complaints.
Metric 3: Readmissions
Thirty-day readmission rate is a proportion: readmissions divided by eligible discharges. The p chart is the standard tool. Each point is a month's rate; the limits vary with the number of discharges that month.
For a p chart, the center line p̄ is total readmissions divided by total discharges over the baseline, and limits for a month with n discharges are
p̄ ± 3 √( p̄(1 − p̄) / n )
Example: a service has a baseline readmission rate of 14%. A month with 250 discharges has limits 0.14 ± 3√(0.14 × 0.86 / 250) = 0.14 ± 0.066, so UCL = 20.6% and LCL = 7.4%. A month with 42 readmissions out of 250 (16.8%) is inside the limits. A month with 55 (22%) is outside.
Two healthcare-specific cautions for the p chart:
Case mix changes the denominator's meaning. If a service starts admitting sicker patients, the readmission rate will rise for reasons unrelated to care quality. Risk-adjusted rates are better inputs to the chart when available.
Very large denominators make the limits very tight. A system-wide chart with 5,000 discharges a month will have limits of ±1.5 percentage points and will signal on trivial fluctuations. This is a known limitation of the p chart with large n; the fix is either to chart at a unit level where n is smaller or to use a Laney p′ chart, which adjusts for the extra variation.
Chart selection summary
Metric | Data type | Chart |
|---|---|---|
Infections per 1,000 device-days | Count with varying exposure | u chart |
Days between rare infections | Time between events | g or t chart |
Falls, medication errors per period | Count with constant exposure | c chart |
Wait time (patients subgrouped by day/week) | Continuous, subgroups | X̄-S chart |
Daily median wait, monthly average LOS | Continuous, one value per period | I-MR chart |
Readmission rate, compliance rate, mortality rate | Proportion | p chart |
Mistakes specific to healthcare
Charting too infrequently. Monthly data means 24 points takes two years. Where possible, chart weekly. For wait times and volumes, daily is feasible and gives a usable baseline in a month.
Starting the chart after the improvement. The baseline has to precede the change, or there's nothing to compare against. Charts should be started as early in a project as possible, even with only a handful of points, using run-chart rules until there are enough for limits.
Recomputing limits every month. Limits are computed from a baseline period and then extended forward unchanged. Recomputing them as each new month arrives absorbs any shift into the limits and hides it.
Ignoring the small-number problem. A unit with two infections a year cannot be monitored on a monthly u chart. Use the days-between chart.
Reporting to committees without the chart. A single number with a comparison to last month invites reaction to noise. The chart, with the limits and the annotations, is the report.
Building the charts
The calculations for u and p charts are simple but tedious when the denominator changes every period, and it's easy to apply the wrong formula. A control chart builder that accepts count and exposure (or numerator and denominator) per period and produces the varying limits automatically removes most of the arithmetic, and makes it practical for a unit-level quality lead to maintain several charts without a statistician.
For staff who are new to the concepts, particularly the idea that a point inside the limits is not a reason to act, the what is a control chart introduction is a short and non-technical starting point.
Summary
Infection rates go on a u chart (or a days-between chart when rare), wait times on an X̄-S or I-MR chart depending on how they're aggregated, and readmission rates on a p chart. The chart replaces the monthly comparison with a question that can actually be answered: is this period's value outside what the process normally produces? Chart early, chart frequently, hold the baseline limits fixed, and bring the chart rather than the number to the meeting.