{"id":"xbar-r-chart","model_id":"X-bar and R chart","name":"Shewhart X-bar and R chart","summary":"The Shewhart X-bar and R chart is a pair of variables control charts for monitoring a process mean and process variability. The X-bar chart plots subgroup means with limits based on the average range, and the R chart plots subgroup ranges with limits based on factors D3 and D4. It is intended for small subgroup sizes, typically up to about 10.","study_notes":["The R chart controls process variability because the sample range is related to the process standard deviation.","The document states the range approach is satisfactory for sample sizes up to around 10; for larger sample sizes subgroup standard deviations are preferable.","A typical sample size is 4 or 5.","For n=2 to 10, the factor table gives A2 = 1.880 to 0.308, D3 = 0 to 0.223, and D4 = 3.267 to 1.777.","For an X̄ chart with 3-sigma limits, the document gives false-alarm probability p = 0.0027.","The document defines A2 = 3/(d2√n), D3 = 1 − 3d3/d2, and D4 = 1 + 3d3/d2."],"safety_notes":[],"sources":[{"url":"https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc321.htm","title":"6.3.2.1. Shewhart X-bar and R and S Control Charts","accessed":"2026-10-25"}],"verification":{"status":"source_checked","checked_at":"2026-10-25","values":8,"method":"llm_source_check"},"updated_at":"2026-10-25","type":"operational","spec":{"kind":"methodology","formula":{"expression":"UCL_X̄ = x̄ + A2(n) R̄; CL_X̄ = x̄; LCL_X̄ = x̄ − A2(n) R̄; UCL_R = D4(n) R̄; CL_R = R̄; LCL_R = D3(n) R̄","variables":[{"symbol":"x̄","name":"average of subgroup means"},{"symbol":"R̄","name":"average subgroup range"},{"symbol":"A2","name":"X-bar chart factor depending on subgroup size"},{"symbol":"D3","name":"R chart lower-limit factor depending on subgroup size"},{"symbol":"D4","name":"R chart upper-limit factor depending on subgroup size"},{"symbol":"n","name":"subgroup size"}]},"related":[]},"url":"https://machineschema.com/operational/xbar-r-chart","license":"https://creativecommons.org/licenses/by/4.0/"}