Variables Sampling (Z1.9)

Variables Sampling

MIL-STD-414 / Z1.9 Tables, Forms 1 and 2, the B-5 Estimator and the OC Curve

What variables sampling decides

An attributes plan counts the defective units in a sample; a variables plan measures the quality characteristic itself and judges the lot from the sample mean and standard deviation. Because a measurement carries far more information than a pass/fail count, a variables plan reaches the same discrimination with a much smaller sample, which is the whole point when a test is costly or destructive. The price is an assumption: the measurements of the lot must be at least approximately normal, and each characteristic needs its own plan. MIL-STD-414 (1957) is the military standard built on that idea; ANSI/ASQ Z1.9 is its civilian descendant, re-aligned in 1980 with the Z1.4 code letters so that the two can be switched between, and the k, M and sample sizes for a given sample size are unchanged. This tool implements Section B of the standard, variability unknown, standard deviation method, which is the part used in practice (unless otherwise specified the standard itself calls for the standard deviation method and Form 2, A6.2).

Code letter, sample size and the master tables

The sample size code letter follows from the lot size and the inspection level in Table A-2 (five levels I to V, level IV unless otherwise specified); the letters B to Q correspond to sample sizes 3, 4, 5, 7, 10, 15, 20, 25, 30, 35, 40, 50, 75, 100, 150 and 200. A specified AQL that is not one of the fourteen table values is read by the ranges of Table A-1 (0.70 to 1.09 is treated as 1.00, and so on). The master tables are then entered by code letter and AQL: Table B-1 gives the acceptability constant k for Form 1 and Table B-3 the maximum allowable percent defective M for Form 2, both for normal and tightened inspection, and Tables B-2 and B-4 give the reduced-inspection plans. Tightened inspection is read in the same tables one AQL column lower (the bottom row of AQLs in the standard), so the sample size stays and the constant becomes stricter. Reduced inspection takes a smaller sample and the constants of the next lower AQL at that smaller size. An arrow in a cell sends the user to the first plan below it, sample size included; a sample size that equals or exceeds the lot means every unit is inspected. The 1957 tables are transcribed here from the standard; one cell, Table B-1 code letter I at AQL 0.25, prints 3.26 where 2.26 is meant (its neighbours are 2.24 and 2.28, and Table B-2 gives 2.26 for the same n = 25 plan).

Form 1: the quality index against k

From the n measurements compute the mean x̄ and the estimate s of the lot standard deviation (with n − 1 in the denominator, laid out in the standard's examples as the sum, the sum of squares, the correction factor (ΣX)²/n, the corrected sum of squares and the variance). For an upper specification limit U the quality index is Q_U = (U − x̄)/s, for a lower limit L it is Q_L = (x̄ − L)/s: the distance from the mean to the limit in units of s. The lot meets the acceptability criterion when Q is equal to or greater than k, and fails when Q is smaller than k or negative (B3.3). Form 1 applies to a single specification limit only.

Form 2: the estimated percent defective against M

Form 2 turns the quality index into an estimate of the lot percent defective beyond the limit, p̂_U or p̂_L, read from Table B-5 with Q and n, and accepts the lot when the estimate is at most M. For a double specification limit (Section B Part II) both estimates are read and added, p = p̂_U + p̂_L, and compared with M when one AQL covers both limits combined (B12.1.1); when the two limits carry different AQLs the three criteria of B12.2.1 apply: p̂_U ≤ M_U, p̂_L ≤ M_L and p at most the larger of the two. The two forms are equivalent for a single limit: M is the estimate that Q = k would give, up to the rounding of the tables, and the tool prints the quality index Q_M at which the estimate equals M beside k so that the equivalence can be seen.

Table B-5 computed, not transcribed

Table B-5 of the standard is the minimum-variance unbiased estimate of the fraction of a normal population beyond a limit derived by Lieberman and Resnikoff (1955): with a = (n − 2)/2 and x = ½ − Q √n / (2(n − 1)), p̂ = I_x(a, a), the regularised incomplete-beta function of the symmetric Beta(a, a) distribution, and p̂ = 0 when x ≤ 0. The tool computes it exactly for any Q and n rather than interpolating a transcription, and the test script checks it against the readings the standard prints in its examples and footnotes (2.19 % at Q = 1.59 and 0.66 % at Q = 1.70 for n = 5; 30.93 % at Q = 0.50 and 5.33 % at Q = 1.60 for n = 50) and against scipy's beta distribution. For n = 4 the estimate is the straight line 100(½ − Q/3), for n = 3 the arcsine form, and at Q = 0 it is 50 % for every n, as the first row of the table shows. A negative quality index is entered by its magnitude and the result taken from 100 % (B13.2.3). The standard reads the table at Q rounded to two decimals and s to the printed digits; the tool shows that reading beside the full-precision one, and at small n the two can differ in the second decimal because the estimate is steep in Q.

The operating characteristic curve

Let the lot's measurements be normal with a fraction p beyond the limit, so that the limit lies z_p = Φ⁻¹(1 − p) standard deviations from the lot mean. Then √n Q = √n (U − x̄)/s is a noncentral t variable with n − 1 degrees of freedom and noncentrality z_p √n, and the probability of acceptance of the Form 1 plan is Pa(p) = P(T′ ≥ k √n), evaluated exactly with Lenth's algorithm AS 243. Form 2 has the same curve with k replaced by Q_M. The standard's own OC curves (Table A-3) are drawn for this single-limit case and the standard notes that they hold closely for a double limit; the exact double-limit curve depends on how the lot mean sits between the limits and is not drawn. For comparison the chart also shows the curve of the same n and k with the lot standard deviation known (Section D, Pa = Φ(√n (z_p − k))), which is steeper, and the OC curve of the Z1.4 (MIL-STD-105E) single attributes plan for the same lot at general inspection level II and the nearest table AQL, with the ratio of the two sample sizes. The AOQL under rectifying inspection is the maximum of p Pa(p) (1 − n/N).

Designing a plan from two OC points

When a plan is wanted whose OC curve passes through (p₁, 1 − α) and (p₂, β), the normal approximation behind the nomographs (Wallis 1947, reproduced in Montgomery's Chapter 16) gives k = (z_β z_p1 + z_α z_p2)/(z_α + z_β) and n = ((z_α + z_β)/(z_p1 − z_p2))² (1 + k²/2), the factor 1 + k²/2 allowing for the estimated standard deviation. The tool reports that plan, its exact noncentral-t risks, and the smallest n at the same k that meets both risks exactly; a nomograph reading lands within about 10 % in n of these values.

Severity of inspection

Part III of Section B governs switching. The process average is the mean of the Table B-5 estimates of the preceding ten lots (Form 1 users compute the estimate for this purpose). Tightened inspection is instituted when that average exceeds the AQL and more than T of the lots have estimates above the AQL, T from Table B-6 by code letter, AQL and number of lots; normal inspection returns when the average under tightened inspection is at or below the AQL. Reduced inspection may be used when the preceding ten lots were all accepted under normal inspection, every estimate was below the limit of Table B-7 (or zero for the number of lots that table names) and production is steady; a rejected lot, an average above the AQL or irregular production ends it. Tables B-6 and B-7 are not transcribed in this tool.

Limits of the tool

Section B only: the range method of Section C (average range of subgroups of five with the c factor) and the known-variability plans of Section D are not implemented, apart from the σ-known OC curve drawn for comparison. The 1957 code letters are used; Z1.9 since 1980 assigns Z1.4's letters to the same sample sizes, so a Z1.9 user should enter the lot size and level and read the sample size, k and M, which are the same. The normality of the measurements is assumed and not tested here; the Distribution Fitting tool can test it on the same data. Mixed variables–attributes plans (A9) are not offered.