# Cpk Confidence Bound Calculator for Small Samples

> Source: https://netray.co/tools/cpk-confidence-bound-sample-size-calculator
> Publisher: Netray (netray.co), AI and integration engineering for ERP and manufacturing systems.
> Section: Free Tools

## Short answer

An observed Cpk of 1.8 from 30 parts does not mean the true value is 1.8. See the confidence interval and how sample size changes it.

## Category

quality

## Industry

Automotive

## Audience

- Supplier quality engineer
- Quality engineer
- Process engineer

## Tool

**Kind:** calculator

**Inputs:**

#### Observed Cpk or Ppk

**Key:** cpk

**Type:** number

**Default:** 1.8

**Min:** 0

**Step:** 0.01

**Help:** The index calculated from your sample.

#### Sample size

**Key:** n

**Type:** number

**Default:** 30

**Min:** 5

**Step:** 1

**Unit:** parts

**Help:** Number of individual readings behind the index.

#### One-sided confidence

**Key:** z

**Type:** select

**Default:** 1.6449

**Options:**

##### 90%

**Value:** 1.2816

##### 95%

**Value:** 1.6449

##### 99%

**Value:** 2.3263

#### Required index

**Key:** target

**Type:** number

**Default:** 1.33

**Min:** 0

**Step:** 0.01

**Help:** The customer minimum you want to demonstrate.

**Outputs:**

#### Standard error of the index

**Key:** se

**Expr:** sqrt(1 / (9 * n) + cpk * cpk / (2 * (n - 1)))

**Format:** decimal

**Description:** Large-sample approximation for a normal, stable process.

#### Observed index

**Key:** obs

**Expr:** cpk

**Format:** decimal

#### Lower confidence bound

**Key:** lower

**Expr:** cpk - z * se

**Format:** decimal

**Primary:** true

**Description:** You can say the true index is at least this value at the chosen confidence.

#### Upper confidence bound

**Key:** upper

**Expr:** cpk + z * se

**Format:** decimal

#### Lower bound minus required

**Key:** margin

**Expr:** lower - target

**Format:** decimal

**Primary:** true

**Description:** Positive means capability is demonstrated with statistical confidence.

#### PPM at lower bound (centered)

**Key:** ppm_low

**Expr:** 2 * normsdist(0 - 3 * max(lower, 0)) * 1000000

**Format:** integer

**Description:** Two-sided centered estimate at the pessimistic index.

**Chart:**

#### Confidence bounds vs sample size

**Type:** line

**X:** n

**From:** 10

**To:** 200

**Steps:** 20

**Series:**

- lower
- obs
- upper

**Disclaimer:** Uses the common normal-approximation formula for the standard error of Cpk. Valid for stable, approximately normal data. Exact intervals use noncentral chi-square methods.

## Content

**Intro:** A capability index from a sample is an estimate, and a small sample can look great by luck. This calculator shows the lower and upper confidence bound on an observed Cpk or Ppk, so you can tell your customer what you can actually prove rather than what the sample happened to show.

**Sections:**

#### How to use it

**Body:** Enter the index from your study and the number of readings behind it. Choose a one-sided confidence level, since you only care that the true index is at least some value. The lower bound is the figure to compare with the customer requirement. The sweep chart shows how the bounds tighten as the sample grows, which helps you decide whether to collect more parts or accept a lower claim. It is also a fast check when a customer reviewer asks whether 30 pieces is enough for a 1.67 requirement.

#### The formula

**Body:** The standard error of an index is approximated by sqrt(1/(9n) + Cpk squared / (2(n-1))). The bound is Cpk plus or minus z times that error, where z is 1.28, 1.64 or 2.33 for 90, 95 or 99 percent one-sided. The first term comes from uncertainty in the mean and the second from uncertainty in the standard deviation. At small n the second term dominates, which is why high observed indices carry wider absolute uncertainty than low ones.

#### Reading the output

**Body:** An observed Cpk of 1.80 from 30 parts has a 95 percent lower bound near 1.3, so you can claim 1.33 with only marginal confidence. At 100 parts the lower bound rises to roughly 1.5. If the lower bound sits under the requirement, the process may still be capable, but you have not demonstrated it. Options are more samples, a tighter process, or agreeing a higher observed target with the customer to offset the uncertainty.

#### Common mistakes

**Body:** Do not treat the bounds as valid for unstable data: if the control chart shows special causes, no interval is meaningful. Do not mix subgroup averages and individuals when counting n; use the number of individual readings. Do not apply the formula to non-normal data without a transformation. Finally, do not forget gage error, which widens true observed variation and is not captured here.

**Cta:** If capability evidence is scattered across spreadsheets and ERP inspection records, Netray can help consolidate it so sample sizes and indices are traceable by part and characteristic.

**Faq:**

#### How many samples do I need to prove a Cpk of 1.33?

**Answer:** To show a lower bound at or above 1.33 at 95 percent confidence, you need an observed Cpk meaningfully above 1.33. With n of 30 the observed value needs roughly 1.7 to 1.8. Use the sweep chart to find your point.

#### Is this formula exact?

**Answer:** No. It is a widely used large-sample approximation. Exact methods use the noncentral chi-square distribution and give slightly different bounds, especially for small samples, but the approximation is good enough for planning.

#### Does it apply to Ppk as well as Cpk?

**Answer:** Yes. The same standard error approximation is commonly applied to both Cpk and Ppk, as long as n is the number of individual readings and the process was stable during the study.

#### Should I use 90, 95 or 99 percent?

**Answer:** Use whatever your customer or internal procedure specifies. If nothing is specified, 95 percent one-sided is a common and defensible choice for supplier submissions.

## Search terms

cpk confidence interval calculator, cpk sample size, cpk lower confidence limit, how many samples for cpk, ppap capability sample size, cpk uncertainty, cpk confidence bound, capability index confidence limits

---
Talk to an engineer: https://netray.co/contact?intent=erp_ai | Machine catalog: https://netray.co/agent/catalog.json