Cpk 1.35 and You Dare to Approve? —— Confidence Lower Limit and Minimum Sample Size Table

A certain automotive parts factory submitted a new project process capability report to the original equipment manufacturer (OEM): 30 samples were drawn, and the calculated Cpk was 1.35. The report was approved based on the client's requirement of "≥1.33." Three months later, during mass production, the actual Cpk dropped to 1.15 for two consecutive months. When the client traced back, they only asked one question: "What does your sample of 30 pieces represent?"

This is not data falsification; it's a matter of interpretation. Cpk is an estimated value, and its reliability is tightly linked to the sample size: a Cpk of 1.35 calculated from 30 pieces and 300 pieces means completely different things. This article provides a "sample size—confidence lower limit" table and four on-site interpretation rules to answer a specific question: To what extent can I confidently report this batch of data?

1. Where Does the Error in Cpk Hide?

The formula for Cpk is min[(USL−x̄)/3s, (x̄−LSL)/3s]. The mean (x̄) in the numerator and the standard deviation (s) in the denominator are both estimated from the sample, and both carry sampling errors.

The error in the mean is easier to understand, but the error in the standard deviation is more subtle. The relative fluctuation of the sample standard deviation is approximately 1/√(2(n−1)): when n=30, it is about 13%, and when n=100, it is about 7%. This means that a σ estimated from 30 samples can vary by 13% in either direction, and the Cpk will also vary by 13% —— a Cpk of 1.33 could easily become 1.15 or 1.55, all within the range of sampling fluctuations.

Therefore, signing off on a single point estimate is like staking your judgment on the luck of a single sample. To determine whether "this 1.35 is reliable," you need to look at its lower limit.

Another often overlooked point is that the client's 1.33 is a "threshold," not a "target." When the measured value is only 0.02 away from the threshold, you actually have no margin to absorb sampling fluctuations, batch differences, and equipment wear —— the probability of such a "barely qualified" value failing during mass production is much higher than the security suggested by the number in the report.

2. How to Calculate the Confidence Lower Limit: One Formula, One Example

In engineering, the following formula is commonly used to estimate the 95% one-sided confidence lower limit of Cpk:

Cpk_L ≈ Cpk − 1.645 × √[ 1/(9n) + Cpk² / (2(n−1)) ]

The term inside the brackets is an approximation of the variance of the estimate, and 1.645 corresponds to the 95% confidence level (use 2.33 for 99%). The prerequisites are threefold: the process is in a state of statistical control, the data is approximately normally distributed, and the samples are independent of each other.

This approximation formula is slightly optimistic for small sample sizes, typically differing by only a few percent in capability units compared to the exact interval, which is sufficient for binary judgments like "should it be approved?" For formal reports that require client verification, it is more prudent to use the exact interval from statistical software or bootstrap methods.

Let's calculate using the example from the beginning, where n=30 and Cpk=1.35:

• 1/(9×30) = 0.0037
• 1.35² / (2×29) = 1.8225 / 58 = 0.0314
• √(0.0037+0.0314) = √0.0351 = 0.187
• 1.645 × 0.187 = 0.308
• Cpk_L = 1.35 − 0.308 ≈ 1.04

The conclusion is straightforward: this report can only prove that "the true capability is likely no less than 1.04," not "no less than 1.33." The client's threshold of 1.33 was not actually met by this data.

3. Reference Table: How Many Pieces to Sample to Prove 1.33

There are three key points to consider when reading this table:

① Reporting 1.33 with n=30 is equivalent to saying "barely usable." The lower limit is 1.03, which is far from 1.33. If the client accepts based on the confidence lower limit, this data will not hold up.

② Relying on "increasing the measured value" to compensate for the sample size shortfall is very costly. To confidently report 1.33 with n=30, you need to measure at least 1.71. For n=100, you need 1.52, and even for n=300, you need 1.43. Insufficient sample size means trading capability margin for a higher measured value.

③ 100 to 125 pieces is the cost-effectiveness turning point. Increasing the sample size from 25 to 100 pieces raises the lower limit by about 0.17. Further increasing from 100 to 300 pieces only raises it by 0.07. This is the statistical basis for the initial process study in PPAP requiring 100 pieces —— adding more samples beyond this point increases costs faster than it provides information.

Based on this, a criterion for acceptance can be established: n≥100 and the lower limit ≥1.33, then it is deemed to meet the standard; n between 30 and 100 and the lower limit ≥1.00, then it is conditionally approved, with data to be completed during the ramp-up phase of mass production; the lower limit <1.00, it does not meet the standard, and the process needs to be improved before discussing capability.

This can also be used in reverse: if the client requires "the lower limit to be no less than 1.33," and your process can only achieve a measured value of 1.45, then 1.45−1.33=0.12 can be considered the allowable error width, and the required sample size can be back-calculated —— approximately 220 pieces. Preparing a report that can be verified is much easier than explaining after the fact.

4. Four Interpretation Rules to Avoid Misusing the Table

① Include all details in the report. Do not just write "Cpk = 1.35"; instead, write "Cpk 1.35 (n=30, 95% lower limit 1.04)." The ≥1.33 in the client's manual usually refers to the point estimate, but you must know the lower limit; for new projects, safety characteristics, and one-sided specifications, it is more prudent to include the lower limit.

② Ensure the three prerequisites are met before using the table. If the control chart shows out-of-control points, the GR&R exceeds 10%, or the distribution is clearly non-normal, the table is not applicable —— fix the data first, not the numbers.

③ The sample must cover the sources of variation. Taking 30 pieces from the same shift, the same batch of materials, and the same tool will not represent the process, regardless of the sample size. Sampling should span shifts, material batches, and tool or mold cycles, and should be done at regular time intervals, not all at once.

④ Do not "pad" the sample size. To increase n from 30 to 300, mixing data from after machine adjustments and material changes will artificially inflate s, resulting in a lower Cpk; selecting only the best batches will artificially deflate s, leading to an inflated lower limit. The sample size is to prove process stability, not to make the numbers look better —— if the process changes, the sample size must be recalculated.

5. A Self-Checklist Before Submission

Before submitting the report, check each item. If any item is not met, do not report the Cpk:

• Sample size ≥30 (report only the mean and trend of variation for sample sizes below 30, not the capability index);
• Sampling spans shifts, material batches, and tool or mold cycles, not all at once;
• No out-of-control points on the control chart, and the process is in a state of statistical control;
• The GR&R of the measurement system for the characteristic is ≤10% (10% to 30% must be noted and explained);
• Confirm whether the specification is bilateral or unilateral, and use Cpu or Cpl for unilateral specifications;
• The report clearly states n, the measured value, and the 95% confidence lower limit.

Only if all six items are met will the number withstand client verification.

6. Two Inapplicable Situations

For one-sided specifications (such as positional tolerance with only an upper limit or strength with only a lower limit), use Cpu or Cpl. The formula structure remains the same, but the min function is no longer used, and the lower limit can still be estimated using this method.

For data that is clearly non-normal (such as positional tolerance, roughness, and fatigue life), this table cannot be directly applied: Cpk itself is already distorted. Perform a Box-Cox transformation and recalculate, or use bootstrap to resample 1000 times and take the 5th percentile. If neither method is feasible, the most stable approach is to return to the control chart —— evidence of capability should not rely solely on a single index.

A Cpk from a small sample is a point, not a line; reporting the lower limit together is the way to keep the judgment in your hands.

Knowledge code: 6.3.2

Version: v20261021

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