5.2.13 Center Line & Control Limit Calculations

Center Line & Control Limit Calculations Purpose of Center Lines and Control Limits Control charts distinguish natural process variation from special causes. Center lines and control limits are the quantitative backbone of this distinction. - Center line: The expected average performance of the process. - Control limits: Statistical boundaries that define the range of common-cause variation. For standard Shewhart charts under stable conditions: - Center line = process average (or target, if specified). - Upper control limit (UCL) and lower control limit (LCL) are usually set at ±3 standard deviations of the plotted statistic. Control limits are not specification limits; they describe what the process is doing, not what it should do. --- Fundamental Concepts Process Distribution and 3-Sigma Limits Control limit formulas assume: - Individual points (or subgroup statistics) follow a stable distribution over time. - For most charts, an underlying normal distribution assumption is used (or an approximate normality through central limit theorem). Core idea: - Statistic being plotted (e.g., subgroup mean, individual value) has: - A center (its expected value). - A standard error (its short-term variation). - Control limits = center ± 3 × standard error. This is applied differently depending on the chart type but follows the same logic. Subgroups and Rational Sampling Control chart calculations depend on how data are grouped: - Rational subgroups: Samples taken so that: - Variation within a subgroup reflects short-term, common-cause variation. - Variation between subgroups can reveal shifts and trends over time. Choice of subgroup size (n) and composition affects: - How accurately the average is estimated. - Which control chart type and formulas are appropriate. - The constants used in calculations (such as A2, D3, D4, d2). --- X-bar and R Chart Calculations X-bar and R charts are used for continuous data with subgroup size n (typically 2–10). They display: - X-bar chart: subgroup means. - R chart: subgroup ranges (max − min). R Chart: Center Line and Limits Given k subgroups of size n: 1. Compute each subgroup range: - Ri = maxi − mini 1. Compute the average range: - (\bar{R} = \dfrac{\sum{i=1}^{k} Ri}{k}) 1. Use control chart constants (D3 and D4) based on subgroup size n. 1. R chart calculations: - Center line: (\bar{R}) - UCL_R = D4 × (\bar{R}) - LCL_R = D3 × (\bar{R}) Notes: - For small n, D3 can be 0, giving LCL = 0. - D3 and D4 are derived from distribution properties of the range and depend only on n. X-bar Chart: Center Line and Limits 1. Compute each subgroup mean: - (\bar{X}i = \dfrac{\sum X{ij}}{n}) 1. Compute the grand mean: - (\bar{\bar{X}} = \dfrac{\sum{i=1}^{k} \bar{X}i}{k}) 1. Use constant A2 based on subgroup size n. 1. X-bar chart calculations: - Center line: (\bar{\bar{X}}) - UCL_{\bar{X}} = (\bar{\bar{X}} + A_2 \times \bar{R}) - LCL_{\bar{X}} = (\bar{\bar{X}} - A_2 \times \bar{R}) Rationale: - The process standard deviation is estimated as: - (\hat{\sigma} \approx \dfrac{\bar{R}}{d_2}) - Standard error of the mean: - (\sigma_{\bar{X}} = \dfrac{\hat{\sigma}}{\sqrt{n}}) - A2 combines the factors 3, 1/√n, and 1/d2 into a single constant: - (A2 = \dfrac{3}{d2 \sqrt{n}}) --- X-bar and s Chart Calculations X-bar and s charts are used when subgroup size is larger (commonly n ≥ 10), or when standard deviation is preferred to range. - X-bar chart: subgroup means. - s chart: subgroup standard deviations. s Chart: Center Line and Limits 1. Compute each subgroup standard deviation: - si (sample standard deviation for subgroup i) 1. Compute the average standard deviation: - (\bar{s} = \dfrac{\sum{i=1}^{k} si}{k}) 1. Use constants B3 and B4 based on subgroup size n. 1. s chart calculations: - Center line: (\bar{s}) - UCL_s = B4 × (\bar{s}) - LCL_s = B3 × (\bar{s}) Notes: - B3 may be 0 for some subgroup sizes, giving LCL = 0. - B3 and B4 are derived from distribution properties of the standard deviation and depend only on n. X-bar Chart with s: Center Line and Limits 1. Compute subgroup means and grand mean (\bar{\bar{X}}) as for the X-bar and R chart. 1. Use constant A3 based on subgroup size n. 1. X-bar chart calculations: - Center line: (\bar{\bar{X}}) - UCL_{\bar{X}} = (\bar{\bar{X}} + A_3 \times \bar{s}) - LCL_{\bar{X}} = (\bar{\bar{X}} - A_3 \times \bar{s}) Rationale: - Process standard deviation estimated as: - (\hat{\sigma} \approx \dfrac{\bar{s}}{c_4}) - Constant A3 incorporates 3, 1/√n, and 1/c4 into a single factor: - (A3 = \dfrac{3}{c4 \sqrt{n}}) --- Individuals and Moving Range (XmR) Chart Calculations XmR charts are used for individual measurements (n = 1) when subgrouping is not practical. - X chart: individual values. - mR chart: moving ranges, usually between consecutive values. Moving Range Chart: Center Line and Limits 1. Form moving ranges: - mRi = |Xi − Xi−1| for i = 2 to N 1. Compute average moving range: - (\bar{mR} = \dfrac{\sum{i=2}^{N} mRi}{N - 1}) 1. Use constant d2 for n = 2 (since each moving range is based on 2 points); commonly d2 = 1.128. 1. mR chart calculations: - Center line: (\bar{mR}) - Typical limits use constants D3 and D4 specialized for moving range: - UCL_{mR} = D4 × (\bar{mR}) (often D4 ≈ 3.267 for n = 2) - LCL_{mR} = D3 × (\bar{mR}) (often D3 = 0 for n = 2) Some formulations compute limits directly as 0 and 3.267 × (\bar{mR}). X (Individuals) Chart: Center Line and Limits 1. Compute average of all individual values: - (\bar{X} = \dfrac{\sum{i=1}^{N} Xi}{N}) 1. Estimate process standard deviation from moving range: - (\hat{\sigma} \approx \dfrac{\bar{mR}}{d_2}) - With n = 2, d2 ≈ 1.128. 1. X chart calculations (3-sigma limits): - Center line: (\bar{X}) - UCL_X = (\bar{X} + 3 \times \hat{\sigma}) - LCL_X = (\bar{X} - 3 \times \hat{\sigma}) Substitute (\hat{\sigma} = \bar{mR} / d2): - UCL_X = (\bar{X} + 3 \times \dfrac{\bar{mR}}{d_2}) - LCL_X = (\bar{X} - 3 \times \dfrac{\bar{mR}}{d_2}) Using d2 = 1.128: - 3 / 1.128 ≈ 2.66 - So: - UCL_X ≈ (\bar{X} + 2.66 \times \bar{mR}) - LCL_X ≈ (\bar{X} - 2.66 \times \bar{mR}) --- Attribute Charts: p, np, c, and u Attribute charts handle count or proportion data. Control limits are based on binomial or Poisson distributions, often approximated by normal theory for larger counts. p Chart: Proportion Nonconforming Used when: - Each subgroup has n units inspected. - Each unit is conforming or nonconforming. - Subgroup size n may vary. 1. For subgroup i: - ni = number inspected. - xi = number nonconforming. - pi = xi / ni (proportion nonconforming). 1. Overall proportion: - (\bar{p} = \dfrac{\sum xi}{\sum ni}) 1. Standard deviation of proportion for subgroup i: - (\sigma{pi} = \sqrt{\dfrac{\bar{p}(1 - \bar{p})}{n_i}}) 1. p chart calculations: - Center line: (\bar{p}) - For each subgroup i: - **UCL_{pi}** = (\bar{p} + 3 \times \sigma{p_i}) - **LCL_{pi}** = (\bar{p} - 3 \times \sigma{p_i}) 1. If LCLp_i is negative, set it to 0. When subgroup size is constant (n): - (\sigma_p = \sqrt{\dfrac{\bar{p}(1 - \bar{p})}{n}}) - Limits are the same for all subgroups. np Chart: Count Nonconforming Used when: - Each subgroup has a constant size n. - Tracking number (not proportion) of nonconforming units. 1. Overall proportion: - (\bar{p} = \dfrac{\sum x_i}{k n}) 1. Average count per subgroup: - (\bar{np} = \bar{p} \times n) 1. Standard deviation for np: - (\sigma_{np} = \sqrt{n \bar{p}(1 - \bar{p})}) 1. np chart calculations: - Center line: (\bar{np}) - UCL_{np} = (\bar{np} + 3 \times \sigma_{np}) - LCL_{np} = (\bar{np} - 3 \times \sigma_{np}) If LCLnp < 0, set LCLnp = 0. c Chart: Count of Nonconformities per Unit Used when: - Counting total nonconformities on a single inspection unit (fixed area, item, or opportunity set). - Sample size (unit size or area) is constant. 1. Let ci be the count of nonconformities on unit i. 1. Average count: - (\bar{c} = \dfrac{\sum c_i}{k}) 1. Under a Poisson assumption, variance equals the mean: - (\sigma_c = \sqrt{\bar{c}}) 1. c chart calculations: - Center line: (\bar{c}) - UCL_c = (\bar{c} + 3 \sqrt{\bar{c}}) - LCL_c = (\bar{c} - 3 \sqrt{\bar{c}}) If LCLc < 0, set LCLc = 0. u Chart: Nonconformities per Unit (Variable Area or Sample Size) Used when: - Counting nonconformities. - The sample size (area, length, opportunity count) varies by observation. 1. For subgroup i: - ci = count of nonconformities. - ni = number of units, area, or opportunity measure. - ui = ci / ni. 1. Overall rate: - (\bar{u} = \dfrac{\sum ci}{\sum ni}) 1. Standard deviation of ui: - (\sigma{ui} = \sqrt{\dfrac{\bar{u}}{n_i}}) 1. u chart calculations: - Center line: (\bar{u}) - For each subgroup i: - **UCL_{ui}** = (\bar{u} + 3 \times \sigma{u_i}) - **LCL_{ui}** = (\bar{u} - 3 \times \sigma{u_i}) If LCLui < 0, set LCLui = 0. --- Using Constants in Control Limit Calculations Control chart constants (A2, A3, B3, B4, D3, D4, d2, c4) are precomputed to simplify formulas. They depend only on subgroup size n and the statistic used. Key points: - A2: Relates average range to limits on X-bar. - A3: Relates average standard deviation to limits on X-bar. - D3, D4: Define limits on R or mR charts. - B3, B4: Define limits on s charts. - d2, c4: Bias correction factors used in estimating σ from R and s. When constructing control limits: - Select the correct constant for your subgroup size and chart type. - Apply it exactly as in the formula; misusing constants yields incorrect limits. - Ensure consistency: if you switch from R to s, use the correct matching constants (A2 vs A3, D4 vs B4, etc.). --- Estimation Choices: Using Data vs Using Known Sigma There are two main ways to compute control limits: - Estimated from data: - Use (\bar{R}), (\bar{s}), or (\bar{mR}) to estimate σ. - Most common in improvement projects. - Based on known or specified σ: - If a reliable σ is known, control limits for X-bar or individuals charts can be: - Center line: known mean μ (or target). - UCL = μ + 3 σ / √n. - LCL = μ − 3 σ / √n. When using known σ: - Ensure that the σ value truly reflects current, stable process performance. - Reassess limits if process conditions change. --- Handling Negative and Impossible Limits In practice: - For attribute charts, negative LCLs are impossible: - Set LCL = 0 in those cases. - For charts where the statistic cannot be negative (R, s, mR): - Use the formula; if LCL < 0, set LCL = 0. - For continuous measurements (e.g., dimensions that can technically be negative), negative limits can be meaningful and are retained if physically possible. Consistency is important: treat all subgroups the same way when adjusting LCLs. --- Recalculating Limits and Dealing with Out-of-Control Points Control limit calculations assume the data used represent a stable process. Out-of-control points distort estimates of the center and spread. Practical steps: - Construct preliminary charts using all data. - Identify and investigate special-cause points. - If special causes are confirmed and removed: - Exclude those points from calculations of center line and spread. - Recalculate limits using only in-control data. - Document which data were used to calculate final control limits. Avoid repeatedly recalculating limits just because new points deviate. Limits should represent a stable process; changing them too often hides true shifts. --- Summary Center lines and control limits are quantitative tools for distinguishing common-cause from special-cause variation. Their calculations follow a consistent structure: - Identify the statistic to chart (mean, individual value, range, standard deviation, proportion, count). - Estimate its center from data (or use a known target). - Determine its standard error based on: - Subgroup size. - Estimated process standard deviation or distribution-specific formulas. - Apply 3-sigma limits using appropriate constants and formulas for the chart type. Mastery of center line and control limit calculations requires: - Correct choice of chart and subgroup structure. - Accurate computation of averages, ranges, standard deviations, and proportions. - Proper use of constants and handling of impossible limits. - Careful use of only stable, in-control data to set final limits. These elements ensure that control charts reliably signal true process changes rather than random noise.