Shewhart Variables Control Chart (X-bar / R)
Shewhart Variables Control Chart (X-bar and R) · Also known as: X-bar and R chart, Shewhart chart, variables control chart, process control chart, Shewhart kontrol kartı
The Shewhart control chart, invented by Walter Shewhart at Bell Labs in the 1920s and set out in his 1931 book, is the foundational tool of statistical process control. It plots a process statistic — typically the subgroup mean (X-bar) and range (R) — over time against a center line and three-sigma control limits, distinguishing the natural common-cause variation inherent in a stable process from special-cause variation that signals something has changed and warrants investigation.
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When to use it
Use a Shewhart X-bar/R chart to monitor a continuous (variables) quality characteristic of a repetitive process over time — manufacturing dimensions, fill weights, cycle times, laboratory measurements — when you can collect rational subgroups and want to detect changes in the mean and spread. It is most effective for detecting moderate-to-large, sudden shifts; for small, sustained shifts the CUSUM or EWMA charts are more sensitive. The method assumes approximately independent observations and rational subgrouping; autocorrelation or poor subgrouping undermines the limits. For count or proportion data rather than measurements, use attributes charts (p, np, c, u) instead.
Strengths & limitations
- Separates common-cause from special-cause variation, guiding when to act and when to leave a process alone.
- Simple, visual, and universally understood on the shop floor and in service operations.
- Three-sigma limits give a low, predictable false-alarm rate for a stable process.
- Foundational and general — the template from which most other control charts derive.
- Insensitive to small, sustained shifts in the mean, which CUSUM and EWMA detect faster.
- Assumes independent observations; autocorrelated data inflate false alarms.
- Relies on rational subgrouping, which can be hard to achieve in some processes.
- Standard rules assume approximate normality of the subgroup statistic.
Frequently asked
What is the difference between control limits and specification limits?
Control limits describe what the process actually does — they are computed from process data and bound common-cause variation. Specification limits describe what the customer requires. A process can be in control (within its control limits) yet incapable (outside specifications), so the two must never be confused or plotted interchangeably.
Why three sigma?
Shewhart chose three-sigma limits as an economic balance between false alarms and missed signals. For a stable, roughly normal process they produce about one false alarm every 370 points, rare enough that a signal is worth investigating without causing constant needless searches.
When should I use CUSUM or EWMA instead?
Shewhart charts react quickly to large shifts but slowly to small, persistent ones because each point is judged in isolation. When detecting small sustained shifts in the mean matters, the CUSUM and EWMA charts, which accumulate information across points, are substantially more sensitive.
Sources
- Shewhart, W. A. (1931). Economic Control of Quality of Manufactured Product. D. Van Nostrand Company. ISBN: 978-0-87389-076-2
- Montgomery, D. C. (2009). Introduction to Statistical Quality Control (6th ed.). John Wiley & Sons. ISBN: 978-0-470-16992-6
How to cite this page
ScholarGate. (2026, June 2). Shewhart Variables Control Chart (X-bar and R). ScholarGate. https://scholargate.app/en/statistics/shewhart-control-chart
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Attributes Control ChartStatistics↔ compare
- CUSUM ChartStatistics↔ compare
- Descriptive StatisticsStatistics↔ compare
- EWMA ChartStatistics↔ compare