--- title: "Constructing and interpreting X-bar and R charts" subject: "Metrology" chapter: 15 key: "semester_3/metrology:x-bar-r-chart-construction" source: https://www.bctengineer.com/q/semester_3/metrology/x-bar-r-chart-construction site: BCT Engineering --- # Constructing and interpreting X-bar and R charts Metrology · Chapter 15: Quality Control Management - Asked in: year not given, 8 marks - Video: [X chart and R chart || With Example || Statistical quality control || Lecture Notes](https://www.youtube.com/watch?v=dkSROYov-ow) — Prithvi Study Point - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/metrology/x-bar-r-chart-construction - Whole chapter: https://www.bctengineer.com/md/semester_3/metrology/chapter-15 ## Question Samples of five shafts were taken from a production line at ten intervals. The sample means $\bar X$ and ranges $R$ (mm) are: Mean: 50.02, 49.99, 50.04, 50.01, 49.99, 50.03, 50.00, 50.12, 49.99, 50.02. Range: 0.06, 0.08, 0.05, 0.07, 0.09, 0.06, 0.04, 0.07, 0.08, 0.05. For $n = 5$: $A_2 = 0.577$, $D_3 = 0$, $D_4 = 2.114$, $d_2 = 2.326$. Calculate the control limits of the $\bar X$ and $R$ charts and say whether the process is in control. If any sample is out of control, remove it and recalculate the limits. Explain the use of a control chart. ## Short answer Grand mean = 50.021 mm, mean range = 0.065 mm. Initial X-bar limits: 49.9835–50.0585 mm; R limits: 0–0.1374 mm. Sample 8 (50.12) out. After removal, revised X-bar limits 49.9728–50.0472 mm, R limits 0–0.1362 mm. The chart detects assignable causes beyond chance variation. ## Answer #### Use of a control chart A control chart is a plot of a sample statistic against time with a centre line and upper and lower control limits (UCL, LCL) at $\pm 3\sigma$. Points inside the limits show only chance variation; a point outside (or a trend) shows an assignable cause that has to be found and removed. $\bar X$ chart monitors the process average and $R$ chart monitors the spread. #### Calculation (all ten samples) $$ \bar{\bar X} = \frac{500.21}{10} = 50.021\ \text{mm}, \qquad \bar R = \frac{0.65}{10} = 0.065\ \text{mm} $$ $\bar X$ chart: $$ UCL = \bar{\bar X} + A_2\bar R = 50.021 + 0.577(0.065) = 50.0585 $$ $$ LCL = 50.021 - 0.0375 = 49.9835 $$ $R$ chart: $$ UCL_R = D_4\bar R = 2.114(0.065) = 0.1374, \qquad LCL_R = D_3\bar R = 0 $$ #### Interpretation All ranges (0.04 to 0.09) are below 0.1374, so the variation within samples is stable. In the $\bar X$ chart, sample 8 has $\bar X = 50.12 > 50.0585$ and is **out of control**; all others lie between 49.9835 and 50.0585. #### Revised limits (without sample 8) $$ \bar{\bar X} = \frac{450.09}{9} = 50.010, \qquad \bar R = \frac{0.58}{9} = 0.0644 $$ $$ UCL = 50.010 + 0.577(0.0644) = 50.0472, \qquad LCL = 50.010 - 0.0372 = 49.9728 $$ $$ UCL_R = 2.114(0.0644) = 0.1362, \qquad LCL_R = 0 $$ The remaining nine samples (49.99 to 50.04) lie within the revised limits, so the process is now in statistical control. The estimated process standard deviation is $\sigma = \bar R/d_2 = 0.0644/2.326 = 0.0277$ mm. **Answer: initial $\bar X$ limits 49.9835 to 50.0585 mm, $R$ limit 0 to 0.1374 mm (sample 8 out); revised $\bar X$ limits 49.9728 to 50.0472 mm, $R$ limit 0 to 0.1362 mm.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does a point outside the control limits indicate? A. An assignable cause that must be found and removed B. Only chance variation, so no action C. The specification limits are wrong D. That the sample size is too large Answer: A. Control limits at +/-3 sigma separate chance variation from assignable causes. (Key point: Use of chart) 2. With X-double-bar = 50.021, R-bar = 0.065, A2 = 0.577, the X-bar limits are: A. 49.9835 to 50.0210 mm B. 49.9835 to 50.0585 mm C. 49.9460 to 50.0960 mm D. 49.9990 to 50.0430 mm Answer: B. A2 R-bar = 0.577 x 0.065 = 0.0375; limits are 50.021 +/- 0.0375. (Key point: Initial X-bar limits) 3. Which sample is out of control and on which chart? A. Sample 8 on the R chart only B. Sample 5 (49.99) on the X-bar chart C. Sample 8 (mean 50.12) on the X-bar chart; the R chart is in control D. No sample; all points are inside Answer: C. 50.12 exceeds UCL 50.0585, while all ranges (0.04 to 0.09) are below UCL_R = 0.1374. (Key point: Out of control) 4. After removing sample 8, the revised X-bar limits are: A. 49.9835 to 50.0585 mm B. 49.9500 to 50.0700 mm C. 50.0000 to 50.0472 mm D. 49.9728 to 50.0472 mm Answer: D. Revised X-double-bar = 50.010 and R-bar = 0.0644, so 50.010 +/- 0.577 x 0.0644 = 50.010 +/- 0.0372. (Key point: Revised limits) ## Related in Metrology - [Error propagation and effect of averaging](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Aerror-propagation-and-averaging) — Metrology, chapter 2 - [Sine bar principle, setting and error](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Asine-bar-setting-and-error) — Metrology, chapter 5 - [Vernier caliper and micrometer readings](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Avernier-and-micrometer-readings) — Metrology, chapter 4 - [Least squares fit for calibration data](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Aleast-squares-calibration-line) — Metrology, chapter 2 - [Process capability and acceptance sampling](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Aprocess-capability-and-sampling-inspection) — Metrology, chapter 15 - [Slip gauges: wringing and building a stack](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Aslip-gauge-wringing-and-stack) — Metrology, chapter 4 ## Related in other subjects - [Settling tank 20 m long, 3.5 m deep; 0.02 and 0.01 mm](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Ased-b79) — Water Supply Engineering, chapter 6, asked 3× - [Settling tank 30 m x 3 m, 0.02 mm at 25°C; 0.01 mm at 10°C](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Ased-as70) — Water Supply Engineering, chapter 6, asked 3× - [Settling tank 30 m x 3 m, 0.02 mm, SG 2.50](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Ased-bk80) — Water Supply Engineering, chapter 6 - [Reconstruction from samples taken at Nyquist rate](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Areconstruction-nyquist-samples) — Communication Systems, chapter 4