# Methods and formulas for Variables Acceptance Sampling (Accept/Reject Lot)

Select the method or formula of your choice.

## Sample size and critical distance

The calculation of sample size, n, and critical distance, k, depends on the number of specification limits given and whether standard deviation is known.

### Single specification limit and known standard deviation

The sample size is given by:

The critical distance is given by:

where:

### Single specification limit and unknown standard deviation

The sample size is given by:

The critical distance is given by:

The standard deviation is estimated from the sample data:

### Double specification limits and known standard deviation

First Minitab calculates z:

Then Minitab finds p* from the standard normal distribution as the upper tail area corresponding to z. This is the minimum probability of defective outside one of the specification limits.

The method Minitab uses for the calculation of sample size and critical distance depends on this value of p*.

Let p1 = AQL, p2 = RQL

• If 2p* ≤ (p1/ 2), then the two specifications are relatively far apart and calculations follow the single limit plans.
• If p1/ 2 < 2p* ≤ p1, then the two specifications are not relatively far apart, but are still not so close that the minimum probability of defective can be found for certain mean values. Minitab performs an iteration to find sample size and critical distance.

Let

μ = μ0+ m * h, where h = σ/100

Let m = 1, 2, ...300. For each μ calculate:

where Φ is the cumulative distribution function of the standard normal distribution. If Prob (X<L) + Prob (X>U) is extremely close to p1, then Minitab uses the larger value between Prob (X<L) and Prob (X>U) to find sample size and accept number.

Suppose Prob (X<L) is the larger value, let pL = Prob (X<L).

The sample size is given by:

The critical distance is given by:

where:

ZpL = the (1 – pL) * 100 percentile of the standard normal distribution

• If p1 < 2p* < p2, then the specifications of the plans must be reconsidered because the minimum probability defective determined by the two specification limits and the standard deviations is larger than the acceptable quality level p1. Consider a plan with a slightly larger probability of defective than p1.
• If 2p* ≥ p2, then the lot should be rejected; the minimum probability of defective determined by the two specification limits and the standard deviation is larger than the rejectable quality level. You can reject the lot without testing any products.

### Double specification limits and unknown standard deviation

First Minitab lets the critical distance be the value as given in the case of two separate single-limit plans:

Then Minitab finds the upper tail area from the standard normal distribution, p*, corresponding to the k as the percentile; and the percentile Zp** from the standard normal distribution corresponding to upper tail area of p* / 2.

The maximum standard deviation (MSD) is given by:

The estimated standard deviation is given by:

Minitab tests whether the estimated standard deviation, s, is less than or equal to the MSD.

If the estimated standard deviation, s, is less than or equal to the MSD, then:

The sample size is given by:

The critical distance is given by:

If the estimated standard deviation, s, is not less than or equal to the MSD, then the standard deviation is too large to be consistent with acceptance criteria and you must reject the lot.

### Notation

TermDescription
Z1the (1 – p1) * 100 percentile of the standard normal distribution
p1Acceptable Quality Level (AQL)
Z2the (1 – p2) * 100 percentile of the standard normal distribution
p2Rejectable Quality Level (RQL)
Zαthe (1 – α) * 100 percentile of the standard normal distribution
Zβthe (1 – β ) * 100 percentile of the standard normal distribution
Xithe ith measurement
the mean of actual measurements
Llower specification limit
Uupper specification limit
σknown standard deviation

## Probability of acceptance

Let p be the probability of defective which is the x value of a point on an OC curve.

### Single specification limit and known standard deviation

Single lower specification limit and known standard deviation
Prob (X < L) = p.
Single upper specification limit and known standard deviation
Prob (X > L) = p.

### Double specification limits and known standard deviation

First Minitab calculates z

Then finds p* from the standard normal distribution as the upper tail area corresponding to z. This is the minimum probability of defective outside one of the specification limits.

The method Minitab uses for the probability of acceptance depends on this value of p*.

Let p1 = AQL, p2 = RQL

• If 2p* ≤ (p1/ 2), then the two specifications are relatively far apart and calculations for sample size and critical distance follow the single limit plans.
• If p1/ 2 < 2p* ≤ p1, then the two specifications are not relatively far apart, but are still not so close that the minimum probability of defective can be found for certain mean values.

For any given p, Minitab finds the mean, μ, of the measurements using a grid search algorithm. Then,

### Double specification limits and unknown standard deviation

When you have both upper and lower specification limits, but do not know the standard deviation, Minitab uses the OC curve for the single-limit plan to approximate the double specification limits case. The OC curve derived for a single-limit plan with specified p1, p2, α, and β is the lower limit of the band of OC curves for a two-sided specification plan with the same p1, p2, α, and β and for most practical cases can be taken as the OC curve for the two-sided plan. See Duncan1.

1. Duncan (1986). Quality Control and Industrial Statistics, 5th edition.

### Notation

TermDescription
nsample size
kcritical distance
σknown standard deviation
Zpthe (1 - p)th percentile from the standard normal distribution
Φthe cumulative distribution function of the standard normal distribution
T

is non-central t distributed with degrees of freedom = n – 1, and the non central parameter,

Llower specification limit
Uupper specification limit

## Probability of rejecting

The probability of rejecting (Pr) describes the chance of rejecting a particular lot based on a specific sampling plan and incoming proportion defective. It is simply 1 minus the probability of acceptance.

Pr = 1 – Pa

where:

Pa = probability of acceptance

## Method for accepting or rejecting a lot

Minitab calculates an accept or reject decision based on your measurements from sampled items and the criteria (sample size and critical distance) of your variables acceptance sampling plan.

First Minitab calculates the mean and standard deviation from your data (if you have not specified a historical standard deviation):

Mean
Standard deviation
Acceptance criteria
###### Note

Minitab uses σ instead of s in the Z calculations when you provide a historical standard deviation.

• If both specifications are given, the Z-values for each specification are calculated. Accept the lot if Z.LSL ≥ k and Z.USL ≥ k; otherwise reject the lot.
• If only one specification is given, the corresponding Z-value is calculated. If lower specification only, accept the lot if Z.LSL ≥ k; otherwise reject the lot. If upper specification only, accept the lot if Z.USL ≥ k; otherwise reject the lot.

### Notation

TermDescription
Ximeasurement data
mean
sestimated standard deviation
σknown standard deviation
nsample size
kcritical distance
Llower specification limit
Uupper specification limit
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