SOP: Constructing X and R Control Charts
GIFSQ — Global Institute for Food Safety & Quality
Principle
No process turns out identical units, hour to hour or day to day. Control charts visualize trends against an upper and a lower limit — the dividing line between acceptable and unacceptable quality — so causes of defective product can be anticipated and corrected. The X chart controls variation in the mean (average) of samples; the R (range) chart shows the difference between the highest and lowest values in a subgroup, i.e. the spread of variance.
Control limits must not be confused with specification limits. Specification limits are the permissible limits of a quality characteristic for each individual unit; control limits evaluate variation from subgroup to subgroup. Control limits are set at ±3 standard deviations from the mean — 99.73% of the normal curve — so when the process is in control, the subgroup values fall between the limits 997 times out of 1,000.
Draw subgroups of 4 to 5 samples: from a statistical point of view, subgroup averages are nearly normal for subgroups of 4 or more, even when drawn from non-normal groups. When the subgroup size exceeds 10, use the σ chart instead of the R chart for controlling dispersion.
Procedure
Select the quality characteristic
↓
Select the method for measuring the variable
↓
Determine the subgroup size to use for the X and R charts
↓
Select the time interval for taking out samples
↓
Ensure the samples taken out are representative
↓
Record any change occurring at the different intervals while samples are taken — change of operators, product quality, etc.
↓
Collect the data
↓
Determine the trial control limits
↓
Establish the revised control limits
↓
Repeat the studies until enough data have been collected for the limits to be reliable
Data example: filled weight of pineapple segments into 401 × 411 cans, four samples taken every hour for 10 hours from the filling line. Subgroup averages (X̄) and ranges (R) per hour give ΣX̄ = 5,473.00 over m = 10 subgroups and ΣR = 101. (The individual cells of the book’s data table are OCR-garbled; the subgroup totals are carried as printed. The next SOP discards the 5th-hour mean and the 2nd- and 6th-hour ranges, which exceed these trial limits.)
Calculation
X̿ = ΣX̄/m
R̄ = ΣR/m
UCLₓ = X̿ + A₂R̄
LCLₓ = X̿ − A₂R̄
UCL_R̄ = D₄R̄
LCL_R̄ = D₃R̄
σ = R̄/d₂
Where,
- X̿ = average of the subgroup means (the process average)
- X̄ = subgroup average
- ΣX̄ = sum of the subgroup averages
- m = number of subgroups
- R̄ = average of the subgroup ranges
- R = subgroup range
- A₂, D₃, D₄, d₂ = control-chart factors for the subgroup size, from the factors table
- σ = process standard deviation
Worked example: subgroup size 4 gives A₂ = 0.729, D₄ = 2.282, D₃ = 0.
X̿ = 5,473.00/10 = 547.3
R̄ = 101/10 = 10.1
UCLₓ = 547.3 + 0.729 × 10.1 = 547.3 + 7.36 = 554.66 g
LCLₓ = 547.3 − 7.36 = 539.94 g
UCL_R̄ = 2.282 × 10.1 = 23.04 g
LCL_R̄ = 0 × 10.1 = 0 g
Note: the book’s control-chart factors table is garbled in the scan and is not reproduced here; the factor values above are carried as printed.
Precautions
- Control limits are not specification limits — they mark subgroup-to-subgroup variation, not the permissible range for individual units.
- Keep subgroup size at 4 to 5; above 10, use the σ chart instead of the R chart.
- Record changes (operators, product quality) at each interval, and repeat the study until the limits are reliable.
References
Ranganna, S., Handbook of Analysis and Quality Control for Fruit and Vegetable Products, 2nd edition.
Enrick, N.L., Quality Control and Reliability, Industrial Press, New York, 1977 — cited in the book.
Besterfield, D.H., Quality Control — A Practical Approach, Prentice-Hall, New Jersey, 1979 — cited in the book.
Kramer, A. & B.A. Twigg, Fundamentals of Quality Control for the Food Industry, Vol. 1, AVI Publishing Co., Westport, 1970 — cited in the book.