Why Does One Batch Trigger an Alarm While the Other Doesn’t, Despite Both Having 3 Nonconforming Pieces? — A Five-Step Method for Selecting Counting Control Charts (p/np/c/u)

When it comes to SPC control charts, most workshops immediately think of the X̄-R chart: measuring dimensions, torque, and calculating ranges, a process they are very familiar with. However, the majority of data on the production floor is not measurement data but counting data, such as "conforming/nonconforming" or "how many defects." This type of data also needs to be monitored, and there are specific control charts for it: p, np, c, and u charts.

The issue is that these four charts look similar but have very different uses. If the wrong chart is selected, the chart will still be drawn, but the alarms will not reflect whether the process has changed; instead, they will depend on the size of the denominator.

1. Counting Data Actually Has Two Types of "Counts"

Clarifying the data itself is half the battle when selecting a chart.

• Number of Nonconforming Products (pieces): A product is either conforming or nonconforming, and the unit of measurement is "pieces," with the denominator being the number of inspected pieces. For example, 6 pieces out of 200 are nonconforming.
• Number of Defects (defects): A product can have multiple defects, and the unit of measurement is "defects," with the denominator being the inspection unit (pieces, meters, square meters). For example, a board has 3 defective solder points.

Both types of data appear to be "counted numbers," but they follow different distribution models and thus require different charts. Treating data with multiple defects per piece as "conforming/nonconforming" is equivalent to discarding the most information-rich part.

2. How to Select the Four Types of Charts

Selecting the chart requires answering two questions: Are you counting "pieces" or "defects"? Is the subgroup size fixed? Answer these two questions, and the chart is determined: pieces + fixed = np, pieces + variable = p, defects + fixed = c, defects + variable = u.

Since p charts and u charts can cover variable subgroup sizes, is it more convenient to use them in all scenarios? Not really. np charts and c charts report "number of pieces" and "number of defects," which are easier for teams to understand and react to quickly. More importantly, when subgroup sizes vary, each point on p and u charts has different control limits, creating a stepped boundary that makes judgment more difficult. Therefore, the principle on the production floor is: keep subgroup sizes fixed if possible, and only use p or u charts when the sampling quantity is naturally determined by production volume, orders, or inspection areas and cannot be fixed.

3. How to Calculate Control Limits and Two Important Thresholds

The center line for all four charts is the "process average," and the control limits are 3 standard deviations above and below the center line. The form of the standard deviation differs for each chart:

• p Chart: Center line p̄ = total number of nonconforming products ÷ total number of inspections; control limits = p̄ ± 3 × √[p̄(1−p̄)/n]. Calculate for each subgroup if the subgroup sizes are unequal, and set the lower limit to 0 if it is negative.
• np Chart: np̄ ± 3 × √[np̄(1−p̄)], used when subgroup sizes are fixed, and the result is directly the "number of pieces."
• c Chart: c̄ ± 3 × √c̄.
• u Chart: ū ± 3 × √(ū/n), where n is the ratio of the current inspection unit to the baseline unit.

Two often-overlooked thresholds:

The subgroup must be large enough. Generally, the expected number of nonconforming products np̄ should be ≥ 5, meaning p̄ should be at least 1% to 2% and n should be no less than 50. If the subgroup is too small, the control limits will be wide like a door, and the lower limit will be 0, making the chart a constant alarm.

Counting charts cannot be used to calculate Cpk. They record whether a product is conforming or nonconforming, not measurement values, and can only answer whether the process is stable, not whether it has sufficient process capability. For capability evaluation, return to measurement data.

4. An Example

A certain assembly line inspects 200 pieces per shift for appearance, continuously recording 20 shifts, totaling 12,000 inspections, with 360 nonconforming pieces, p̄ = 3.0%.

Upper control limit = 3.0% + 3 × √(3% × 97% ÷ 200) = 3.0% + 3.6% = 6.6%, lower limit is negative, set to 0.

• Shift 6: 6 nonconforming pieces (3.0%), stable;
• Shift 11: 9 nonconforming pieces (4.5%), which appears to be 50% higher than normal, but it is still within 6.6%, so it is a normal fluctuation and parameters do not need to be adjusted;
• Shift 17: 14 nonconforming pieces (7.0%), exceeding the upper control limit, triggering an alarm — at this point, check if the material or mold has been changed, or if a new employee has started operating the machine.

If Shift 11 had been treated as an anomaly without looking at the chart, the result would often be to misadjust a process that was already under control. The value of control charts lies in distinguishing between "scary-looking" and "truly out of control."

5. Five Steps to Implementation

1. Define the Scope: First, determine whether you are counting "pieces" or "defects," and the inspection unit (pieces, meters, or square meters). Record multiple defects per piece as the number of defects.
2. Define Subgroups and Frequency: Fixed sampling quantities facilitate the use of np charts; if they are unequal, use p charts and calculate limits for each subgroup. The subgroup size must satisfy np̄ ≥ 5.
3. Select the Chart: Determine p/np/c/u based on the answers to the above two questions, and do not force the use of X̄-R charts for conformity rates.
4. Establish and Regularly Review Limits: Calculate the center line using data from a stable period; recalculate after changes in process, product, or inspection standards. Control limits are not set once and used for three years.
5. Identify Anomalies and Respond: The same criteria for identifying anomalies (such as consecutive points, trends, and chains) apply. When subgroup sizes are unequal, each point has its own control limits, and judgments are based on these limits. Follow the response plan in the control plan after an alarm and document the actions taken.

6. Three Common Misuses

• Using Targets as Control Limits: Directly drawing a control line for "nonconforming rate not exceeding 1%" results in either constant alarms or no alarms at all, masking real variations.
• Subgroup Too Small: Inspecting only 20 pieces per shift, with p̄ = 3%, the upper control limit is close to 16%, making the chart useless. Either increase the sample size or extend the statistical period to use cumulative data.
• Multiple Defects per Piece but Using p Chart: Only recording "this piece is nonconforming" and discarding information about 5 defects on a single piece means that improvements will not be reflected in the chart. In this case, use a c chart or u chart.

Counting data is the most cost-effective and dense data source on the production floor, but it is also the most wasted. By clearly distinguishing between "pieces" and "defects," matching subgroup sizes with the appropriate chart type, and correctly calculating and regularly reviewing control limits, these four charts can truly help you monitor the process.

Counting "pieces" or "defects," and whether the subgroup size is fixed — two questions to determine the chart, don't let the denominator decide the alarm.

Knowledge code: 6.3.1

Version: v20261002

Author: QTank QTank is dedicated to providing systematic professional knowledge, methodologies, and practical tools for quality management practitioners, helping companies continuously improve their quality capabilities.