Process Capability (Cp & Cpk)

What do you think, Prabin?

In the context of an Annual Product Quality Review (APQR) for pharmaceutical products, there isn’t a universally mandated number of batches required for calculating Cp and Cpk values. However, industry best practices and statistical principles provide guidance:

Recommended Sample Size for Cp and Cpk Calculations

• Minimum Sample Size: While there’s no strict minimum, a sample size of at least 30 batches is often recommended to achieve a statistically significant estimation of process capability indices. This aligns with the Central Limit Theorem, which suggests that sample means approximate a normal distribution as the sample size increases.
• Larger Sample Sizes: For more reliable and confident assessments, especially in processes with higher variability, a sample size of around 100 batches can provide a more accurate estimation of Cp and Cpk values. This larger dataset helps in capturing the true variability and performance of the process over time.

Regulatory Expectations

Regulatory guidelines, such as those from the FDA and EMA, emphasize the importance of comprehensive data analysis in APQRs but do not specify exact batch numbers for capability studies. The focus is on ensuring that the process remains in a state of control and consistently produces products meeting quality specifications.

Practical Considerations

In practice, the number of batches included in the Cp and Cpk calculations should be sufficient to:

• Capture the inherent variability of the manufacturing process.
• Identify any trends or shifts in process performance.
• Support meaningful statistical analysis and decision-making.

It’s essential to ensure that the data used for these calculations comes from a stable process. If the process is not in statistical control, the Cp and Cpk values may not accurately reflect the true capability of the process.

If you’re preparing for Six Sigma certification and wish to deepen your knowledge in this area, consider our comprehensive courses:

• Pass Your Six Sigma Green Belt
• Pass Your Six Sigma Black Belt

Hi Amit,

Yes, you can absolutely calculate the process capability of manual processes. While manual processes may introduce more variability due to human factors, the fundamental statistical principles of process capability still apply. Here’s how and what to consider:

1. Feasibility of Calculating Capability in Manual Processes

Manual processes often have higher variability, but as long as you can collect measurable data over time, you can assess capability using Cp and Cpk. The process needs to be:

• Stable (in control using tools like control charts)
• Consistent (repeated measurements under the same conditions)
• Quantifiable (the output must be measurable on a continuous or attribute scale)

2. Key Considerations and Rules

1. Ensure Statistical Control: Before computing Cp or Cpk, confirm that the process is stable using control charts. If special cause variations are present, capability metrics will not be valid.
2. Collect Adequate Data: A reasonable number of data points (often at least 30) is necessary to calculate meaningful statistics.
3. Consider Operator Influence: In manual processes, variability often comes from human performance. It can be useful to stratify the data by operator to understand and minimize this effect.
4. Time and Condition Consistency: Measure the process under consistent conditions (e.g., time of day, tools used) to isolate natural variation from external influence.

3. Cp and Cpk in Manual Processes

Once data is collected and the process is deemed stable, calculate Cp and Cpk the same way you would for automated processes:

Cp = (USL - LSL) / (6 * σ)
Cpk = min[(USL - μ) / (3 * σ), (μ - LSL) / (3 * σ)]

Manual processes tend to have wider variation, so Cp and Cpk values might be lower than automated ones—but the goal remains the same: identify, measure, and reduce variability.

If you’d like structured learning with examples and practice questions, consider our certification prep courses:

• Pass Your Six Sigma Green Belt
• Pass Your Six Sigma Black Belt

Best, Ted.

Alex, what have you tried here?

To calculate the Ppk (Process Performance Index), we use the formula:

Ppk = min[(USL – μ) / (3σ), (μ – LSL) / (3σ)]

• Mean (μ) = 68
• Upper Spec Limit (USL) = 82
• Lower Spec Limit (LSL) = 54
• Spread = 24 units ⇒ Standard deviation (σ) = 24 / 6 = 4

Now, calculating both components:

• (USL – μ) / (3σ) = (82 – 68) / (3 × 4) = 14 / 12 = 1.17
• (μ – LSL) / (3σ) = (68 – 54) / (3 × 4) = 14 / 12 = 1.17

Ppk = min(1.17, 1.17) = 1.17

So, the Ppk value is 1.17.

To understand more about Ppk and how it fits into process capability analysis, check out:

• Ppk – Process Performance Index

Also consider enrolling in these courses to build a strong foundation in Six Sigma metrics and analysis:

• Pass Your Six Sigma Green Belt
• Pass Your Six Sigma Black Belt

Sabarish, can you show your calculation here?

You’re seeing a situation where all measured values are close to the nominal (around 49), yet the Cp value is low (~0.3). Let’s break down why this happens.

Given:

• Target (nominal) = 49
• Tolerance = ±1 → Spec limits: 48 to 50
• Measured values: tightly packed within 48.9 to 49.2
• Sample size = 125 (subgroup size n = 5)
• Calculated Cp ≈ 0.3

Why Cp Is Low Despite Good Values

Cp measures the potential capability based on spread (not centering) of your data relative to the allowable tolerance width.

Cp = (USL - LSL) / (6 × σ)

So if:

• Your data has very low variation (tight spread)
• But the calculated standard deviation is large (possibly due to incorrect estimation from R̄/d2 or subgroup method)

Then Cp will be low—even if all parts are near the target. This usually means the estimated σ is much larger than it should be.

Steps to Diagnose and Correct

1. Double-check the formula used to calculate σ — are you using R̄/d2 properly if working with subgroups?
2. Check if the variation within subgroups is actually low — a single outlier can skew σ
3. Verify your control chart type — are you mixing subgroup and individual analysis incorrectly?

Insight:

If the data is very tightly clustered and σ is overestimated, Cp will incorrectly appear low. A histogram and control chart can help confirm if this is a calculation artifact.

Learn More:

• SPC – Statistical Process Control
• Pass Your Six Sigma Green Belt
• Pass Your Six Sigma Black Belt

Summary: Cp is low not because your data is bad—but because the estimated standard deviation is too high. Refine your σ calculation, and you’ll likely see Cp improve to reflect the tight control you’re observing.

You’re working with individual measurements (n = 1 per subgroup), which is common in scenarios where only one measurement per unit is available. This means you’re using an Individuals and Moving Range (I-MR) approach.

Steps to Calculate Cp and Cpk for Individual Data (n = 1)

1. Gather Your Data

You already have 250 individual values. Great!

2. Calculate the Mean (μ)

μ = Average of all 250 values

3. Calculate the Moving Range (MR)

MRi = |Xi - Xi-1| for i = 2 to 250

Then find the average: MR̄ = average of 249 moving ranges

4. Estimate the Standard Deviation (σ) using:

σ = MR̄ / d2

For individual measurements, d2 ≈ 1.128 (from control chart constants)

5. Use the Cp and Cpk Formulas:

Cp = (USL - LSL) / (6 × σ)
Cpu = (USL - μ) / (3 × σ)
Cpl = (μ - LSL) / (3 × σ)
Cpk = min(Cpu, Cpl)

Tip:

You can use Excel or any statistical software to calculate the MR and perform the above steps. Plotting an I-MR chart will also help visualize process behavior.

Recommended Resources:

• SPC – Statistical Process Control
• Pass Your Six Sigma Green Belt
• Pass Your Six Sigma Black Belt

Summary: You’re using an Individuals chart method; estimate σ using Moving Range and apply standard Cp/Cpk formulas from there.

Thanks for the question!

In pharmaceutical manufacturing, Cpk = 0.97 for assay values over 10 batches suggests that your process is not quite meeting the typical industry expectations.

Generally, a Cpk ≥ 1.33 is considered acceptable for most industries, and in critical sectors like pharma, many companies aim for Cpk ≥ 1.67 to ensure consistent product quality and compliance. A Cpk of 0.97 indicates the process is not capable of consistently staying within specifications.

Impact of Sample Size on Cpk

The number of batches (sample size) plays a significant role in the reliability of the Cpk estimate:

• With only 10 batches, your estimate of standard deviation and mean may not be stable, making Cpk calculations less reliable.
• Small sample sizes can either inflate or deflate Cpk values depending on the natural variability in the data.
• As sample size increases, the estimate becomes more representative of the true process capability.

So, while your current Cpk value indicates a need for process improvement, you should consider collecting more data to get a more accurate picture. Also, run control charts to check if the process is stable before relying heavily on the Cpk value.

To master these critical quality concepts, consider enrolling in one of our courses:

• Pass Your Six Sigma Green Belt
• Pass Your Six Sigma Black Belt

This is a good question. I’ve added this to my list of improvements. Stay tuned…

Hi Dany,

I only have the capacity to answer these kind of questions in the paid forum. We’d love to have you join!

Best, Ted.

Thanks for the question! I’ve added this as a question in the Pass Your Six Sigma Green Belt question set. Sign up or log in for access.

Cpk is typically used for continuous (variable) data; however, for attribute data—like pass/fail, defects, or non-conformities—you can still estimate a sigma level and then derive a Cpk-like interpretation. Here’s how it works in a manufacturing context:

Example Scenario (Attribute Data)

Suppose you manufacture 10,000 units and 80 are defective.

Step 1: Calculate the Yield

Yield = (Total Units - Defectives) / Total Units
      = (10,000 - 80) / 10,000 = 0.992 or 99.2%

Step 2: Convert Yield to Z-Score (Short-Term Sigma Level)

Z = NORMSINV(0.992) ≈ 2.41

This is your short-term Z value. If you want the long-term sigma level (as per Six Sigma methodology), subtract 1.5 sigma:

Zlong-term = 2.41 - 1.5 = 0.91

Step 3: Estimate Cpk (Optional and Approximate)

While Cpk isn’t technically defined for attribute data, you can approximate it from Z:

Cpk ≈ Z / 3
Cpk ≈ 2.41 / 3 ≈ 0.803 (short-term)

Interpretation

A Cpk of ~0.8 indicates a process that is not consistently capable of meeting quality requirements. This metric gives you an estimate of process performance even though the data type is attribute.

To master these techniques for certification or career growth, check out our in-depth training programs:

• Pass Your Six Sigma Green Belt
• Pass Your Six Sigma Black Belt

Glad it helped, Krishna!

Yes, a wide range of specification limits does impact the Ppk value, and here’s why:

Understanding the Ppk Formula

Ppk = min[(USL - Mean) / (3σ), (Mean - LSL) / (3σ)]

Where:

• USL = Upper Specification Limit
• LSL = Lower Specification Limit
• Mean = Process Mean
• σ = Standard Deviation of the process

Impact of a Wider Specification Range

• If you widen the USL and LSL while keeping the process mean and variation (σ) the same, both numerators in the Ppk formula increase.
• This means the Ppk value will increase, making the process look more capable even though the actual variation and centering of the process haven’t changed.

Why This Matters

A higher Ppk indicates that the process output is well within the specification limits, so wider specs make the process appear more capable. However, this can be misleading if the spec limits are too loose relative to customer or functional needs.

Summary

• Wider specification limits = higher Ppk (if process variation stays constant)
• Narrower specification limits = lower Ppk

Further Reading

• Pass Your Six Sigma Green Belt
• Pass Your Six Sigma Black Belt

Hey, Mark,

This is a very insightful and well-posed question, and you’re absolutely correct to question whether traditional Cpk analysis is a good fit for your discrete batch compounding process.

What Cpk Assumes

Cpk is intended for continuous, stable processes—typically high-volume, repeatable manufacturing. It assumes:

• A steady-state process with inherent variation
• Normal distribution of the data over time
• Large enough data sets to reflect consistent process behavior

Your compounding process is:

• Discrete and batch-based, with frequent adjustments based on test results
• Not truly continuous; every batch may have slightly different input conditions
• Tested and corrected until in-spec, so your process never allows drift beyond limits
• Runs infrequently (4–5 times/year), making longitudinal studies difficult

Making the Case

Yes, you can make a strong case that traditional Cpk analysis does not apply. Instead, consider proposing:

1. Capability Reporting by Batch

Report the mean and range of each property for each lot, along with the specification window.

2. Performance Summary Over Time

Provide a summary of all batches over a 12- or 24-month period, highlighting that 100% have conformed to specification limits.

3. Alternative Metrics

• Process Yield (e.g., % of batches first-pass conforming)
• Range charts or boxplots to illustrate within-lot consistency and between-lot spread

4. Control Plan Assurance

Explain your control plan: material verification, in-process checks, and recipe adjustments to guarantee spec conformity.

Next Steps

It’s very reasonable to educate your customer that Cpk isn’t meaningful in your context. But you’re still providing a highly controlled, compliant product! If needed, invite a technical discussion or QA review to align on alternative quality metrics.

This approach shows your commitment to quality and process control, even if the classical Cpk doesn’t apply.

Hi George,

I’m not familiar with that symbol in that context. Can you share an example?

Hi Sathesh,

Thank you for the question.

See the article here.

Best, Ted.

Great, Franz – how can I best help?

Excellent question! To understand the process’s actual capability, we need to interpret what Cp = 1 and Cpk = 0.78 imply:

What Cp = 1 Means

Cp measures the potential capability of the process assuming it is perfectly centered between the specification limits. A Cp of 1 means that the process spread (6σ) exactly matches the specification width. This indicates that in theory, the process is capable of meeting specifications—but only if it is perfectly centered.

What Cpk = 0.78 Means

Cpk accounts for how centered the process is. A Cpk of 0.78 indicates that the process is not centered within the specification limits, and the actual performance is worse than its potential. It means the process mean is closer to one of the specification limits, increasing the likelihood of defects.

Actual Process Capability

In practical terms, the actual capability of the process is described by Cpk, not Cp. Therefore, the actual process capability is 0.78.

This suggests that:

• The process is producing more variation relative to one of the spec limits.
• There’s a higher probability of producing out-of-spec products than if the process were centered (Cpk = Cp).
• To improve, you may need to re-center the process or reduce variation.

Additionally, if you’re interested in preparing for your certification, consider our comprehensive courses:

• Pass Your Six Sigma Green Belt
• Pass Your Six Sigma Black Belt

Well, let’s look at the math.

Cpl = (Process Mean – LSL)/(3*Standard Deviation)
Cpu = (USL – Process Mean)/(3*Standard Deviation)

Cpk is merely the smallest value of the Cpl or Cpu denoted: Cpk= Min (Cpl, Cpu)

Can you construct a scenario where you would get a Cpk in that range?

Excellent question. You’re exploring a nuanced topic that intersects statistical theory with real-world capability analysis. Here’s a breakdown of your approach and some deeper context.

Your Approach (Truncating the Distribution)

1. Identify the 10th and 90th percentiles from the normal distribution
2. Use their corresponding X values to define a truncated range
3. Assume this new range represents ±3σ → derive new σ
4. Use the new σ to recalculate Cp and Cpk with the same mean

This is conceptually valid as an approximation, and it’s a reasonable exploratory method. But it’s important to understand that:

• Once you truncate the data, it is no longer normally distributed
• The new distribution is a truncated normal distribution, which has different mean and variance

Mathematical Formulation

The variance of a truncated normal distribution is not simply based on the new range. It is derived from:

σ²_truncated = σ² × [1 + (αϕ(α) - βϕ(β)) / (Φ(β) - Φ(α)) - ((ϕ(α) - ϕ(β)) / (Φ(β) - Φ(α)))²]

Where:

• α = (a – μ)/σ (lower bound standardized)
• β = (b – μ)/σ (upper bound standardized)
• ϕ = standard normal PDF
• Φ = standard normal CDF

So to do it rigorously, you should recalculate the mean and variance of the truncated distribution using this formulation, and then use that to compute Cp and Cpk.

Alternative Approaches

• Monte Carlo Simulation: Generate data, truncate it, and empirically calculate new std dev and Cp/Cpk
• Use statistical libraries: Python (SciPy), R (truncnorm package) for precise truncated normal modeling

Your manual approach is valid as an exploratory method, but to make it analytically accurate, you should use the truncated normal distribution’s actual variance and mean adjustments.

Well, in what cases do you need multiple batches? What kind of data are you processing?

When preparing an Annual Product Quality Review (APQR)Recommended Minimum Number of Batches

• Minimum: 25–30 batches. This aligns with standard Six Sigma practice for reliable statistical analysis
• Ideal: All batches produced in the review period. (Thinking usually over 12 months, as APQR aims to assess overall process performance for the year.)

I’d aim for all eligible batches from the year in your APQR; at the very least, use 30 batches to ensure meaningful Cp/Cpk analysis.

Thank you for your question! Let’s analyze the values you’ve provided:

• Cp = 4.395
• Cpk = 1.688
• USL = 42.4
• LSL = 42
• 6σ = 0.091

First, verify Cp:

Cp is calculated as (USL - LSL) / 6σ. Plugging in your numbers:

(42.4 - 42) / 0.091 = 0.4 / 0.091 ≈ 4.395, which matches your Cp. So this value is correct.

Next, evaluate Cpk:

Cpk is the minimum of (USL - μ) / 3σ and (μ - LSL) / 3σ. Since the process is centered between a very tight spec range (only 0.4 units wide), a Cpk of 1.688 suggests the process mean is not exactly in the center but is slightly closer to one limit than the other. While the Cp shows the process potential capability, Cpk shows the actual performance considering centering. Both can be accurate together if the process is off-center.

These values are internally consistent and possible given a tightly controlled process with a very narrow tolerance band. Such high Cp values are rare and often found in extremely high-precision processes like semiconductor manufacturing.

Larger is better. The larger Cpk is, the less likely any item will be outside the specification limits.

Sounds like you should take action to look for common cause variation, address that, and then re-check the capability to see if you’ve increased the stability of your process.

Thanks for your question! If you’ve only been provided with the minimum and maximum values of a dataset, unfortunately, you can’t accurately calculate Cpk. The Cpk (Process Capability Index) relies on both the process mean and the process standard deviation (σ), which require more detailed data than just the range.

While the range (max – min) gives a rough sense of data spread, it doesn’t reflect how data is distributed within that range, especially if there are outliers. To calculate Cpk properly, you need subgroup data or at least a reliable estimate of the process mean and standard deviation. That way, you can apply the formulas:

Cpk = min[(USL – μ) / (3σ), (μ – LSL) / (3σ)]

If you’re working from limited data and still want to assess capability, you might consider other approaches like estimating σ from a range using X-bar and R charts if subgroup data is availabl, but this still requires more than just two values.

Let me know if you’d like help finding alternative ways to estimate capability based on the data you have!