Cpk 1.33 Explained: Cp vs Cpk, Sample Size, Japanese Specs
A Japanese customer's drawing or quality agreement says "Cpk ≥ 1.33" (in Japanese documents, often written 工程能力指数 1.33以上, kotei noryoku shisu, process capability index). Suppose your team runs the numbers and gets 1.31 in one spreadsheet but 1.35 in your SPC software (a hypothetical case). Then the customer's quality assurance team asks how many parts you measured, which standard deviation you used, and whether the process was stable. Suppliers in Thailand, India, or the US who ship to Japanese automakers and Tier 1s can expect questions like these.
The index itself is simple. Cpk compares the distance from the process mean to the nearer specification limit with three standard deviations of the process. But the same process can produce different Cpk values depending on how you estimate the standard deviation and how much data you use. And 1.33 is not a universal acceptance threshold: its meaning depends on which standard, manual, or customer requirement you are working under.
This guide covers the Cp and Cpk formulas, a worked example with fictional data, what 1.33 means in parts per million, where the 1.33 and 1.67 criteria actually come from (JIS, AIAG PPAP, customer-specific requirements), Cpk vs. Ppk, sample size and confidence intervals, and the checks to run before you calculate. It ends with a reporting checklist, questions to ask the Japanese customer, and a sample inquiry email. All example data in this article is fictional, and every calculated value has been rechecked.
- A process capability index compares the width of the specification with the spread of the process. Cp = (USL − LSL) ÷ 6σ; Cpk = distance from the mean to the nearer limit ÷ 3σ
- Cp looks only at spread. Cpk also reflects centering, so it drops as the mean moves away from the center of the tolerance. With a one-sided specification, use only the one-sided Cpk formula
- Cpk 1.33 means the nearer limit is 4σ from the mean. For a stable, normally distributed process, that works out to roughly 3 to 6 parts per 100,000 out of specification
- 1.33 and 1.67 are not fixed thresholds set by one standard; the drawing or quality agreement decides. Under AIAG PPAP (4th edition), an initial study result from 1.33 to 1.67 needs the customer's review
- Sample size: the NIST handbook says capability studies generally need at least 100 values; the AIAG PPAP manual asks for at least 25 subgroups and 100 readings. A Cpk of 1.33 from 30 parts has a 95% confidence interval of 0.97 to 1.69
Contents
- What Are Cp and Cpk? The Formulas First
- Worked Example: Shaft OD 10.000 ± 0.020 mm (Fictional Data)
- What Cpk 1.33 Means: Index vs. Out-of-Spec Rate
- Where Do 1.33 and 1.67 Come From? JIS, Textbooks, AIAG PPAP
- Cpk vs. Ppk: Which Standard Deviation You Used
- How Many Samples Do You Need? Sample Size and Cpk Confidence Intervals
- Three Checks Before You Calculate
- Before You Submit: Checklist and Questions for the Japanese Customer
- Limits of This Guide
- FAQ
- Sources
What Are Cp and Cpk? The Formulas First
A process capability index tells you how small the process spread is compared with the specification width. You need only three formulas. USL is the upper specification limit, LSL the lower specification limit, and σ (sigma) the process standard deviation.
Cp divides by 6σ. Cpk and the one-sided index divide by 3σ.
In Japan, these terms are defined in the statistics vocabulary standard JIS Z 8101-2. It treats process capability as an estimate for a process that has been confirmed to be in a state of statistical control (stable). Cp is the index tied to the tolerance width, and Cpk is the smaller of the upper and lower capability indices.
The specification width comes from the tolerance on the drawing. If the drawing relies on a general tolerance note rather than a stated tolerance, see our guide to JIS B 0405 general tolerances to work out the actual limits.
Related: Datum Selection in GD&T: JIS vs ASME on Japanese Drawings
Cp Measures Spread; Cpk Also Measures Centering
Cp depends only on the ratio of tolerance width to spread, so it stays the same even when the mean drifts off center. Cpk is calculated from the distance between the mean and the nearer limit, so it drops as the mean moves off center.
The spread stays the same; only the mean moves by 1σ.
Schematic of a process whose tolerance width is 10σ (Cp = 1.67). The curves represent a normal distribution.
You can link the two with the off-center ratio k: Cpk = Cp × (1 − k). Here k is the distance between the tolerance center and the mean, divided by half the tolerance width. When the mean sits on the center, k is 0 and Cp equals Cpk.
In practice, read the two together. If Cp is high but Cpk is low, adjusting the process aim (target value) can close much of the gap. If Cp itself is low, moving the mean will not get you there; you have to reduce the sources of variation, such as equipment, fixtures, materials, or process conditions.
One-Sided Specifications: Use Only One Side
Some characteristics have only an upper limit, such as surface roughness or runout. Others have only a lower limit, such as tensile strength or plating thickness. With no tolerance width, Cp cannot be calculated. Use only the formula for the side that has a limit and report that value.
Characteristics where zero is best, such as runout or flatness, tend not to be symmetric; the distribution usually has a long tail on one side. That breaks the normal-distribution assumption discussed later, so check the shape of the distribution before you apply the formula.
Worked Example: Shaft OD 10.000 ± 0.020 mm (Fictional Data)
Take a shaft outer diameter of 10.000 ± 0.020 mm (LSL 9.980 mm, USL 10.020 mm). Suppose you pulled 5 consecutive parts as one subgroup, 25 times at intervals, for 125 parts in total, and measured each to 0.001 mm. The numbers are made up for illustration and do not come from a real process.
A spreadsheet is enough. Decide how you will collect the data before you start.
The mean of the 125 parts is 10.0043 mm and the standard deviation (STDEV.S) is 0.00408 mm. The smallest reading is 9.993 mm and the largest is 10.014 mm, so nothing is out of specification. From these two figures, the indices work out as follows.
Cp is 1.63, but Cpk is only 1.28.
Fictional example. These values use the standard deviation of all the data. Values based on within-subgroup variation appear later, under Cpk vs. Ppk.
Cp is 1.63 but Cpk is 1.28 because the mean sits 0.0043 mm above the tolerance center. The off-center ratio k is 0.0043 ÷ 0.020 = 0.215, and 1.63 × (1 − 0.215) = 1.28, which matches the relationship above.
If the mean could be brought back to the tolerance center, Cpk would rise to 1.63. So in this case, the first step is to correct the aim, before working on reducing variation.
Use STDEV.S, which divides by n − 1, because you are estimating the whole process from a sample.
What Cpk 1.33 Means: Index vs. Out-of-Spec Rate
A value of 1.33 means the nearer specification limit is four standard deviations (4σ) from the mean, since 4 ÷ 3 = 1.33. In the same way, 1.00 corresponds to 3σ, 1.67 to 5σ, and 2.00 to 6σ.
If you assume the data follow a normal distribution, you can convert the index into a probability of falling outside the limits. The table below uses the upper-tail probabilities of the standard normal distribution.
Each step from 1.00 to 1.33 to 1.67 cuts the out-of-spec rate by one to two orders of magnitude.
ppm = parts per million. Assumes a stable, normally distributed process. "Both sides" applies when the mean is centered and Cp equals Cpk; "one side" counts only the nearer limit. 0.67, 1.33, and 1.67 are calculated as 2/3, 4/3, and 5/3 (using exactly 1.33 gives about 33 ppm one-sided and 66 ppm two-sided).
Roughly, an index of 1.00 means 1 to 3 parts per 1,000 out of specification; 1.33 means 3 to 6 per 100,000; and 1.67 means 3 to 6 per 10 million.
The NIST/SEMATECH e-Handbook of Statistical Methods gives nearly the same figures: 0.27% rejects at Cp = 1.00 and 64 ppm at Cp = 1.33. The table also shows how much centering matters. If a process with Cp = 1.33 shifts by 1σ, Cpk falls to 1.00 and the out-of-spec rate goes from about 63 ppm to about 1,350 ppm, more than 20 times higher.
These are calculated values that hold only when the assumptions hold. They do not guarantee an actual defect rate. Tail probabilities at the parts-per-million level are especially sensitive: a small departure from the normal shape changes them a lot.
Where Do 1.33 and 1.67 Come From? JIS, Textbooks, AIAG PPAP
People often say "1.33 or higher passes," but the number has more than one source. These sources fall into four groups.
The same 1.33 and 1.67 mean different things in different documents.
Which one applies is set by the drawing, specification, or quality agreement.
JIS Z 9020-2 and the Japanese Five-Level Scale
The Japanese standard that addresses the number directly is JIS Z 9020-2, the Shewhart control chart standard. Its 2016 edition is an identical (IDT) adoption of ISO 7870-2:2013, so the same guidance should appear in the ISO standard (we checked only the JIS text). The 2016 edition treats Cp = 1.33 as the minimum generally acceptable value, because sampling variation is always present and almost no process is perfectly stable.
The same clause says the minimum requirement is for the supplier and customer to agree on. The standard was revised in 2023 and 2025; the current edition is JIS Z 9020-2:2025. What we describe here comes from the 2016 edition, whose text we could check, so confirm the relevant clause in the current edition before you quote it.
Japanese quality textbooks and company guides widely use a five-level scale (Figure 6). The wording differs slightly between sources, and the underlying standard or reference is unclear. The vocabulary standard JIS Z 8101-2:2015 contains only definitions and formulas, not the scale. Treat it as a customary rule of thumb in Japan, not as a requirement. You may still see it in a Japanese customer's training materials or internal standards, so ask which scale their documents use.
IATF 16949, AIAG PPAP, and Customer Requirements
In the automotive industry, IATF 16949 clause 9.1.1.1 covers process capability. It requires suppliers to maintain the process capability (Cpk) or process performance (Ppk) results specified in the customer's part approval process.
The AIAG Production Part Approval Process (PPAP) manual, 4th edition (2006), sets the acceptance criteria for initial process studies. Note that a result between 1.33 and 1.67 is not a pass; it is a range where you must contact the customer to review the result. Below 1.33, the process does not meet the acceptance criteria.
Customer-specific requirements add another layer. Ford's published requirements (for use with the AIAG PPAP 4th edition) set Ppk ≥ 1.67 as the acceptance level for both initial and final process capability. The supplier PPAP manual of Enerpac, a hydraulic tool maker, separates criteria: 1.33 or higher for significant characteristics and 1.67 or higher for safety-related ones.
Douglas Montgomery's widely used textbook, Introduction to Statistical Quality Control, also recommends minimum values for two-sided specifications: 1.33 for existing processes, 1.50 for new processes, and 1.67 for new processes with safety or critical characteristics.
On the SPC side, AIAG and the German VDA QMC released a joint AIAG & VDA SPC Manual on July 1, 2026. Its published summaries mention harmonized minimum capability thresholds, but we could not confirm the specific values. If your customer cites this manual, ask them which thresholds and index types apply.
The numbers look alike, but who set them, and why, differs. Neither 1.33 nor 1.67 is an absolute acceptance threshold; both come from the contract and the drawing. Which characteristics need a higher index depends on the severity rated in the FMEA and on the special characteristics the customer designates.
Related: AIAG-VDA FMEA for Japanese Customers: DFMEA vs PFMEA
Cpk vs. Ppk: Which Standard Deviation You Used
Automotive forms list Ppk next to Cpk. The formulas have the same shape; the difference is the standard deviation in the denominator.
The difference is whether you divide by within-subgroup variation or by overall variation.
The within-subgroup vs. overall split is the terminology used in the AIAG PPAP and SPC manuals and in statistical software. JIS Z 8101-2 separates the two by whether the process has been confirmed stable, and it also allows the standard deviation of all data to be used for the process capability index.
Now calculate the within-subgroup variation for the same fictional data. The average range (maximum − minimum) of the 25 subgroups is 0.00916 mm. Dividing by d2 = 2.326 (the constant for a subgroup size of 5) gives 0.003938 mm (about 0.00394 mm).
Using this value and the unrounded mean of 10.004344 mm gives Cp = 1.69 and Cpk = 1.325. The 1.63 and 1.28 in Figure 4 used the standard deviation of all data, so in AIAG terms they are Pp and Ppk.
The same data therefore produce two different numbers, 1.325 and 1.28, depending on how you estimate the standard deviation. Rounding adds another trap: 1.325 displays as 1.33 at two decimal places, but the unrounded value falls just short of 1.33, so you cannot claim it meets a "1.33 or higher" requirement.
The choice of standard deviation and the rounding alone can change how a result looks against the criterion. Judge borderline values on the unrounded number, and report a few extra decimal places to avoid misunderstanding.
This matters with Japanese customers in particular. Many Japanese introductory textbooks calculate "Cp" from the standard deviation of all data. JIS Z 8101-2 lists both estimates, from all data and from within-subgroup variation. So the question is not which one is correct but which one you used, and you should state it on the report. If the customer's template just says "Cpk," confirm which standard deviation they expect.
A note in the same standard says that for a stable process the two estimates converge, so comparing them shows how stable the process is. A large gap between Cpk and Ppk signals that the mean is moving between subgroups. Use the control chart to look for causes such as changeovers, material lots, or operator shifts.
How Many Samples Do You Need? Sample Size and Cpk Confidence Intervals
The mean and standard deviation in the formula are estimates from a sample. That makes Cpk an estimate too: take new data and you get a different value. The less data you have, the more it can swing.
The NIST handbook gives an approximate confidence interval for Cpk that is commonly used in practice: Cpk ± z × √(1 ÷ (9n) + Cpk² ÷ (2(n − 1))). For a 95% confidence interval, z = 1.96. The figure below applies it to an estimate of 1.33 at different sample sizes.
With 30 parts, you cannot rule out a true value below 1.00.
Calculated with the approximation the NIST handbook describes as commonly used in practice. Assumes a stable, normally distributed process; the approximation applies when the sample size is 25 or more. n is the total number of values.
A result of 1.33 from 30 values has a 95% confidence interval of 0.97 to 1.69. With 100 values the interval is 1.13 to 1.53, and even with 300 it is still 1.22 to 1.44. For the fictional Ppk of 1.28 (125 values), it is 1.11 to 1.45.
Turned around: to say the lower end of the interval is at least 1.33, the estimate needs to be about 1.81 with 30 values, about 1.67 with 50, and about 1.53 even with 125. This helps explain why customers ask for a higher index when only limited pre-production data is available.
So how many values are enough? These are the figures with clear conditions and sources:
- NIST handbook: about 50 independent values for the estimates to be valid, and generally at least 100 for capability studies
- AIAG PPAP manual (4th edition): for initial process studies of characteristics that can be tracked on control charts, at least 25 subgroups and at least 100 readings from consecutive parts of a production-representative run
- Ford customer-specific requirements: at least 25 subgroups with a subgroup size of at least 5, for at least 125 measurements; until capability can be shown from a 125-piece sample, 100% inspection or error-proofing is required
- JIS Z 9020-2 (2016 edition): confirm stability over at least 25 subgroups before starting the capability assessment
All of these come with conditions, and none of them sets a single number that is always enough. Equally important is what range of variation the data cover. One hundred parts made back to back in one hour do not include variation from material lots, changeovers, or shift changes.
Some characteristics, such as those checked by destructive testing, cannot reach the required count. In that case, agree on the measurement and evaluation method with the customer first, and report the index with its confidence interval.
Three Checks Before You Calculate
The Cp and Cpk formulas are simple, but the result means something only under certain conditions. Check three things in order before you calculate: measurement variation, process stability, and the shape of the distribution. If any of them fails, the index does not reflect what the process can really do.
1. Measurement Variation Is Small Enough
Observed variation mixes the variation of the parts with the variation of the measurement. In terms of variance, observed σ² = process σ² + measurement σ². A coarse measurement system makes the process look less capable than it is.
From this relationship, if measurement variation is 30% of the observed total (as a ratio of standard deviations), Cp comes out about 5% lower than the true value. That corresponds to a Gage R&R of 30% of total variation: an observed 1.33 means the process itself is about 1.39. At 10%, the difference is about 0.5%.
If Gage R&R is expressed as a percentage of tolerance, the math changes. If the measurement spread (6σ) is 30% of the tolerance, an observed 1.33 means the process itself is about 1.45, so the observed value is about 8% low. This effect grows as the index gets higher, so check which ratio the Gage R&R result uses before you rely on it.
The AIAG Measurement Systems Analysis (MSA) manual gives the usual guidelines: under 10% is generally acceptable, 10% to 30% may be acceptable depending on the application, and over 30% is unacceptable. For the study procedure and a worked example, see our guide to Gage R&R acceptance criteria. The accuracy of the instrument itself is a separate matter, covered by calibration and traceability; see our guide to calibration intervals and traceability records.
2. The Process Is Stable
Under the JIS definition, process capability applies to a process in a state of statistical control. An index from an unstable process describes only how the process performed during the study period, and you cannot use it to predict future production.
Use control charts to check. JIS Z 9020-2 (2016 edition) says to find and remove assignable causes, confirm the process has stayed stable over 25 subgroups, and only then start the capability assessment. In the fictional data, neither the X-bar chart nor the R chart has any points outside the control limits.
Related: 7 QC Tools in Japanese Quality Reports: How to Read Them
3. The Distribution Can Be Treated as Normal
The probabilities in Figure 5 assume a normal distribution. The NIST handbook states that Cp and Cpk assume normally distributed data. Start with a histogram and check for a single peak that is roughly symmetric.
If there are two peaks, split the data by machine, mold cavity, material lot, and so on (stratify) and look again. Mixed data inflate the standard deviation and hide which stream has the problem.
Some characteristics, such as runout and flatness mentioned earlier, are not normally distributed to begin with. NIST suggests either transforming the data toward normality or using indices designed for non-normal data. Under PPAP, acceptance criteria for one-sided specifications and non-normal distributions are agreed with the customer.
Before You Submit: Checklist and Questions for the Japanese Customer
A bare index tells the reader nothing about how far to trust it. As shown above, the number changes with the calculation conditions, so include the following with every capability report. If the customer specifies a form or software, follow it.
Capability Report Checklist
- Characteristic and specification: drawing number, USL and LSL, two-sided or one-sided
- Sample size and sampling plan: subgroup size and number of subgroups, period, and the lots, machines, and shifts covered
- Standard deviation used (overall or within-subgroup) and the index name (Cpk or Ppk)
- Mean, standard deviation, minimum, and maximum, with a histogram and control chart
- Unrounded index values, with a few extra decimal places for borderline results
- Measuring instrument and measurement system results (Gage R&R date and result)
- Distribution check, and the calculation method if the data are not normal
- Software or spreadsheet used, and how it was verified
Ford's requirements go as far as asking that capability calculation software be validated with standard input data sets. For an in-house spreadsheet, checking it once against a hand calculation or another software package catches formula errors early.
If the index falls short, do not drop inconvenient data without a documented reason. Under PPAP, below 1.33 does not meet the acceptance criteria, and IATF 16949 requires a reaction plan for characteristics that are not capable, including containment and 100% inspection where needed. Compare Cp and Cpk to narrow down the fix: correct the aim if the mean is off, or stratify the data to find the cause if the spread is too wide. If the customer asks for a formal corrective action report, see our guide to writing an 8D report.
Questions to Ask the Japanese Customer
- Which characteristics need a capability study, and what is the required index and value for each (for example, Cpk ≥ 1.33, Ppk ≥ 1.67)?
- Is the requirement "1.33 or higher" or "above 1.33," and are results judged on rounded or unrounded values?
- Do you expect within-subgroup or overall standard deviation? When your template says "Cpk," which one does it mean?
- Which document governs: the drawing, the quality agreement, your supplier quality manual, or AIAG PPAP? Which edition?
- How many subgroups and parts, and over what period, lots, and machines?
- Is there a separate requirement for initial studies and for ongoing production?
- How should one-sided specifications and non-normal characteristics be evaluated?
- What happens if the result is in the 1.33 to 1.67 range or below 1.33: interim 100% inspection, a reaction plan, or a later restudy?
- Do you specify a report form or calculation software?
Sample Inquiry Email
Adapt the part numbers and details. Short, numbered questions are easier for a Japanese quality team to answer and to translate internally if needed.
Subject: Process capability requirements for [part number]
Dear [Name],
We are preparing the process capability study for [part number] for the PPAP submission. To make sure our report matches your requirements, could you please confirm the following?
1. The characteristics that require a capability study, and the required index and value for each (e.g., Cpk ≥ 1.33 or Ppk ≥ 1.67).
2. Whether the index should be based on within-subgroup or overall standard deviation.
3. The required number of subgroups and parts, and the production period to cover.
4. How to evaluate one-sided or non-normal characteristics.
5. Whether you have a required report form or calculation software.We plan to start data collection on [date]. Thank you for your support.
Best regards,
[Your name]
[Company]
Capability indices take more effort in data collection and assumption checks than in the calculation itself. For where capability studies fit in the overall flow, from pre-production quality planning to handling complaints after shipment, see our overview of the quality assurance process at Japanese plants.
Limits of This Guide
- Criteria differ by customer, industry, and characteristic. The drawing, quality agreement, and customer-specific requirements always take priority over the general figures here.
- The JIS Z 9020-2 content is from the 2016 edition. Check the current 2025 edition before quoting it.
- The AIAG PPAP criteria are from the 4th edition. We could not confirm the capability thresholds in the AIAG & VDA SPC Manual released in July 2026.
- The ppm figures and confidence intervals assume a stable, normally distributed process. They are not a guaranteed defect rate.
- All example data are fictional.
FAQ
Should the Cpk target be 1.33 or 1.67?
It depends on the drawing or quality agreement, not on which number is "correct." Under AIAG PPAP, an initial study result from 1.33 to 1.67 needs customer review, and some companies require 1.67 only for safety characteristics.
Can I calculate Cpk in Excel?
Yes. Get the mean and standard deviation with AVERAGE and STDEV.S, calculate the upper and lower values, and take the smaller one as the result. Note that this gives Ppk in AIAG terms; for Cpk, use the within-subgroup standard deviation (for example, average range ÷ d2). Check the distribution and stability with a histogram and control chart as well.
What should I do if Cpk is below 1.33?
Compare Cp and Cpk first. If Cp is high but Cpk is low, correcting the aim can close much of the gap. If Cp itself is low, work on the sources of variation such as machines, fixtures, material, and process conditions, and agree on interim containment with the customer.
What is 工程能力指数 in English?
工程能力指数 (kotei noryoku shisu) is the process capability index; Cp and Cpk are its best-known forms. プロセスパフォーマンス指数 is the process performance index (Pp, Ppk). Japanese documents use the same abbreviations.
Sources
- JIS Z 8101-2:2015 Statistics — Vocabulary and symbols — Part 2: Applied statistics (Japanese), definitions of process capability, capability indices, and performance indices: kikakurui.com
- JIS Z 9020-2:2016 Control charts — Part 2: Shewhart control charts (Japanese), clause 9: kikakurui.com; current edition JIS Z 9020-2:2025 (JSA)
- NIST/SEMATECH, e-Handbook of Statistical Methods, 6.1.6 What is Process Capability?
- Ford Motor Company, Customer-Specific Requirements for use with AIAG PPAP Fourth Edition (2024, IATF Global Oversight)
- Enerpac, Supplier PPAP Manual
- Pretesh Biswas, AIAG Production Part Approval Process (summary of PPAP 4th edition 2.2.11)
- Minitab, Potential (within) capability for Normal Capability Analysis
- SPC for Excel, Acceptance Criteria for Measurement Systems Analysis
- IATF 16949 clause 9.1.1.1 summary (Japanese): IATF16949 tettei kaisetsu
- Mitutoyo, Quality control basics (Japanese), one-sided formulas
- Kaminashi, What is a process capability index? (Japanese), an example of the five-level scale
- Douglas C. Montgomery, Introduction to Statistical Quality Control (Wiley). The original book was not checked directly; values were cross-checked against Outsourced Pharma (citing the 7th edition, 2012) and Wikipedia: Process capability index
- DQS, Release of AIAG & VDA SPC Manual 1st Edition (July 2026)
Original article in JapaneseThis guide is adapted from a MONOSCOPE article written for readers in Japan: 工程能力指数Cp・Cpkとは｜計算式・1.33の目安・必要なデータ数（n数）
About the editor
MONOSCOPE EditorFOUNDER & EDITORSpent about ten years at a plant equipment maker and a plant engineering company, proposing, designing, and commissioning equipment, then moved into web marketing. Now a web consultant for B2B manufacturers, helping them explain their products and technology. Licensed in Japan as an Energy Manager and a Third-Class Chief Electrical Engineer.
Has run an engineering blog since 2017 (500+ articles) and a Japanese YouTube channel with about 20,000 subscribers. Each English guide is written from the Japanese shop-floor point of view and checked against the cited standards.
