Np chart: Meaning, Criticisms & Real-World Uses

What Is Np Chart?

An Np chart is a type of control chart used in statistical process control (SPC) to monitor the number of nonconforming items in samples of a constant size over time. This tool falls under the broader category of quality control and is specifically applied to attribute data, where items are classified into one of two categories, such as "defective" or "non-defective"¹⁵⁷, ¹⁵⁸. The Np chart helps identify whether a process is stable and predictable or if unusual variations are occurring that require investigation. It tracks the actual count of nonconforming units rather than the proportion, which can be more intuitive for many users¹⁵⁵, ¹⁵⁶.

History and Origin

The concept of control charts, including the foundational principles behind charts like the Np chart, was pioneered by Walter A. Shewhart at Bell Telephone Laboratories in the 1920s¹⁵³, ¹⁵⁴. On May 16, 1924, Shewhart introduced the first control chart, laying the groundwork for modern statistical process control¹⁵¹, ¹⁵². His innovation allowed for the distinction between two types of variation in a process: common cause variation (inherent and natural to the process) and special cause variation (due to specific, identifiable factors)¹⁴⁹, ¹⁵⁰. This ability to differentiate between normal fluctuations and significant deviations revolutionized quality management and process improvement¹⁴⁸. Shewhart’s work profoundly influenced figures like W. Edwards Deming, who further popularized the use of SPC, including attribute charts, in industries worldwide.
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Key Takeaways

The Np chart monitors the actual count of nonconforming units in samples of a constant size.
*¹⁴⁵, ¹⁴⁶ It is an attribute control chart, meaning it deals with binary data (e.g., pass/fail, defective/non-defective).
*¹⁴³, ¹⁴⁴ Np charts are based on the binomial distribution, assuming each unit inspected has only two possible outcomes.
*¹⁴¹, ¹⁴² A primary purpose of the Np chart is to identify "out-of-control" signals, indicating the presence of special cause variation in a process.
*¹³⁹, ¹⁴⁰ Effective use of Np charts supports process improvement by detecting significant shifts in defect rates and enhancing overall quality assurance.

¹³⁸## Formula and Calculation

The Np chart uses the total number of nonconforming units in a sample of fixed size. The central line and control limits are derived from the binomial distribution.

The formulas for an Np chart are as follows:

Central Line (CL):

CL = n \bar{p}

Where:

(n) = constant subgroup (sample) size
(\bar{p}) = average proportion of nonconforming units (total nonconforming units / total units inspected)

Upper Control Limit (UCL):

UCL = n \bar{p} + 3 \sqrt{n \bar{p}(1 - \bar{p})}

Lower Control Limit (LCL):

LCL = n \bar{p} - 3 \sqrt{n \bar{p}(1 - \bar{p})}

If the calculated LCL is a negative value, it is conventionally set to zero, as the number of defective items cannot be less than zero. ¹³⁶, ¹³⁷These control limits define the expected range of variation for the number of nonconforming items when the process is in statistical control. The (3) in the formula refers to three standard deviations from the mean, a common practice in control charts.
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Interpreting the Np Chart

Interpreting an Np chart involves analyzing the plotted points relative to the central line and the upper and lower control limits. When all data points fall within the control limits and show a random pattern, the process performance is considered stable and in statistical control, meaning only common cause variation is present.
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Conversely, points that fall outside the control limits signal an "out-of-control" condition, indicating the presence of special cause variation. For instance, a point above the Upper Control Limit (UCL) suggests an unexpected increase in the number of nonconforming units, while a point below the Lower Control Limit (LCL) might indicate an unexpected improvement. ¹³¹, ¹³²Beyond individual points, analysts look for non-random patterns, such as consecutive points on one side of the central line or clear trends, which also suggest an out-of-control process. Identifying these signals prompts further data analysis to pinpoint and address the root causes of the variation.
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Hypothetical Example

Consider a financial institution that processes a high volume of new account applications daily. Management wants to monitor the number of applications containing errors that require re-work (nonconforming applications). They decide to use an Np chart, sampling 100 applications each day for 20 consecutive days.

Step 1: Data Collection
Over the 20 days, the total number of applications sampled is (20 \times 100 = 2,000). The total number of nonconforming applications found during this period is 80.

Step 2: Calculate Average Proportion (p-bar)
The average proportion of nonconforming applications ((\bar{p})) is:

\bar{p} = \frac{\text{Total nonconforming applications}}{\text{Total applications inspected}} = \frac{80}{2000} = 0.04

Step 3: Calculate Central Line (CL)
Since the sample size (n) is constant at 100:

CL = n \bar{p} = 100 \times 0.04 = 4

The central line for the Np chart is 4. This means, on average, 4 nonconforming applications are expected per sample of 100.

Step 4: Calculate Control Limits (UCL and LCL)

UCL = 4 + 3 \sqrt{100 \times 0.04 (1 - 0.04)} = 4 + 3 \sqrt{4 \times 0.96} = 4 + 3 \sqrt{3.84} \approx 4 + 3 \times 1.9596 \approx 4 + 5.8788 \approx 9.88

LCL = 4 - 3 \sqrt{100 \times 0.04 (1 - 0.04)} = 4 - 5.8788 \approx -1.88

Since the LCL is negative, it is set to 0.

Step 5: Plotting and Interpretation
The Np chart would show a central line at 4, an UCL at approximately 9.88, and an LCL at 0. Each day's count of nonconforming applications would be plotted. If on Day 15, for example, 12 nonconforming applications were found, this point would fall above the UCL, indicating an out-of-control signal. This would prompt the institution to investigate what caused the sudden increase in errors on that day, potentially leading to process improvement.

Practical Applications

Np charts are valuable tools across various industries for monitoring and enhancing process performance. In manufacturing, they are widely used to track the number of defective products in batches, helping companies maintain quality control and reduce waste. ¹²⁸, ¹²⁹For example, a factory producing electronic components might use an Np chart to monitor the number of defective units in hourly samples, allowing for timely intervention if defect rates rise unexpectedly.
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In the financial services sector, Np charts can be applied to monitor internal processes and ensure compliance. They can track the number of errors in tasks like processing wire transfers, handling customer emails, or managing capital movements. ¹²⁶By using an Np chart to track error rates, financial institutions can identify deviations from expected patterns, which is critical for compliance and risk management. ¹²⁵For instance, if an Np chart shows a spike in errors related to loan application processing, it can prompt a review of the verification steps or workload distribution, leading to targeted improvements. ¹²⁴Furthermore, while less common for direct financial market data like stock prices, Np charts can be adapted for compliance reporting, tracking the number of instances where certain financial ratios or operational metrics fall outside predefined regulatory thresholds, thereby supporting proactive regulatory adherence.
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Limitations and Criticisms

While Np charts are effective for monitoring attribute data with constant sample sizes, they have certain limitations. One significant constraint is their applicability only to situations where the sample size is constant for each subgroup. ¹²¹, ¹²²If sample sizes vary, a different type of control chart, such as a p-chart, is generally more appropriate.
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Another limitation is that Np charts are not well-suited for processes with very small sample sizes, as this can lead to inaccurate control limits and an unreliable representation of the process. ¹¹⁷, ¹¹⁸Similarly, if the average number of nonconforming units ((n\bar{p})) is too small (e.g., less than 5), the approximation using the normal distribution for control limits may not be valid, and the actual binomial distribution should be used.
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Np charts can also be sensitive to outliers and extreme values, which may lead to false signals for out-of-control conditions. ¹¹⁴Furthermore, they may not effectively detect gradual trends or small shifts in a process until a point falls outside the control limits. ¹¹³This means a slow degradation in process performance might go unnoticed for some time. There can also be challenges in ensuring that the assumption of independent nonconformities (where the occurrence of one nonconforming unit doesn't affect the probability of others) holds true, as violations can lead to overdispersion, causing the chart to show too many out-of-control points. ¹¹¹, ¹¹²Researchers continue to explore methods to enhance the economic and statistical design of control charts, including the Np chart, especially when process parameters are estimated rather than known, to minimize expected costs and improve detection capabilities.
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Np chart vs. p-chart

Both the Np chart and the p-chart are types of control charts used in statistical process control to monitor attribute data, specifically the occurrence of nonconforming units. ¹⁰⁸, ¹⁰⁹The primary difference lies in what they plot and when they are used.

Feature	Np Chart	p-chart
What it plots	The number of nonconforming units.	¹⁰⁶, ¹⁰⁷ The proportion or fraction of nonconforming units.
Sample size	Requires a constant sample (subgroup) size.	¹⁰², ¹⁰³ Can accommodate varying sample (subgroup) sizes.
Interpretation	Often more intuitive, as it deals with concrete counts.	⁹⁹ Involves proportions, which can be less direct for some.
Control Limits	Center line and control limits are fixed if (n) is constant.	⁹⁷ Center line is straight, but control limits vary if (n) is not constant.

While both charts are rooted in the binomial distribution for modeling binary outcomes, the choice between them hinges on the nature of data collection. If the sample size is consistent across all observations, an Np chart is simpler to use and interpret because it directly displays the counts. ⁹⁵However, if the sample size naturally varies from one inspection period to another, a p-chart is the appropriate choice to accurately monitor the proportion of nonconformities.
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FAQs

What kind of data is used with an Np chart?

An Np chart is used with attribute data, which classifies items into two distinct categories, such as "conforming" or "nonconforming," "pass" or "fail," or "defective" or "non-defective". ⁹², ⁹³It does not measure continuous variables like length or weight.

When should I use an Np chart instead of a p-chart?

You should use an Np chart when your sample (subgroup) size is constant for each data point collected. ⁹⁰, ⁹¹If the sample size varies, a p-chart is the more suitable control chart to use. ⁸⁸, ⁸⁹The Np chart is often preferred for its direct display of the count of defective items, which can be easier to understand than a proportion for many users.
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Can an Np chart detect process improvements?

Yes, an Np chart can detect process improvements. If the number of nonconforming units consistently decreases and eventually falls below the calculated Lower Control Limit (LCL), it can signal a significant, sustained improvement in the process performance. ⁸⁶Such a signal indicates a positive shift that warrants investigation to understand and standardize the changes that led to the improvement.

What does it mean if a point falls outside the control limits on an Np chart?

If a point falls outside the control limits (either above the Upper Control Limit or below the Lower Control Limit) on an Np chart, it indicates that the process is "out of control". ⁸⁴, ⁸⁵This suggests the presence of special cause variation—an unusual, non-random event or factor that is influencing the process and needs to be identified and addressed.

What happens if the sample size is too small for an Np chart?

If the sample size is too small, the control limits calculated for an Np chart may not be accurate, leading to unreliable interpretations of process performance. In⁸², ⁸³ such cases, the assumptions underlying the calculation of control limits (often relying on a normal approximation to the binomial distribution) may not hold, potentially resulting in false signals or a failure to detect real process issues.[^1⁸⁰, ⁸¹^](https://www.numberanalytics.com/blog/ultimate-guide-np-charts), ² ³, ⁴ ⁵, ⁶ ⁷[⁸](⁷⁸, ⁷⁹ https://www.numberanalytics.com/blog/practical-np-chart-manual)[⁹](https://www.6sigma.us/six-sigma-in-focus/np-chart/), ¹⁰ ¹¹, [^1⁷⁶, ⁷⁷2^](https://www.numberanalytics.com/blog/practical-np-chart-manual)[¹³](https://www.advantive.com/solutions/spc-software/quality-advisor/data-analysis-tools/np-chart/), [¹⁴](https://www.spcforexcel.com/knowledge/attribute-control-charts/np-control-charts[⁷⁴](https://www.qualitymag.com/articles/96349-a-brief-history-of-statistical-process-control), ⁷⁵/)[¹⁵](https://www.quora.com/Whats-the-difference-between-a-P-Chart-and-an-NP-Chart-I-understand-the-mathematical-differen[⁷²](https://www.leanblog.org/2024/05/celebrating-100-years-of-shewharts-control-charts-a-century-of-quality-management/), ⁷³ces-I-dont-understand-when-to-use-one-over-the-other-and-why)¹⁶ ¹⁷[^1⁷⁰, ⁷¹8^](https://www.numberanalytics.com/blog/ultimate-guide-np-charts)[¹⁹](https://www.numberanalytics.com/blog/practical-np-chart-manual)[²⁰](h⁶⁹ttps://www.numberanalytics.com/blog/practical-np-chart-manual)²¹, ²² ²³, [²⁴](https://www.a[⁶⁶](https://www.6sigma.us/six-sigma-in-focus/np-chart/), ⁶⁷dvantive.com/solutions/spc-software/quality-advisor/data-analysis-tools/np-chart/)[²⁵](https://www.6sigma.us/six-s[⁶⁴](https://www.6sigma.us/six-sigma-in-focus/np-chart/), ⁶⁵igma-in-focus/np-chart/), ²⁶ ²⁷, [²⁸](https://sixsigma[⁶²](https://www.6sigma.us/six-sigma-in-focus/np-chart/), ⁶³studyguide.com/attribute-chart-np-chart/)²⁹, ³⁰[³¹](https://pmc.ncbi.nlm.nih.gov/art[⁶⁰](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/NP_Charts.pdf), ⁶¹icles/PMC7141697/)³², ³³ ³⁴ ³⁵ ³⁶, ³⁷ ³⁸, ³⁹ ⁴⁰, ⁴¹ ⁴², ⁴³[⁴⁴](ht⁵⁷, ⁵⁸tps://blog.minitab.com/en/2-use-cases-for-control-charts-in-finance)⁴⁵ ⁴⁶ ⁴⁷ ⁴⁸ ⁴⁹, ⁵⁰ ⁵¹ ⁵², ⁵³ ⁵⁴, ⁵⁵