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Variables Sampling Calculator: Quality Index vs k (Form 1)

When you can measure instead of just pass or fail, variables sampling reaches the same protection with a much smaller sample. The k-method turns your mean and standard deviation into quality indices and a single accept or reject test. This calculator does the arithmetic and shows what happens as the process centre drifts.

Your numbers

Sample size n

pieces

Sample mean

Average of the n measurements.

Sample standard deviation s

Upper specification limit

Lower specification limit

Acceptability constant k

Take this from the master table of your standard for your sample size code letter and AQL. The default is illustrative only.

Your results

Smaller quality index

Decision (1 = accept, 0 = reject)

Accept only if both QU and QL are at least k.

Upper quality index QU

(USL - mean) / s

Lower quality index QL

4.6667

(mean - LSL) / s

Margin over k

0.28

Approx. percent nonconforming

2.2752

Normal-distribution estimate; the standard uses a beta-based estimate, so treat this as a guide.

Approx. PPM nonconforming

22,752

Use the constant k and sample size from your licensed copy of ANSI/ASQ Z1.9 or ISO 3951. This tool does the arithmetic only. Assumes the characteristic is approximately normal.

Save results:

Smaller quality index as the sample mean shifts

Sweeps Sample mean from 9.97 to 10.07, holding your other inputs fixed.

Automate variables acceptance in your ERP

Optional: a review of how to run quality index calculations on inspection results as they are posted.

Optional. The tool and every export work without this. No spam.

How to use it

Measure the sample the standard calls for, then enter the count, the mean and the sample standard deviation, plus the upper and lower specification limits. Look up k for your sample size code letter and AQL in your standard and type it in. For a one-sided specification, set the unused limit to a value far outside the process and ignore that index. The result tells you accept or reject and gives a rough percent nonconforming.

Use the sample standard deviation with n - 1 in the divisor.
Check normality first with a quick plot; skewed data breaks the method.
Keep units consistent between measurements and limits.

The method behind it

The quality indices are QU = (USL - mean) / s and QL = (mean - LSL) / s. They count how many standard deviations the sample mean sits from each limit. The lot is accepted when both indices are at least k. Larger k means a stricter plan. Under the double specification case, standards also check a maximum standard deviation or use a combined estimate; confirm which form your standard prescribes. The percent nonconforming shown here uses the normal tail areas as a simple guide.

Worked example

Ten shafts measure a mean of 10.020 mm with s = 0.015 mm against limits 9.95 and 10.05. QU = 0.030 / 0.015 = 2.0 and QL = 0.070 / 0.015 = 4.67. With k = 1.72 the smaller index of 2.0 passes. Now drag the mean toward 10.03: QU falls to 1.33 and the lot is rejected even though every part may still be in tolerance. That is the point: variables sampling reacts to a drifting average before many parts fail.

Common mistakes

The most frequent error is using the population sigma form when sigma is not truly known, or the reverse. Others include mixing a k value from a different AQL, rounding s too coarsely, and measuring with a gauge whose resolution is too coarse for the tolerance. A poor gauge inflates s and makes good lots fail.

Frequently Asked Questions

What is the k-method in variables sampling?

It converts the sample mean and standard deviation into quality indices, the distance from the mean to each specification limit in standard deviations, and accepts the lot if each index is at least the constant k from the standard's table.

Where do I find k?

In the master tables of ANSI/ASQ Z1.9 or ISO 3951-1, found by sample size code letter and AQL. Those tables are copyrighted, so this tool asks you to enter the value rather than reproduce them.

Why can a lot with all parts in tolerance be rejected?

Because the decision is about the estimated fraction nonconforming in the whole lot. A mean close to a limit or a large spread predicts defectives in the unmeasured parts, even if the measured sample is clean.

When should I not use variables sampling?

When the characteristic is not roughly normal, when it is pass or fail, or when the sample is measured with a gauge too coarse to give a meaningful standard deviation. Attribute sampling is safer in those cases.

If your measurements are captured in SAP QM, MES or an ERP quality module, we can automate these calculations on each inspection lot. Ask for a review.

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