What is the difference between a run chart and a control chart?

A run chart plots data over time around the median and uses run rules to spot non-random patterns. A control chart adds a center line at the mean and upper and lower control limits calculated from the data (about 3 standard deviations), so you can also see single points that signal special-cause variation.

When should I use an XmR (I-MR) chart instead of a p-chart?

Use an individuals (XmR) chart when you have one value per period, such as a weekly median wait time or a monthly satisfaction score. Use a p-chart when each period is a proportion with a known denominator, such as readmissions out of discharges; its limits change with the denominator.

Are control limits the same as targets or specification limits?

No. Control limits come from the process data and describe what the process is currently doing. Targets and specification limits are set by people. A process can be stable (in control) and still miss its target.

Why do p-chart limits get wider in some months?

Because the limits depend on the denominator: p̄ ± 3√(p̄(1 − p̄)/n). With the same 12% average, a month with 160 discharges has an upper limit of 19.71%, but a month with only 40 has an upper limit of 27.41%. Wider limits reflect more uncertainty, not a less stable process.

Free tool · CPHQ® exam topic

Control chart calculator

Paste one value per period to draw an individuals (XmR) chart with its control limits and special-cause signals. Tracking a proportion? Use the p-chart calculator below it.

Mean (center line): 32.8
UCL: 45.54
LCL: 20.06
Average moving range: 4.79
Moving range UCL: 15.65

Special-cause signals: Point 14 falls outside the control limits. Investigate what changed before acting on individual points.

p-chart limits for a proportion

Readmissions, falls with injury, bundle compliance: when each period is a count out of a denominator, use a p-chart. Enter the totals for your baseline periods and this period’s numbers.

Result

This period: 20.63%
Center line (p̄): 12.00%
Upper control limit: 19.71%
Lower control limit: 4.29%

This period’s 20.63% is above the upper control limit: a special-cause signal worth investigating.

Formulas

For an individuals (XmR, or I-MR) chart, NIST estimates the process standard deviation from the average moving range:

Moving range MRᵢ = |xᵢ − xᵢ₋₁|, the absolute difference between consecutive points.
Center line = x̄, the mean of all points.
Control limits = x̄ ± 3 × MR̄ ÷ 1.128 (that is, x̄ ± 2.66 × MR̄).
Moving range chart upper limit = 3.267 × MR̄.

For a p-chart, the center line is p̄ (all events ÷ all denominators) and the limits for a period with denominator n are p̄ ± 3√(p̄(1 − p̄)/n), with the lower limit floored at 0.

Worked example

An emergency department tracks its weekly median door-to-provider time. The 20 weeks above have a mean of 32.8minutes and an average moving range of 4.79, so the limits are 20.06 to 45.54 minutes. Week 14, at 47 minutes, is above the upper limit: a special-cause signal. The team traces it to an EHR downtime that week. Every other week moves within the limits, which is common-cause variation: reacting to any one of those weeks would be tampering.

To reduce the average itself, the team has to change the process (for example, a new triage model). When the change works, the chart shows a shift: 8 or more weeks in a row below the old mean. Once the shift is sustained, recalculate the limits from the new weeks only.

Signals this tool flags

A point outside the control limits: a single special-cause signal.
A run of 8 or more consecutive points on one side of the center line: a sustained shift.

Other rule sets (Western Electric, Nelson) add trend and zone rules, and some organizations use 6 or 7 points for a shift. Agree on one set before you start reviewing charts, and don’t react to each point that is merely above or below average: that is tampering with common-cause variation.

Test yourself: a CPHQ-style question

In statistical process control, what do the upper and lower control limits (UCL/LCL) represent?

Regulatory thresholds set by accreditors
Performance goals set by leadership
The expected range of variation (±3 standard deviations) in a stable process
The 90th and 10th percentile benchmarks

Show answer

C. The expected range of variation (±3 standard deviations) in a stable process

Control limits are calculated from the process data itself and sit about 3 standard deviations from the mean. Points outside them signal special-cause variation that needs investigating.

Questions like this come up in the Health Data Analytics domain of the CPHQ® exam. The 249 questions in the app each come with a written explanation.

FAQ

What is the difference between a run chart and a control chart?: A run chart plots data over time around the median and uses run rules to spot non-random patterns. A control chart adds a center line at the mean and upper and lower control limits calculated from the data (about 3 standard deviations), so you can also see single points that signal special-cause variation.
When should I use an XmR (I-MR) chart instead of a p-chart?: Use an individuals (XmR) chart when you have one value per period, such as a weekly median wait time or a monthly satisfaction score. Use a p-chart when each period is a proportion with a known denominator, such as readmissions out of discharges; its limits change with the denominator.
Are control limits the same as targets or specification limits?: No. Control limits come from the process data and describe what the process is currently doing. Targets and specification limits are set by people. A process can be stable (in control) and still miss its target.
Why do p-chart limits get wider in some months?: Because the limits depend on the denominator: p̄ ± 3√(p̄(1 − p̄)/n). With the same 12% average, a month with 160 discharges has an upper limit of 19.71%, but a month with only 40 has an upper limit of 27.41%. Wider limits reflect more uncertainty, not a less stable process.

Sources

Checked against these sources on October 29, 2026. For education and CPHQ® exam preparation only, not a substitute for your organization’s policies, official measure specifications or professional judgment. Calculations run in your browser; nothing you enter is sent anywhere.