BIZOMAT MAKEOS · Quality
Lot Acceptance from Process Capability (Fraction Defective, Sampling Plan Risk)
Given how capable my process is, what fraction of parts fall outside tolerance, and how often will my customer’s sampling plan reject a lot?
Link between process capability and incoming inspection: z-values, Cpk and the fraction of parts outside the limits for a normal process (as the parts out of tolerance tool), then the probability that a lot with that fraction defective is accepted by a single sampling plan of n parts with acceptance number c (as the sampling plan operating characteristic tool), the share of lots rejected and the average outgoing quality, against the acceptance probability you need.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Process mean | Mean of the process. | |
| Standard deviation | Standard deviation of the process. | |
| Lower specification limit | Lower specification limit. | |
| Upper specification limit | Upper specification limit. | |
| Sampling plan | ||
| Sample size | items | Parts sampled from each lot. |
| Acceptance number | defects | Largest number of defectives in the sample for the lot to pass. |
| Least probability of acceptance wanted | % | Probability of a lot passing that you want to reach. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Process mean
- What it is
- The average of the measured characteristic from your process data, such as a capability study or control chart.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10.02 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moving the mean towards a limit raises the fraction defective and lowers Cpk.
- Get it right
- Use the same unit for the mean, standard deviation and both limits.
What this means: Standard deviation
- What it is
- The standard deviation of the measured characteristic, showing how much parts vary around the mean.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.03 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger standard deviation lowers the z-values and Cpk and raises the fraction defective.
- Get it right
- Use the long-term standard deviation if lots are built over time, not just a short-run value.
What this means: Lower specification limit
- What it is
- The lower specification limit for the characteristic, from the drawing or specification.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 9.9 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter the limit itself, not the tolerance; it must be below the upper limit.
What this means: Upper specification limit
- What it is
- The upper specification limit for the characteristic, from the drawing or specification.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10.1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter the limit itself, in the same unit as the mean.
Sampling plan
What this means: Sample size
- What it is
- The number of parts drawn at random from each lot and inspected under the single sampling plan.
- Units and format
- Enter a number in items: a count of items. The form accepts 1 items to 5000 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger sample with the same acceptance number lowers the chance a lot with defects is accepted.
- Get it right
- It must be larger than the acceptance number; take it from the sampling plan agreed with the customer.
What this means: Acceptance number
- What it is
- The largest number of defective parts allowed in the sample for the lot still to be accepted.
- Units and format
- Enter a number in defects: defects. The form accepts 0 defects to 1000 defects. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 defects as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger acceptance number raises the probability of acceptance.
- Get it right
- It is a count of defectives in the sample, not a percentage.
What this means: Least probability of acceptance wanted
- What it is
- The probability of lot acceptance you want to reach, compared with the probability the sampling plan gives for this process.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 1% to 99.999%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 95% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Type the percentage itself (90 for 90 %).
Method and formulas
- z to each limit = distance ÷ σ; Cpk = smaller z ÷ 3; fraction defective from the normal distribution (as the parts out of tolerance tool).
- Probability of acceptance = Σ (k = 0 … c) C(n, k) p^k (1 − p)^(n − k) (as the sampling plan operating characteristic tool).
- Lots rejected = 1 − Pa; average outgoing quality ≈ p × Pa.
- Example: Cpk 0.88889 gives 3,862.1 ppm out of tolerance; the n = 50, c = 1 plan accepts 98.384% of lots.
Limitations
- Stable normal process; tails of real processes are often heavier.
- Binomial sampling from large lots.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Predict how often lots will be rejected at goods-in
- Set a capability target to keep lots passing
- Show the cost of an off-centre process
Who it is for
- Quality engineers
- Supplier quality engineers
- Production managers
What you get
- z-values, Cpk and fraction outside tolerance
- Probability of acceptance and lots rejected
- Average outgoing quality
Interpreting the result
- Centring the process usually cuts rejected lots faster than tightening the spread.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 128–133. Topic reference; no source tables or recommended settings are reproduced.
- ISO 2859-1 Sampling procedures for inspection by attributes
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Accepting or rejecting a lot
In Accept or Reject by the Numbers
What it covers: Decide about lots by sampling: designing a plan for producer's and consumer's risk, operating characteristics, lot acceptance from process capability, and what a sample adds to prior data.
Method explained here · BIZOMAT original book
Thresholds, false calls and acceptance decisions
In Eddy-Current Inspection by the Numbers
What it covers: Probability of detection and false calls, guard bands from measurement uncertainty, capability with confidence bounds, lot acceptance, and testing a suspect reading before discarding it.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.
Open the Lot Acceptance from Process Capability (Fraction Defective, Sampling Plan Risk) calculator