5.2.6 NP Chart

NP Chart Introduction An NP chart is a control chart used to monitor the number of nonconforming units in a process when each unit is classified as either conforming or nonconforming. It is part of the family of attribute control charts and is applied when the sample size can vary from one subgroup to another. An NP chart helps answer: - Is the process proportion of defective units stable over time? - Are there signals that the defect rate has changed? - Is the process capable of maintaining a targeted nonconformance level? Understanding NP charts requires clarity on attribute data, binomial assumptions, and how to construct and interpret the chart. When to Use an NP Chart Data and Measurement Conditions An NP chart is appropriate when: - Type of data: Each unit is judged as either nonconforming or conforming. - Example: a part is either defective or not; a form is either acceptable or rejected. - Metric tracked: The number of nonconforming units in each sample (subgroup). - Not the number of defects per unit, but the count of units that fail at least one criterion. - Underlying distribution: Binomial distribution assumptions are reasonable. - Sample size: - Sample size can vary over time. - Each subgroup must have a known sample size (n_i), which may differ between subgroups. If sample size is constant, an NP chart is simpler to read than a P chart. If sample size varies substantially, NP charts are still usable, but P charts (proportion nonconforming) are often preferred. However, both rely on the same binomial basis. Assumptions and Preconditions For correct use of an NP chart: - Binary classification: - Each unit is either nonconforming or conforming, with no partial credit. - Independent units: - The nonconformance status of one unit does not systematically affect another. - Constant underlying proportion (when in control): - The true process proportion of nonconforming units is stable over time in the absence of special causes. - Stable inspection and definition: - The definition of nonconforming does not change. - Inspection and measurement methods are consistent over time. Violating these conditions can lead to misleading control limits and false signals. Core Concepts Behind NP Charts Binomial Model and the NP Statistic For each subgroup (i): - (n_i): sample size in subgroup (i). - (np_i): number of nonconforming units in subgroup (i). - (pi = \dfrac{npi}{n_i}): observed proportion nonconforming in subgroup (i). Under binomial assumptions with true process proportion (p): - Expected number nonconforming in subgroup (i): (E[npi] = ni p). - Variance: (Var(npi) = ni p (1 - p)). The NP chart directly plots the statistic (npi) for each subgroup rather than the proportion (pi). Relationship to P Chart NP and P charts are closely related: - P chart plots the proportion nonconforming (p_i). - NP chart plots the number nonconforming (np_i). Key points: - Both assume a binomial process. - Both use the same central proportion (\bar{p}) as the process estimate. - NP chart control limits scale with each subgroup size (n_i). Choice considerations: - NP chart can be more intuitive when counts are small whole numbers. - P chart is often favored when subgroup sizes vary widely. Constructing an NP Chart Step 1: Collect and Organize Data Collect data in rational subgroups over time: - For each time period (i): - Record sample size (n_i). - Count nonconforming units (np_i). Organize as: - Subgroup index (e.g., day, batch, shift) - Sample size (n_i) - Number nonconforming (np_i) - Proportion nonconforming (pi = npi / n_i) (for calculation, even if not plotted) Ensure samples are representative of typical process conditions and subgrouping reflects natural production or service intervals. Step 2: Estimate the Average Proportion Nonconforming Compute the overall average proportion nonconforming: [ \bar{p} = \frac{\sum{i=1}^{k} npi}{\sum{i=1}^{k} ni} ] Where: - (k) is the number of subgroups. - (\sum np_i) is the total number of nonconforming units observed. - (\sum n_i) is the total number of units inspected. (\bar{p}) is the best estimate of the underlying process proportion, assuming the process was mostly stable during data collection. Step 3: Calculate Center Line and Control Limits For each subgroup (i): - Center line (CL): [ CLi = ni \bar{p} ] - Standard deviation of (np_i): [ \sigma{npi} = \sqrt{n_i \bar{p}(1-\bar{p})} ] - Upper control limit (UCL): [ UCLi = ni \bar{p} + 3\sqrt{n_i \bar{p}(1-\bar{p})} ] - Lower control limit (LCL): [ LCLi = ni \bar{p} - 3\sqrt{n_i \bar{p}(1-\bar{p})} ] Because the statistic is a count: - If calculated (LCLi < 0), set (LCLi = 0). - Round limits appropriately for counts (e.g., to nearest whole number) while keeping logic consistent. Step 4: Plot the NP Chart To construct the chart: - Horizontal axis: subgroup index (time order). - Vertical axis: number of nonconforming units (np_i). - Plot: - Individual points at (np_i) for each subgroup. - The center line at (ni \bar{p}) for each subgroup (may appear as a varying line if (ni) varies). - UCL and LCL for each subgroup, calculated using each (n_i). If sample sizes are constant, the center line and limits are horizontal. Interpretation of NP Charts Identifying Out-of-Control Signals The NP chart is used to detect special causes of variation. Common signals include: - Point beyond control limits: - Any subgroup count (np_i) above UCL or below LCL suggests a special cause. - Runs and trends: - Series of consecutive points on one side of the center line. - Steady upward or downward patterns in the number nonconforming. - Shifts in level: - Several consecutive points consistently higher or lower than historical performance, even if all are within control limits. - Unusual patterns: - Cycles aligning with known shifts (e.g., specific shifts, days, or suppliers). - Frequent near-limit points on one side of the chart. These patterns indicate that the underlying proportion nonconforming may have changed, or there are systematic effects entering the process. Common Interpretation Pitfalls Avoid misinterpretation by recognizing: - Stable but high defect level: - A process can be statistically in control but still produce an unacceptably high number of nonconforming units. - NP charts detect instability, not adequacy to a target or specification. - Ignoring varying sample sizes: - When (n_i) varies, control limits should be recalculated for each subgroup. - Comparing raw counts across very different subgroup sizes without limits can be misleading. - Overreacting to random variation: - Single points near but within limits are often due to common cause variation. - Use defined rules rather than intuition alone. Linking NP Chart Behavior to Process Changes When signals appear: - Compare out-of-control subgroups to in-control ones: - What changed in materials, methods, machines, environment, or people? - Consider whether: - New procedures, tools, or conditions coincided with the shift. - Inspection or classification criteria were altered. - Once a special cause is identified, remove or control it, then continue monitoring with the NP chart. Choosing Between NP and Other Attribute Charts NP vs P Chart Both are binomial-based and suitable for nonconforming unit data. - NP chart: - Plots counts (np_i). - Useful when communicating in terms of number of bad units. - More intuitive when sample sizes are equal or only mildly different. - P chart: - Plots proportion nonconforming (p_i). - Naturally normalizes for varying sample sizes. - Easier to compare performance across different sample sizes. When sample sizes vary moderately but remain reasonably large, either chart can be used; selection depends on user preference and communication needs. NP vs C and U Charts C and U charts are for defect counts, not defective units: - C chart: - Number of defects per constant area or unit (Poisson). - U chart: - Defects per unit with potentially varying area or opportunity (Poisson-based). NP chart is appropriate when: - The concern is whether each unit is acceptable or not. - Units are classified as nonconforming if they have one or more defects. - The focus is on the count of nonconforming units, not the number of defects. Practical Considerations and Limitations Minimum Sample Sizes and Approximation The NP chart relies on a normal approximation to the binomial: - The approximation is more valid when: - (ni \bar{p} \ge 5) and (ni (1-\bar{p}) \ge 5). - If the expected number of nonconforming units is very small: - Control limits may collapse toward zero. - The chart becomes less sensitive and more discrete in behavior. In such cases, alternative approaches or different subgrouping strategies may be needed to obtain actionable information. Dealing with Very High or Very Low Defect Rates - Very low defect rates: - Many subgroups may have zero nonconforming units. - Out-of-control points, when they occur, will stand out clearly but may be rare. - Very high defect rates: - Limits may approach the sample size. - While the chart detects instability, the overall performance may still be unacceptable. In both extremes, ensure that: - The attribute definition is meaningful. - The chosen subgroup size supports effective detection of changes in the process. Maintaining Chart Validity Over Time To keep the NP chart useful: - Recalculate (\bar{p}) and control limits when: - A significant, sustained process improvement or degradation is confirmed. - The process, materials, or methods change in a systematic way. - Avoid mixing fundamentally different process conditions in the same baseline unless the goal is to reflect all typical variation. Document: - How nonconforming units are defined. - Sampling strategy and frequencies. - Any changes in process or chart calculations. NP Chart in Ongoing Process Monitoring Establishing Baseline Performance Use initial data to: - Estimate (\bar{p}) and control limits. - Confirm that early points mostly fall within the limits and show no systematic patterns. - Identify and investigate any early special causes to avoid building limits on unstable data. Once a stable baseline is established, the NP chart becomes: - A monitoring tool to detect deviations from that baseline. - A visual feedback mechanism for process owners and teams. Using NP Charts to Evaluate Actions When process changes are implemented: - Continue plotting NP chart data with existing limits to see: - Immediate shifts in number of nonconforming units. - Changes in pattern (e.g., reduction in scatter or elimination of out-of-control signals). - After confirming a sustained new level of performance: - Re-estimate (\bar{p}). - Reset control limits to reflect the improved or altered process. Monitoring before and after changes helps distinguish: - Random fluctuation from real process shifts. - Short-lived improvements from sustained performance changes. Summary NP charts monitor the stability of the number of nonconforming units in a process using binomial-based control limits. They are appropriate when: - Data are counts of units classified as conforming or nonconforming. - Sample sizes are known and can vary. - The process can reasonably be modeled by a binomial distribution. To construct an NP chart: - Collect subgroup sample sizes (ni) and nonconforming counts (npi). - Estimate the average proportion nonconforming (\bar{p}). - Calculate center line and control limits for each subgroup using (n_i) and (\bar{p}). - Plot counts over time against these limits. Interpretation focuses on: - Points beyond control limits. - Runs, trends, and patterns that indicate special causes. - Distinguishing stable but poor performance from unstable processes. NP charts are closely related to P charts and complement C and U charts for attribute data. When used correctly, they provide a practical and rigorous method to detect meaningful changes in the rate of nonconforming units and to monitor process performance over time.