# Required Cpk for a PPM Target Calculator

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

## Short answer

Work backwards from a PPM target to the Cp and Cpk you need and the largest standard deviation your process can have, then compare with your current mean and sigma.

## Category

quality

## Audience

- Quality engineer
- Process engineer
- Supplier quality

## Tool

**Kind:** calculator

**Inputs:**

#### Upper spec limit (USL)

**Key:** usl

**Type:** number

**Default:** 25.05

**Step:** 0.001

**Unit:** mm

#### Lower spec limit (LSL)

**Key:** lsl

**Type:** number

**Default:** 24.95

**Step:** 0.001

**Unit:** mm

#### Process mean

**Key:** mean

**Type:** number

**Default:** 25.004

**Step:** 0.001

**Unit:** mm

#### Process standard deviation

**Key:** sigma

**Type:** number

**Default:** 0.012

**Min:** 0.0001

**Step:** 0.001

**Unit:** mm

**Help:** Short-term sigma from a stable process.

#### Target defects

**Key:** target_ppm

**Type:** select

**Default:** 10

**Options:**

##### 2,700 PPM (Cpk 1.00 centered)

**Value:** 2700

##### 1,000 PPM

**Value:** 1000

##### 100 PPM

**Value:** 100

##### 63 PPM (Cp 1.33 centered)

**Value:** 63

##### 10 PPM

**Value:** 10

##### 1 PPM

**Value:** 1

**Outputs:**

#### Tolerance width (USL - LSL)

**Key:** tol_width

**Expr:** usl - lsl

**Format:** decimal

#### Current Cp

**Key:** cp

**Expr:** tol_width / (6 * sigma)

**Format:** decimal

#### Current Cpu

**Key:** cpu

**Expr:** (usl - mean) / (3 * sigma)

**Format:** decimal

#### Current Cpl

**Key:** cpl

**Expr:** (mean - lsl) / (3 * sigma)

**Format:** decimal

#### Current Cpk

**Key:** cpk

**Expr:** min(cpu, cpl)

**Format:** decimal

**Primary:** true

#### Estimated current PPM out of spec

**Key:** ppm_actual

**Expr:** (1 - normsdist((usl - mean) / sigma) + normsdist((lsl - mean) / sigma)) * 1000000

**Format:** decimal

**Primary:** true

**Description:** Normal model, both tails.

#### Z needed per tail if centered

**Key:** z_req

**Expr:** normsinv(1 - target_ppm / 2000000)

**Format:** decimal

#### Required Cp if perfectly centered

**Key:** req_cp

**Expr:** z_req / 3

**Format:** decimal

**Description:** Cp needed so that total defects equal the target with the mean at mid-tolerance.

#### Required Cpk if the whole target falls on one side

**Key:** req_cpk_one

**Expr:** normsinv(1 - target_ppm / 1000000) / 3

**Format:** decimal

**Description:** A conservative Cpk for an off-center process, all defects on the nearest limit.

#### Maximum sigma if centered

**Key:** max_sigma_centered

**Expr:** tol_width / (2 * z_req)

**Format:** decimal

#### Maximum sigma at the current mean

**Key:** max_sigma_now

**Expr:** min(usl - mean, mean - lsl) / normsinv(1 - target_ppm / 1000000)

**Format:** decimal

#### Sigma reduction needed at the current mean

**Key:** cut_pct

**Expr:** max(0, (sigma - max_sigma_now) / sigma * 100)

**Format:** percent

**Chart:**

#### Cp and Cpk versus process standard deviation

**Type:** line

**X:** sigma

**From:** 0.004

**To:** 0.03

**Steps:** 27

**Series:**

- cp
- cpk

**Disclaimer:** Assumes a stable, normally distributed process and sigma estimated from a sufficient sample. Real PPM can differ for skewed characteristics such as position or flatness, and Cpk estimated from small samples carries wide uncertainty. Use as a planning aid.

## Content

**Intro:** Customers say 'Cpk 1.33' but what they really care about is defects. This calculator converts a PPM target into the Cp and Cpk you need, the maximum standard deviation your process can run, and how far your current process is from the goal.

**Sections:**

#### How to use it

**Body:** Enter the spec limits and the current process mean and standard deviation, then choose a PPM target. The calculator returns the current Cp, Cpk and estimated PPM, and the capability you would need. Required Cp assumes a perfectly centered process; required Cpk assumes the worst case where all allowed defects fall on the nearer limit. The chart sweeps sigma to show how fast Cp and Cpk collapse as variation grows, which is useful when deciding whether to reduce variation or move the mean.

**Bullets:**

- Use a standard deviation from a stable, in-control process.
- Select the PPM target from the customer quality agreement.
- Compare required Cp to current Cp to judge spread, and Cpk to judge centering.
- Use the sigma reduction figure to scope an improvement project.

#### The math

**Body:** For a normal process, defects beyond a limit at distance d from the mean are 1 - Phi(d / sigma), and Cpk = d_min / (3 sigma), so a Cpk of 1.0 means the nearest limit is three standard deviations away. For a centered process the defects split between two tails, so the Z per tail for a total PPM target T is the inverse normal of 1 - T / 2,000,000, and the required Cp is that Z divided by 3. For example 63 PPM gives Z of about 4.0 per tail and Cp of 1.33. For a single tail the required Cpk is the inverse normal of 1 - T / 1,000,000 divided by 3.

#### Reading the default

**Body:** With a 0.100 mm tolerance, a mean of 25.004 and sigma 0.012, Cp is about 1.39 and Cpk about 1.28, and the estimated PPM is around 63. Against a 10 PPM target, the required centered Cp is about 1.47 and the maximum centered sigma is about 0.0113. At the current mean the allowed sigma is about 0.0108, a reduction of roughly 10 percent. The tool also shows why centering matters: moving the mean to 25.000 lifts Cpk to Cp, without reducing variation at all.

#### Limits of the method

**Body:** The normal model understates risk for features with natural boundaries and skew. For these use a non-normal capability method or empirical PPM from large samples. Remember that Cpk is itself an estimate: with 30 samples, a calculated 1.33 could be between roughly 1.0 and 1.7. Measurement error is also included in observed sigma, so a poor gauge makes the process look worse than it is. Run a Gage R&R first, then evaluate capability.

**Cta:** If your customers ask for capability on many part numbers, Netray can help build a private agent that calculates Cpk from inspection results in your ERP or MES and flags features that miss their PPM target.

**Faq:**

#### What PPM does Cpk 1.33 correspond to?

**Answer:** For a centered process, Cpk 1.33 means the limits are four standard deviations from the mean, about 63 PPM in total across both tails. If the process is off-center and the nearer limit is at four sigma, the nearest tail contributes about 32 PPM.

#### Why is required Cpk different from required Cp?

**Answer:** Cp ignores centering, so a required Cp assumes the mean is at the middle of tolerance and the defects split between two tails. Cpk uses the nearest limit, so for the same PPM target the required Cpk is lower in value when all defects are assigned to one side.

#### Can I use this with a six sigma target of 3.4 PPM?

**Answer:** Six sigma quotes 3.4 PPM with a 1.5 sigma long-term shift, which corresponds to a short-term Cpk of 2.0. This calculator assumes no shift, so a 3.4 PPM target gives a required centered Cp of roughly 1.6. Choose the 1 PPM option for a tougher target.

#### How do I reduce the sigma needed?

**Answer:** Find the dominant sources of variation, such as tool wear, fixturing, temperature, material lot and measurement error, then address them in that order. Centering the mean on nominal is usually the cheapest first step because it improves Cpk without reducing spread.

## Search terms

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