# Cpk Confidence Interval Calculator

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

## Short answer

Turn a single Cpk number and a sample size into a lower and upper confidence bound, and see how the interval narrows as sample size grows.

## Category

quality

## Industry

Manufacturing

## Audience

- Quality engineer
- Supplier quality
- Six Sigma Black Belt

## Tool

**Kind:** calculator

**Inputs:**

#### Observed Cpk

**Key:** cpk

**Type:** number

**Default:** 1.5

**Min:** 0

**Step:** 0.01

**Help:** The Cpk computed from your sample.

#### Sample size

**Key:** n

**Type:** number

**Default:** 50

**Min:** 2

**Step:** 1

**Unit:** parts

**Help:** Number of individual measurements used to compute Cpk.

#### Confidence level (two-sided)

**Key:** z

**Type:** select

**Default:** 1.96

**Options:**

##### 90% (z 1.645)

**Value:** 1.645

##### 95% (z 1.960)

**Value:** 1.96

##### 99% (z 2.576)

**Value:** 2.576

#### Required Cpk

**Key:** target

**Type:** number

**Default:** 1.33

**Min:** 0

**Step:** 0.01

**Help:** Customer or internal minimum, often 1.33 or 1.67.

**Outputs:**

#### Standard error of Cpk

**Key:** se

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

**Format:** decimal

#### Interval half width

**Key:** half

**Expr:** z * se

**Format:** decimal

#### Lower confidence bound

**Key:** lower

**Expr:** max(0, cpk - half)

**Format:** decimal

**Primary:** true

**Description:** The Cpk you can claim with the chosen confidence.

#### Upper confidence bound

**Key:** upper

**Expr:** cpk + half

**Format:** decimal

**Primary:** true

#### Lower bound minus required Cpk

**Key:** margin

**Expr:** lower - target

**Format:** decimal

**Description:** Positive means the lower bound clears the requirement.

**Chart:**

#### Cpk confidence bounds vs sample size

**Type:** line

**X:** n

**From:** 10

**To:** 300

**Steps:** 30

**Series:**

- lower
- upper

**Disclaimer:** Uses the common large-sample normal approximation (Bissell) and assumes normal, stable, independent data. For small samples or non-normal data use exact or bootstrap methods.

## Content

**Intro:** A Cpk of 1.50 from 30 parts does not mean the process is 1.50. It means the true value is probably somewhere in a band around it. This calculator shows that band and tells you whether your lower bound still clears the customer requirement.

**Sections:**

#### How to use the calculator

**Body:** Enter the Cpk you calculated, the number of individual measurements behind it, the confidence level and the requirement you must meet. The tool returns the standard error, the lower and upper bounds and the margin between your lower bound and the requirement. The chart sweeps the sample size from 10 to 300 so you can see the sample size at which the lower bound crosses the requirement line and plan a larger study if needed.

**Bullets:**

- Use individual readings, not subgroup averages, for n.
- Use 95% two-sided as the default for supplier submissions.
- A negative margin means you need more data or a better process.

#### The formula behind it

**Body:** The widely used approximation for the standard error of Cpk is the square root of 1/(9n) plus Cpk squared divided by 2(n-1). The interval is Cpk plus or minus z times that standard error, where z is 1.645, 1.96 or 2.576 for 90, 95 and 99 percent. It assumes the data come from a stable, normally distributed process and that the sample is random. Because the second term grows with Cpk squared, higher capability needs proportionally more data to pin down.

#### Worked example

**Body:** Take Cpk 1.33 from 50 parts at 95% confidence. The standard error is the square root of (0.00222 + 0.01807), about 0.142, so the half width is about 0.28 and the interval runs from roughly 1.05 to 1.61. The lower bound is below 1.33, so a customer who demands proof at 95% confidence would not be satisfied. At about 300 parts the same Cpk gives a lower bound near 1.22, still short, which shows why a Cpk right at the requirement can never be proven.

#### Common mistakes

**Body:** Treating the observed Cpk as the truth is the main error. Others are computing Cpk from an unstable process, where no interval is meaningful, mixing several machines or cavities into one sample, and using subgroup counts instead of the number of readings. If the data are skewed, transform them or use a non-normal capability method before computing an interval. Always run a control chart first.

**Bullets:**

- Check stability with an X-bar and R or I-MR chart first.
- Do not pool cavities or spindles into one sample.
- Report both the point estimate and the lower bound.

**Cta:** If your capability studies live in an ERP quality module or spreadsheets, we can help build a private report that computes Cpk and its confidence bound directly from inspection results.

**Faq:**

#### How many samples do I need for a reliable Cpk?

**Answer:** Many customers ask for at least 30 to 50 readings, but the interval will still be wide. To prove a Cpk of 1.33 at 95% confidence you generally need an observed value well above 1.33, or several hundred readings. Use the chart to see the trade-off for your numbers.

#### Should I use a one-sided or two-sided interval?

**Answer:** For proving capability you only care about the lower bound, so a one-sided 95% bound (z 1.645) is common. This tool shows a two-sided interval, so select 90% two-sided to read the lower bound as a one-sided 95% limit.

#### Does this work for Ppk?

**Answer:** The same approximation is often applied to Ppk, since both are ratios of distance to spec over three standard deviations. Sample size should then be the total number of readings across the whole study period, not the number per subgroup.

#### Why is the lower bound lower than my requirement even at Cpk 1.5?

**Answer:** With small n the band is simply wide. At n of 30 and Cpk 1.5 the half width is about 0.4, so the lower bound is near 1.1. Collect more readings or improve the process so the observed Cpk is higher.

## Search terms

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