15.1 Control Charts for Variables and Attributes

Control charts are powerful tools for monitoring and improving quality in manufacturing and service processes. They help detect variations, allowing businesses to maintain consistent output and reduce defects.

For measurable characteristics like length or weight, we use variable control charts. For countable features like defects, we use attribute control charts. Both types help identify out-of-control processes and non-random patterns, guiding quality improvement efforts.

Control Charts for Variables

Control charts: variables vs attributes

• Control charts for variables used when quality characteristics measurable on continuous scale (length, weight, temperature, time)
• Control charts for attributes used when quality characteristics categorical or counted (number of defects, proportion of defective items)
• Common types of control charts for variables: X-bar chart, R chart, S chart
• Common types of control charts for attributes: p-chart, np-chart, c-chart, u-chart

X-bar and R charts

• X-bar chart monitors process mean over time
  • Center line: $\bar{\bar{X}} = \frac{\sum_{i=1}^{k} \bar{X}_i}{k}$, where $\bar{X}_i$ is sample mean and $k$ is number of samples
  • Upper control limit (UCL): $\bar{\bar{X}} + A_2 \bar{R}$
  • Lower control limit (LCL): $\bar{\bar{X}} - A_2 \bar{R}$
  • $A_2$ is constant based on sample size, and $\bar{R}$ is average range
• R chart monitors process variability over time
  • Center line: $\bar{R} = \frac{\sum_{i=1}^{k} R_i}{k}$, where $R_i$ is sample range
  • UCL: $D_4 \bar{R}$
  • LCL: $D_3 \bar{R}$
  • $D_3$ and $D_4$ are constants based on sample size
• Points outside control limits indicate out-of-control process
• Non-random patterns (trends, shifts, cycles) suggest process instability

Control Charts for Attributes

P-charts and c-charts

• p-chart monitors proportion of defective items in sample
  • Center line: $\bar{p} = \frac{\sum_{i=1}^{k} p_i}{k}$, where $p_i$ is proportion of defective items in sample $i$
  • UCL: $\bar{p} + 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$
  • LCL: $\bar{p} - 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$, where $n$ is sample size
• c-chart monitors number of defects per unit in sample
  • Center line: $\bar{c} = \frac{\sum_{i=1}^{k} c_i}{k}$, where $c_i$ is number of defects in sample $i$
  • UCL: $\bar{c} + 3\sqrt{\bar{c}}$
  • LCL: $\bar{c} - 3\sqrt{\bar{c}}$
• Points outside control limits indicate out-of-control process
• Non-random patterns suggest process instability

Out-of-control patterns

• Out-of-control points are points outside UCL or LCL
• Non-random patterns include:
  1. Trends: Continuous increase or decrease in points
  2. Shifts: Abrupt change in level of process
  3. Cycles: Repeating patterns over time
  4. Clustering: Points grouped close to center line or control limits
  5. Mixing: Points alternating between high and low values
• Investigating and addressing out-of-control points and patterns crucial for process improvement