Statistical Process Control and Process Capability Guide

Principles of Statistical Process Control

• Statistical Process Control (SPC) is a method utilized to ensure that a process is under control and capable.

• The primary goals of SPC include:

  • Improving process efficiency.

  • Minimizing or avoiding defects.

  • Improving productivity while simultaneously enhancing quality.

• SPC is most useful and desirable under specific conditions:

  • Measurability: The process must have quantitative, measurable outputs. If a process cannot be measured, sample statistics cannot be computed.

  • Repetitiveness: The process must be ongoing and repetitive to allow for the collection of time-ordered data.

  • Occasional Shifts: SPC adds the most value when a process has occasional drifts or shifts. If a process is already perfectly stabilized, the cost of monitoring may exceed the benefit.

Random vs. Nonrandom Variability

• Every process tends to have variation. For example, a bag of potato chips labeled as having a net weight of $9.0\,ounces$ will not contain exactly $9.000\,ounces$ in every single instance; some deviation is always anticipated.

• There are two distinct categories of variation:

• Random Variation (Common Variation):

  • This is variation caused purely by chance.

  • It is imposed by countless minor factors that are difficult to identify or isolate.

  • Even if these minor factors are eliminated, the impact on reducing overall variation is negligible.

  • It is considered unavoidable within the current process setup.

  • An example includes the inherent differences in variation between an older machine and the latest equipment.

• Nonrandom Variation (Assignable Variation):

  • This variation has specific, assignable causes.

  • It results in sizable deviations that can be identified and corrected.

  • Common categories of assignable causes include:

    • Faulty Machinery: Equipment needing repair.

    • Defective Inputs: Raw materials or components received from suppliers that do not meet quality standards.

    • Human Factors: Poorly trained employees, use of incorrect manuals, physical fatigue after long hours, or simple carelessness.

Process Control Charts and Boundaries

• A process control chart is a time-ordered plot of sample statistics obtained from an ongoing process.

• Its primary purpose is to distinguish between random and nonrandom variabilities by monitoring process output.

• Control Limits:

  • UCL (Upper Control Limit): The upper boundary of acceptable random variation.

  • LCL (Lower Control Limit): The lower boundary of acceptable random variation.

• Determining Process Status:

  • In Control (Stable): If all plotted sample statistics fall within the UCL and LCL, the process is stable and exhibits only random variation.

  • Out of Control (Unstable): If any point falls on or outside the UCL or LCL, the process is unstable. This indicates the presence of nonrandom variation due to assignable sources.

• Procedural Response to Instability:

  1. Stop the ongoing process immediately.

  2. Identify the assignable cause of variation.

  3. Take corrective action to prevent future defects.

  4. Resume the revised process and collect new data to ensure stability has been restored.

Measurement Types: Attribute vs. Variable

• Before a control chart can be created, the type of measurement must be identified:

• Attribute Measurement:

  • Refers to characteristics that can be counted rather than measured on a continuous scale.

  • Measurements usually result in discrete, categorical outcomes such as "good" or "bad," "defective" or "not defective," and "happy" or "unhappy."

  • The sample statistic used is typically a proportion ($p$) or fraction of occurrences.

• Variable Measurement:

  • Refers to characteristics measured on a continuous scale.

  • Common examples include length, weight, volume, and time.

  • The sample statistics used are the sample mean ($\bar{x}$) to measure central tendency and the range ($R$) or standard deviation ($\sigma$) to measure variability.

Constructing p-Charts for Attribute Measurements

• A p-chart is used when data consists of the fraction or proportion of defectives in a sample.

• Key Definitions:

  • Sample Size ($n$): The quantity of randomly selected observations from a process (e.g., selecting $200$ data entries per day for inspection).

  • Sample Number ($N$): The total number of groups or days data was collected (e.g., $24\,consecutive\,days$).

• Formulas for p-Charts:

  • Mean Proportion ($\bar{p}$): $\bar{p} = \frac{\text{Total Number of Defectives}}{n \times N}$

  • Standard Deviation of the Proportion ($\sigma_p$): $\sqrt{\frac{\bar{p}(1 - \bar{p})}{n}}$

  • Upper Control Limit ($UCL_p$): $\bar{p} + z \times \sqrt{\frac{\bar{p}(1 - \bar{p})}{n}}$

  • Lower Control Limit ($LCL_p$): $\bar{p} - z \times \sqrt{\frac{\bar{p}(1 - \bar{p})}{n}}$

  • Note on LCL: A proportion cannot be negative. If the calculation for $LCL_p$ results in a negative value, it must be converted to $0$.

• Confidence Levels and Z-values:

  • Most control charts use a $99.73\%$ confidence level, which corresponds to $3\,$ standard deviations ($z = 3$) according to the empirical rule of normal distribution.

Constructing x-bar and r-Charts for Variable Measurements

• When measuring variables, both an x-bar chart and an r-chart are constructed together to provide a holistic view of process stability.

• x-bar Chart: Measures changes in the central tendency (mean) of the process.

• r-Chart: Measures changes in process variability (dispersion) using the range.

• Calculating Statistics:

  • Mean of the sample means ($\bar{\bar{x}}$).

  • Mean of the sample ranges ($\bar{R}$).

• Control Limit Formulas (using Three-Sigma Factors):

  • x-bar Chart:

    • $UCL_{\bar{x}} = \bar{\bar{x}} + A_2\bar{R}$

    • $LCL_{\bar{x}} = \bar{\bar{x}} - A_2\bar{R}$

  • r-Chart:

    • $UCL_R = D_4\bar{R}$

    • $LCL_R = D_3\bar{R}$

• Factors ($A_2, D_3, D_4$): These factors are predetermined based on the sample size ($n$).

  • Example for $n = 3$: $A_2 = 1.023$, $D_3 = 0$, $D_4 = 2.574$.

  • Example for $n = 4$: $A_2 = 0.729$, $D_3 = 0$, $D_4 = 2.282$.

  • Example for $n = 10$: $A_2 = 0.308$, $D_3 = 0.223$, $D_4 = 1.777$.

Determining Overall Process Stability

• A process is considered overall in control ONLY if both the x-bar chart and the r-chart are in control.

• If either chart (or both) shows points outside the control limits, the entire process is deemed out of control.

• Points falling below an LCL are still considered out of control. While this might indicate unusually "good" performance (e.g., extremely low defect rates), the process is still unstable. Managers should investigate the cause of this "good" variation to potentially standardize those behaviors for the future.

Process Capability and Design Specifications

• Once a process is stable (in control), it must be evaluated for capability—its ability to meet design specifications.

• Design Specifications: These are set by designers to meet customer expectations. They consist of:

  • Nominal Value (Target): The goal for the product (e.g., $9.0\,ounces$ for a chip bag).

  • Tolerance: The allowable deviation above and below the target (e.g., $\pm 0.5\,ounces$).

  • Specification Limits: The result of the target and tolerance ($USL = 9.5$, $LSL = 8.5$).

• Process Width: The actual spread of the process output, typically measured as six standard deviations ($6\sigma$), accounting for $99.73\%$ of all expected output.

Process Capability Ratio ($C_p$)

• The Process Capability Ratio ($C_p$) compares the width of the design specifications to the width of the process variability.

• Formula: $C_p = \frac{\text{Specification Width}}{\text{Process Width}} = \frac{USL - LSL}{6\sigma}$

• Interpretation of $C_p$ values:

  • C_p > 1.0 (Capable): The process variability is narrower than the design specifications. This is the most desirable state, as virtually all output will meet specifications.

  • $C_p = 1.0$ (Barely Capable): The process variability exactly matches the specification width. Exactly $99.73\%$ of output meets specifications, meaning $2,700$ out of every $1,000,000$ units will be defective.

  • C_p < 1.0 (Incapable): The process variability is wider than the specifications. A significant portion of output will fail to meet design standards.

• Solutions for Incapable Processes:

  1. Redesign the Process: Aim to reduce the standard deviation ($\sigma$) through simplification, standardization, mistake-proofing (fail-safes), or automation.

  2. Use Alternative Processes: Switch to an existing process that is more precise.

  3. 100% Inspection: Inspect all products at the end of the line to ensure no non-conforming items reach the customer.

  4. Relax Specifications: Review whether the tolerance can be increased without harming customer satisfaction.

Questions & Discussion

Question: Do we sum up how many good and bad items were received? What is the point of attribute measurement? Response: Yes, through attribute measurement, we count the frequency of defective or non-defective items. We then use that frequency to find a proportion or fraction. For example, if you survey $100\,customers$ and $10$ are unhappy, the proportion is $10/100 = 10\%$. If we collect $100\,parts$ and $20$ are defective, the proportion is $20\%$. These proportions are then plotted on the control chart.

Question: I thought the LCL could not be negative. Is that correct? Response: That is correct. If the calculated LCL is a negative value, it is automatically converted to zero because you cannot have a negative number of defectives. The lowest possible point is zero.

Question: Does an out-of-control point below the LCL mean there are variables we can account for to make things more efficient? Response: Yes. If points fall below the LCL (meaning exceptionally low defects), it is a "good" reason for a process to be out of control. Managers should stop and investigate why the performance is so good to see if that behavior can be standardized across the whole process.

Question: What is the sample size in the restaurant temperature example where three dishes were picked over ten days? Response: The sample size ($n$) is $3$. It is the quantity of randomly selected observations per subgroup (dishes). The number of days ($10$) is the sample number ($N$), not the sample size.