# Expected PPM Out of Specification Calculator

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

## Short answer

Predict parts per million outside tolerance from a normal process mean, standard deviation and spec limits, and see what an off-centre mean costs.

## Category

quality

## Industry

Discrete Manufacturing

## Audience

- Quality engineer
- Supplier quality
- Six Sigma Black Belt

## Tool

**Kind:** calculator

**Inputs:**

#### Process mean

**Key:** mean

**Type:** number

**Default:** 10.01

**Step:** 0.001

#### Process standard deviation

**Key:** sd

**Type:** number

**Default:** 0.02

**Min:** 0

**Step:** 0.001

**Help:** Use within (short-term) or overall sigma consistently.

#### Upper spec limit (USL)

**Key:** usl

**Type:** number

**Default:** 10.08

**Step:** 0.001

#### Lower spec limit (LSL)

**Key:** lsl

**Type:** number

**Default:** 9.92

**Step:** 0.001

**Outputs:**

#### Z to upper limit

**Key:** zu

**Expr:** (usl - mean) / sd

**Format:** decimal

#### Z to lower limit

**Key:** zl

**Expr:** (mean - lsl) / sd

**Format:** decimal

#### PPM above USL

**Key:** ppm_u

**Expr:** (1 - normsdist(zu)) * 1000000

**Format:** number

#### PPM below LSL

**Key:** ppm_l

**Expr:** (1 - normsdist(zl)) * 1000000

**Format:** number

#### Total expected PPM out of spec

**Key:** ppm

**Expr:** ppm_u + ppm_l

**Format:** number

**Primary:** true

#### Expected yield

**Key:** yield

**Expr:** 100 - ppm / 10000

**Format:** percent

**Primary:** true

#### Cpk equivalent

**Key:** cpk

**Expr:** min(zu, zl) / 3

**Format:** decimal

#### Cp

**Key:** cp

**Expr:** (usl - lsl) / (6 * sd)

**Format:** decimal

**Chart:**

#### Expected PPM as the mean shifts

**Type:** line

**X:** mean

**From:** 9.95

**To:** 10.05

**Steps:** 21

**Series:**

- ppm

**Disclaimer:** Assumes a normal distribution and a stable process. Real tails are often heavier than normal; treat very low PPM as indicative only.

## Content

**Intro:** Cp and Cpk are hard to explain to a buyer; parts per million out of tolerance is not. This calculator converts your mean, standard deviation and spec limits into expected PPM for each side, overall yield and the equivalent Cpk.

**Sections:**

#### How to use it

**Body:** Enter the process mean, the standard deviation and the two specification limits in the same units. For one-sided tolerances, put a very wide value for the unused limit, such as a figure ten sigma away. Use the within-subgroup sigma if you want potential short-term PPM, or the overall sigma for what the customer will receive over time. The chart shows how PPM rises as the mean drifts off centre.

**Bullets:**

- Same units for all four inputs
- Enter a far limit for one-sided specs
- Use overall sigma to forecast what customers see

#### The method

**Body:** Each tail is calculated from the standard normal distribution. The upper tail is 1 minus the cumulative probability at Z = (USL - mean) / sigma, and the lower tail is 1 minus the cumulative probability at Z = (mean - LSL) / sigma. Multiply by one million for PPM. Cpk is the smaller Z divided by 3 and Cp is the tolerance width over six sigma. A centred process with Cpk of 1.33 gives about 63 PPM; Cpk of 1.0 gives about 2,700.

#### Reading the example

**Body:** With the defaults the process is 0.010 above centre with a sigma of 0.020. Cp is 1.33, but Cpk drops to 1.17 and about 233 PPM falls out of spec, almost all on the upper side. Re-centring to 10.000 brings Cpk back to 1.33, which is why the chart is shaped like a bowl. Centring is often cheaper than reducing variation, so check it before launching a variation project.

#### Common mistakes

**Body:** Reporting PPM from a short-term sigma and calling it the field defect rate is optimistic, since drift and tool wear are not in it. Mixing units, such as a mean in millimetres and limits in inches, gives nonsense. Skewed distributions, for example flatness or runout bounded at zero, are not normal and this method will misstate the tail. Use a distribution fit or an empirical count instead.

**Cta:** If you want PPM and capability reported per characteristic from your ERP inspection results, we can scope a private, read-only report.

**Faq:**

#### How do I convert Cpk to PPM?

**Answer:** For a centred normal process, Z equals 3 times Cpk, and PPM is twice the tail area beyond that Z. Cpk 1.0 is about 2,700 PPM, 1.33 about 63 PPM and 1.67 about 0.6 PPM. This tool does it for off-centre processes too.

#### What is the difference between Cp and Cpk here?

**Answer:** Cp compares the tolerance width to the process spread and ignores centring. Cpk uses the nearer limit, so it penalises an off-centre mean. A big gap between them means re-centring will help.

#### Should I add the 1.5 sigma shift?

**Answer:** That is a Six Sigma convention for long-term sigma level reporting, not part of this calculation. If you want it, reduce the effective Z by 1.5 manually or use a sigma level calculator.

#### Why does the tail PPM look so small?

**Answer:** The normal model decays very quickly. Real processes have outliers from setup errors, material lots and measurement problems that normal theory does not predict, so validate with actual reject counts.

## Search terms

ppm out of spec calculator, expected defect rate from mean and sigma, ppm from cpk, normal distribution defect calculator, parts per million defective calculator, process yield from sigma, off center process ppm

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