Advancing QE Skills (33) | Counting Sampling vs. Measurement Sampling: Which is More Cost-Effective When?

The IQC of a motor manufacturing company conducts incoming inspection on the motor shaft diameter φ20±0.05. The quality department has been using a counting sampling plan: a batch size of 1200 pieces, AQL 1.0, general inspection level II, and according to the table, n=198, Ac=4. Two inspectors check over 3000 pieces daily using go/no-go gauges, leading to fatigue and complaints from the assembly line at the end of the month about "shaft diameters being too large to fit." The QE suggests switching to measurement sampling—using micrometers to read measurements and making judgments based on mean and standard deviation, reducing the sample size from 198 pieces to 26 pieces. The inspection supervisor's first reaction is to disagree: can such a small sample pass customer audits? The second reaction is skepticism: why should the conclusions from 26 pieces and 198 pieces be equally reliable for the same batch?

Both questions are valid. The answer is not that "measurement sampling is more advanced," but rather that the protection levels of the two plans can be aligned on the same OC curve, with completely different costs and prerequisites. Without a clear understanding of these costs, changing the plan is like gambling with risk.

1. The Information Difference Between the Two Sampling Methods

Counting sampling outputs only one bit of information per item inspected: conforming or nonconforming. The entire plan's information is the sample size n and the acceptance number Ac, and the acceptance probability is solely related to the nonconforming rate p of the batch.

Measurement sampling outputs a measurement value for each item inspected, carrying two pieces of information: the distribution location (mean) and dispersion (standard deviation). The judgment is not based on the number of nonconforming items but on how many standard deviations the mean is from the specification limit:

• Known σ (k method): (USL − x̄) ／ σ ≥ k for acceptance;
• Unknown σ (M method): (USL − x̄) ／ s ≥ M, where M is found in the standard table.

This is the root of why measurement sampling can reduce the sample size: counting sampling must capture enough nonconforming items from the tail to make a judgment, and tail events are rare, requiring a larger n to "hit" them. Measurement sampling directly measures the position, without waiting for the product to actually exceed the specification.

Given two quality anchor points (95% acceptance probability at AQL, 10% at LQ), for a single-sided specification with known σ, the plan can be calculated in closed form:

n = ((z(1−α) − z(β)) ／ (z(1−p1) − z(1−p2)))²
k = z(1−p1) − z(1−α) ／ √n

2. Sample Size Comparison Under the Same Protection Level

Taking α=5%, β=10%, LQ=4×AQL, single-sided specification, and binomial distribution, the actual measurement results of the two plans balanced by the same set of anchor points are as follows:

Two points must be remembered:

1. The tighter the AQL, the more cost-effective measurement sampling is. At AQL 2.5%, only a 4-fold reduction is achieved, while at AQL 0.65%, a 12-fold reduction is possible. The reason is that the rarer the tail events, the more counting sampling relies on a large sample to capture nonconforming items, whereas measurement sampling is not limited by this.
2. The two OC curves are truly aligned, not just approximate. For AQL 1.0%, when the process mean shifts upward by 0.5σ (nonconforming rate increases from 1.0% to 3.4%), the acceptance probability of the counting plan is 0.196, and that of the measurement plan is 0.188—protection levels are almost identical, but the sample size differs by 7.6 times.

When σ is unknown, the sample size needs to increase by 25% to 35%, and the k value is slightly larger. This is not out of conservatism but because the judgment statistic follows a t-distribution when s replaces σ, and the tails are thicker with fewer degrees of freedom, requiring compensation.

3. Four Admission Conditions for Measurement Sampling

Measurement sampling is not simply "use it if you have numbers." If any of the four conditions are not met, reverting to counting sampling is more prudent:

1. Distribution is approximately normal. Perform an Anderson-Darling test with n≥30, p>0.05; a quick on-site criterion is skewness absolute value ≤1 and kurtosis absolute value ≤3.5. For severely skewed distributions (such as surface roughness or lifespan), perform a logarithmic or Box-Cox transformation first. If the distribution is still not approximately normal after transformation, use the percentile method or revert to counting.
2. Measurement system is qualified. GR&R ≤10% of tolerance can be used directly; 10% to 30% can be used but the k value must be tightened by 5% to 10%; >30% prohibits the use of measurement sampling—measurement variation will be mistaken for process variation, leading to an actual β much higher than the designed value.
3. σ is stable. Calculate s from 20 to 25 historical subgroups, with the maximum value divided by the minimum value ≤1.3. If σ itself is drifting (due to batch changes, tool changes, or supplier changes), the judgment basis becomes unreliable.
4. Single-item measurement cost is controllable. See the cost threshold in the next section; this is the most easily overlooked condition.

4. Reducing Sample Size Does Not Equal Reducing Cost

Total inspection cost = n × single-item inspection cost + expected loss from misjudgment. Measurement sampling reduces n, not the single-item cost, so the true criterion is:

Only when c_measurement ／ c_counting ≤ n_counting ／ n_measurement, is measurement sampling truly cost-effective.

For AQL 1.0%, the ratio threshold is approximately 7.6. Three examples:

• Shaft diameter using a micrometer: Counting 198 pieces × 4 seconds = 13.2 minutes, equivalent to 19.8 yuan at 90 yuan/hour; Measurement 26 pieces × 25 seconds = 10.8 minutes, equivalent to 16.3 yuan. Measurement is slightly more cost-effective and conveniently leaves 26 sets of data for calculating Cpk, making this a good deal.
• Form and position tolerance using a CMM: 2 minutes/piece × 26 = 52 minutes, equivalent to 78 yuan; Counting using a dedicated gauge requires only 19.8 yuan. Measurement is 4 times more expensive and is only worth doing if the customer mandates the submission of process capability evidence.
• Destructive tests like salt spray: Single-item cost is over 200 yuan, and the counting plan of 198 pieces is simply not feasible. In this case, it's not about "which one saves more," but a small sample measurement plus reliability criteria must be used.

5. Five Common Misconceptions

1. Downgrading measurement data. Even with caliper readings, the SIP only records "conforming/nonconforming" and uses a counting table to set n=198. This results in measuring the same characteristic 7 times more and discarding all distribution information.
2. Forcing the use of k method tables when σ is unknown. Using s to replace σ and still looking up k values results in a systematically loose judgment, with the actual β possibly increasing from 10% to over 20%. The M method (s method) table must be used, and the sample size must be increased accordingly.
3. Mixing single-sided and double-sided specifications. A characteristic with only an upper limit (impurity content) is incorrectly used with a double-sided table, increasing n and decreasing k, doubling the cost; while a double-sided specification like φ20±0.05 only judges the upper limit, allowing the entire batch to pass if the lower limit is exceeded. Double-sided specifications must be judged in both directions, taking the n that satisfies both.
4. Ignoring the erosion of the OC curve by GR&R. A beautifully calculated plan fails in actual testing, and upon review, it is found that the gauge repeatability accounts for 35% of the tolerance. Perform MSA first, then determine the sampling plan; the order cannot be reversed.
5. Serial rejudgment. Measurement judges nonconforming, and then 198 pieces are counted to "confirm," assuming it is stricter. In reality, it is the product of two OC curves: it is indeed stricter at LQ, but the probability of good batches being rejected at AQL increases from 5% to nearly 10%, and no one can clearly define where this composite curve is anchored. To tighten the plan, recalculate by changing n or Ac, do not serially combine two plans.

6. Self-Check List

• The characteristic is a measurement value and can be read stably, not judged by visual inspection or feel.
• Normality has been tested (AD test p>0.05 or skewness/kurtosis within limits), otherwise, transformation has been performed or reverted to counting.
• GR&R ≤10% of tolerance, or k values have been tightened according to the 10% to 30% range; >30% has not used measurement sampling.
• Whether σ is known has been confirmed: use the k method if known, use the M method and increase the sample size by 25% to 35% if unknown.
• The ratio of single-item measurement cost to counting cost has been compared with n_counting ／ n_measurement and documented.
• Whether the specification is single-sided or double-sided has been clearly marked, and double-sided specifications have been judged in both directions.

Under the same protection level, the measurement sample size is only a fraction of the counting sample size.

Knowledge code: 6.2.2

Version: v20261019

Author: QTank QTank is dedicated to providing systematic professional knowledge, methodologies, and practical tools for quality management practitioners, helping enterprises continuously improve their quality capabilities.