Cpk 1.53, Zero Rejects in 125 Parts, and 731 ppm Shipped: Statistical Process Control, MSA and DOE Worked From the Arithmetic

A 25-subgroup study on a turned shaft returned Cpk 1.53, a flat range chart and not one out-of-spec part in 125 — while the process was drifting 0.080 mm across the shift and shipping an estimated 731 ppm. This guide works the arithmetic that catches it: the R-bar/d2 and A2/D4 constants, why Cp uses within-subgroup sigma while Ppk uses the overall s, the sample-size penalty that forces you to observe 1.70 to certify 1.33 with 30 parts, a Gage R&R where a caliper scores 33.9% GRR and ndc 4 while a micrometer scores 15.3% and ndc 9, the exact binomial p-chart limit that differs from the normal approximation by two whole defects, why ISO 2859-1 AQL 1.0 at code K accepts a 5%-defective lot 12.4% of the time, and a replicated 2^3 factorial that separates feed from speed with a 0.031 micron confidence half-width.

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Cpk 1.53, Zero Rejects in 125 Parts, and 731 ppm Shipped

A job shop turns a shaft to Ø12.00 ± 0.10 mm on a CNC lathe, 5,000 pieces a month. The quality engineer pulls 125 parts — 25 subgroups of five consecutive pieces, one every twenty minutes across a shift — and runs the numbers. The range chart is flat. The X-bar chart shows eleven of twenty-five subgroup means outside the control limits, but nobody looks at that page, because the headline number is good: Cpk 1.53. Every one of the 125 parts is inside tolerance. Zero rejects. The report goes into the PPAP file.

The same data, read properly, says the process will ship about 731 ppm out of spec, and the reason is on the same page: the subgroup means run from 11.989 mm at the start of the shift to 12.060 mm at the end. The range chart stays in control because the instantaneous spread never changed — the machine's repeatability was fine the whole time. What changed was the machine's location, by 0.080 mm, which is 80% of the entire tolerance band, and the X-bar chart caught it in eleven separate signals that were filed as "chart noise."

That is the whole problem with statistical process control as it is usually practised in small and mid-sized shops. Not the statistics — the arithmetic is forty years old and unambiguous — but the sequence. Capability is computed on data whose stability was never established, with a gauge whose error was never measured, against a sample size too small to support the claim being made, and then compared to a threshold that was invented for a different question.

This guide works the arithmetic in order: capability, control, measurement, sampling, experiment. Every number below comes from one consistent worked example so the pieces can be checked against each other.

1. What the indices actually measure

Four indices share two formulas. The distinction that matters is not "short-term versus long-term" — that framing is a proxy — but which standard deviation you put in the denominator.

C_p and P_p measure spread only. C_{pk} and P_{pk} measure spread and centring. The difference between the C pair and the P pair is the denominator:

When the process is stable, the two estimates converge. When they diverge, the gap is a measurement of how much the process moved during the study — and that movement is usually the expensive part, because it happens on a timescale long enough that nobody notices it in a spot check.

For the shaft example, with n = 125 observations:

The 0.0800 in the numerator is the distance from the grand mean (12.0202) to the nearer limit (12.100). Now the same data with the overall standard deviation, s = 0.02865 mm:

C_{pk} 1.53. P_{pk} 0.93. Same parts, same gauge, same shift. One number says the process comfortably meets a 1.33 target; the other says it does not meet it at all. The ratio s/\hat{\sigma}_{within} = 1.647 is the whole story: 65% more variation existed across the shift than existed inside any five-part subgroup.

Converting indices to parts per million

For a normal distribution, the two-sided fraction nonconforming at a given C_{pk} is:

C_{pk} · Distance to limit · ppm (centred) · Meaning

0.67 · 2.00σ · 44,400 · Not capable; sort 100%

1.00 · 3.00σ · 2,700 · Marginal; ~3 parts per thousand

1.33 · 3.99σ · 66 · Conventional minimum for series production

1.67 · 5.01σ · 0.54 · Usual PPAP target on special characteristics

2.00 · 6.00σ · 0.002 · "Six sigma" in the literal sense

Two cautions on that table, both routinely ignored.

First, the ppm figures assume the process is centred on the mean it will actually run at forever. Real processes drift. The convention that made "1.33" famous assumes a 1.5σ mean shift over the long term, which turns a 3.99σ margin into 2.49σ and 66 ppm into about 12,400 ppm — 1.24%, or 62 parts a month at 5,000 pieces. The index did not get worse. The assumption underneath it was wrong.

Second, ppm is a property of the distribution, not of the parts you inspected. In the shaft example, 0 of 125 parts were out of spec while the expected fallout was 731 ppm — because the parts that would fail are the ones near the end of the shift, and a 125-piece sample can easily contain none of them. Sampling does not find rare events; the model does.

The 2026 change that matters

The harmonised AIAG–VDA SPC manual (the "Yellow Volume", issued 2026) restructures this hierarchy in a way that will show up in IATF 16949 audits. Its position:

The practical consequence: the shaft study above cannot report C_{pk} at all under the new framework. It reports P_{pk} = 0.93 and an out-of-control X-bar chart, and the reaction plan starts.

Non-normal data

Diameter, length, thickness and roundness are usually close enough to normal that the indices behave. Flatness, cylindricity, surface roughness, and anything bounded at zero are not. For those, ISO 22514-2 defines eight time-dependent distribution models (A through D, with sub-classes such as C3 for a systematic trend like tool wear) and gives the capability formulas per model. Applying a normal C_{pk} to a Rayleigh-distributed roundness measurement produces a number, but not a meaningful one — the right move is a quantile-based index or a transformation, and ISO 22514-2 is the reference that says which.

2. Control charts: where the limits come from

A control chart is not a tolerance chart. The limits are computed from the process's own variation, so they answer a different question: is the variation I am seeing consistent with the variation I saw before? A process can be perfectly in control and still produce nothing but scrap — in control means predictable, not acceptable.

The X-bar and R chart

For subgroups of size n, with k subgroups:

The constants are not arbitrary. d_2 is the expected range of n standard normal variates, so \bar{R}/d_2 is an unbiased estimator of \sigma when subgroups are homogeneous. A_2 = 3/(d_2\sqrt{n}) sets the X-bar limits at three standard errors of the subgroup mean. The R-chart constants come from the distribution of the range, which is skewed — that asymmetry is why D_3 = 0 for n \le 6: the lower limit is mathematically below zero, so the chart is one-sided.

n · d_2 · A_2 · D_3 · D_4

2 · 1.128 · 1.880 · — · 3.267

3 · 1.693 · 1.023 · — · 2.574

4 · 2.059 · 0.729 · — · 2.282

5 · 2.326 · 0.577 · — · 2.114

6 · 2.534 · 0.483 · — · 2.004

7 · 2.704 · 0.419 · 0.076 · 1.924

8 · 2.847 · 0.373 · 0.136 · 1.864

9 · 2.970 · 0.337 · 0.184 · 1.816

10 · 3.078 · 0.308 · 0.223 · 1.777

Applying n = 5 to the shaft study:

The X-bar chart has 5 points below 11.9969 and 6 above 12.0436, in a monotonic run: 11.989, 11.984, 11.993 … 12.054, 12.055, 12.047, 12.046, 12.060. The range chart has zero signals, and the largest range in the study is 0.060 mm against a limit of 0.0856. Every Western Electric rule in the book fires on the X-bar page — eight points on one side, a trend of more than seven, points beyond 3σ — and none fires on the range page.

That pattern has a name and a cause. Range in control, mean out of control means the machine is repeating well but not staying put: thermal growth of the ballscrew, tool wear, bar-stock diameter variation between bundles, coolant temperature. The fix is a warm-up cycle, a tool-wear offset schedule, or a pre-shift warm-up plus a mid-shift reset — not tighter feeds and speeds, and not a new machine.

Rational subgrouping

The X-bar chart can only detect what happens between subgroups. If every subgroup is five consecutive parts produced within ninety seconds, the chart is blind to anything that changes over hours — and \hat{\sigma}_{within} will look excellent because it excludes the very variation that is hurting you. Sample five consecutive parts to monitor repeatability. Sample five parts spread across the shift to monitor stability. They answer different questions, and the shaft study needed the latter to catch a drift that a consecutive-parts study would have hidden inside the subgroups.

The formal version of this is ISO 22514-2's model classification. Model A has constant location and dispersion (stationary). Model C3 has constant dispersion with a systematic location trend — exactly the shaft. Model D has both changing, which is what a multi-spindle or multi-cavity process looks like when the streams run at different means. The model determines both the chart type and the capability formula; picking the wrong model is how a drifting process gets a passing grade.

Individuals charts for low volume

A job shop making twelve parts a lot cannot form subgroups of five. The I-MR chart (individuals and moving range) handles n = 1:

The 2.66 is 3/1.128 — three standard errors of a single observation estimated from the average moving range of two. It is less powerful than an X-bar chart (the limits are wider for the same true \sigma because a single observation carries no averaging benefit), which is precisely why grouping into subgroups is worth the effort when the volume exists.

Short-run and small-shift detection

Shewhart limits are tuned for shifts of 2σ or more. Their average run length at zero shift is:

So a 3σ chart raises a false alarm every 370 subgroups, and detects a 1σ shift in about 44 subgroups — 22 hours at one subgroup every thirty minutes. For a shop watching a 0.010 mm tool-wear trend, that is far too slow. Two alternatives:

For a five-person shop, the practical rule: Shewhart X-bar/R for characteristics that fail abruptly (dimensions after a setup), EWMA for characteristics that fail slowly (tool wear, thermal drift, plating thickness, resin viscosity). Using Shewhart on a wear trend guarantees that every chart looks fine until the day it does not.

The false-alarm arithmetic nobody budgets for

Every rule you enable multiplies the false-alarm rate. The AIAG–VDA 2026 manual is explicit that stacking Western Electric rules "to be strict" is wrong: each additional rule increases sensitivity and, in the same motion, the type I error rate. With independent subgroups and m rules, the probability of at least one false signal in m chances is:

At \alpha \approx 0.0027 per rule and nine rules applied to a 25-point chart, the expected number of false signals is not zero and the chart will cry wolf on a stable process often enough that operators learn to ignore it. That is the actual failure mode of SPC in job shops: not that the math is wrong, but that the chart has been trained to be ignored.

3. Capability is a confidence statement, not a number

A C_{pk} from 30 parts and a C_{pk} from 1,000 parts are not the same claim, even when they print the same digits. The estimate's uncertainty depends on the sample size, and for the usual normal-theory interval the lower confidence bound on C_{pk} scales as:

with \chi^2_{0.05,\,n-1} the 5th percentile of the chi-square distribution. Solving for the observed value needed to certify a target:

n · \chi^2_{0.05} · Factor · Observed C_{pk} needed for 95% LCB ≥ 1.33 · 95% LCB if observed C_{pk} = 1.33

30 · 17.71 · 0.781 · 1.702 · 1.039

50 · 33.93 · 0.832 · 1.598 · 1.107

100 · 77.05 · 0.882 · 1.508 · 1.173

125 · 99.28 · 0.895 · 1.486 · 1.190

200 · 167.4 · 0.917 · 1.450 · 1.220

400 · 353.7 · 0.942 · 1.413 · 1.252

A 30-piece study reporting C_{pk} = 1.33 has a 95% lower bound of 1.04 — statistically indistinguishable from "not capable." To certify 1.33 with 30 parts you must observe 1.70. This is why AIAG PPAP specifies 25 subgroups of at least five, and a population of at least 300 parts to draw from: 125 measurements is the smallest dataset where the interval is narrow enough that the acceptance decision means something. The 2026 manual's threshold-raising rule is the same arithmetic applied explicitly.

Two further requirements that PPAP and AS9103 both impose, and that shops routinely skip:

  1. The measurement system must be validated before the study starts. A capability study run on an unqualified gauge measures the gauge as much as the process. Section 4.
  2. The data must cover the full range of expected variation — all cavities, all streams, all shifts, all tools, the actual operating environment. A study run on one favourable afternoon is a study of that afternoon.

The one-sided case

For characteristics with a single limit — flatness, runout, surface roughness, concentricity — the two-sided formulas do not apply. Use the one-sided index:

and report the relevant one. Averaging a one-sided characteristic into a two-sided C_{pk} by inventing a lower limit of zero is a common error that flatters the result, because the "lower limit" is usually six standard deviations away and the \min() function then ignores the limit that actually binds.

4. Measurement system analysis before anything else

The observed variance is the sum of the true process variance and the measurement variance:

Measurement error therefore inflates the denominator of every capability index, and the process looks worse than it is — but the same error also misclassifies good parts as bad and bad as good, which is worse. The relationship between the reported and true index is:

A gauge at %GRR = 30% means a reported C_{pk} of 1.33 is really 1.39; at %GRR = 50%, it is really 1.54 — the process is half again as capable as the report says, and you have been scrapping good parts for months. At %GRR = 10% the correction is 0.5% and can be ignored.

The Gage R&R arithmetic

A crossed Gage R&R: 10 parts × 3 appraisers × 3 trials, with parts selected to span the process range. The average-and-range method uses three constants — K_1 from the number of trials, K_2 from the number of appraisers, K_3 from the number of parts — each the reciprocal of d_2^* for that count:

with K_1 = 0.5908 for 3 trials, K_2 = 0.5231 for 3 appraisers, K_3 = 0.3146 for 10 parts. The subtraction inside AV can go negative — that means appraiser effect is smaller than the noise in its own estimate, and AV is set to zero rather than reported as an imaginary number.

Same ten parts, same three operators, two gauges:

Quantity · 0–25 mm caliper, 0.01 mm resolution · Micrometer, 0.001 mm resolution

EV (repeatability) · 0.0123 mm · 0.0051 mm

AV (reproducibility) · 0.0025 mm · 0.0000 mm

GRR · 0.0126 mm · 0.0051 mm

PV (part variation) · 0.0350 mm · 0.0332 mm

TV · 0.0372 mm · 0.0336 mm

\%GRR of study variation · 33.9% · 15.3%

ndc · 3.9 · 9.1

\%GRR of tolerance (0.20 mm) · 37.7% · 15.4%

The caliper fails on both criteria: %GRR above 30% means the measurement system needs improvement, and ndc = 3.9 means it can distinguish fewer than four categories across the part range — it cannot see the process it is being used to control. The micrometer at 15.3% is in the conditionally acceptable band and ndc 9 comfortably exceeds the minimum of 5.

The acceptance thresholds (AIAG MSA, 4th edition) are worth stating precisely because they are quoted loosely:

For job shops, %GRR of tolerance usually matters more than %GRR of study variation. If the study population is narrow — ten parts pulled from one batch — PV is small, TV is small, and %GRR looks terrible even for a good gauge. Dividing by the tolerance instead asks the question the drawing actually asks: what fraction of the permitted band is consumed by measurement error? A gauge consuming 15% of the tolerance leaves 85% for the process, which is workable. One consuming 38% leaves 62%, and no amount of process improvement will recover it.

Attribute agreement

For go/no-go gauges, visual inspection, and colour matching there is no continuous variable to analyse. Use attribute agreement analysis: 10 parts (with known reference values), 3 appraisers, 3 trials, and compute agreement within appraiser, between appraisers, and against the reference. Kappa adjusts for chance agreement:

with P_o the observed agreement and P_e the agreement expected by chance. \kappa > 0.75 is generally usable; below 0.4 the inspection decision is closer to a coin toss than a measurement, and the sorting operation is not doing what the paperwork claims.

5. Counting data: p-charts and the AQL illusion

Not every characteristic is a dimension. Weld porosity, missing hardware, wrong label, solder bridging — these are counted, not measured, and the right chart is a p-chart (fraction nonconforming) or u-chart (defects per unit).

For a lot of n = 200 with \bar{p} = 0.018 (1.8%, or 3.6 defects expected per lot):

That is 9.2 defects — so a lot with ten defectives signals. But the normal approximation is poor here: with an expected count of 3.6, the binomial distribution is visibly skewed, and the exact binomial upper limit at the equivalent one-sided 0.00135 tail is 11 defects, not 9. Under the true 1.8% rate, a lot with 10 defectives occurs with probability 0.37% — a routine fluctuation, not a signal. The normal-approximation chart raises a false alarm once every ~273 lots; the exact chart does not. The 2026 AIAG–VDA manual makes this explicit: use the binomial or Poisson cumulative distribution directly rather than the normal approximation.

What AQL actually promises

ISO 2859-1 is the attribute sampling standard behind almost every incoming-inspection clause in Indian supply contracts. Read the definitions carefully:

Incoming defect rate · Probability of acceptance

0.5% · 99.6%

1.0% · 96.3%

2.0% · 75.9%

3.0% · 48.1%

5.0% · 12.4%

8.0% · 0.8%

A plan indexed at AQL 1.0 accepts a lot that is 5% defective one time in eight. Its average outgoing quality limit — the worst average quality the plan will pass, assuming rejected lots are screened and resubmitted — is about 1.5% at an incoming rate near 2%. If a customer's drawing requires 0.5% outgoing, an AQL 1.0 plan does not deliver it, and no amount of inspection paperwork changes the arithmetic. This is why critical characteristics are not controlled by acceptance sampling at all: they are controlled by process capability, error-proofing, or 100% automated inspection.

6. Design of experiments: stop tuning one factor at a time

The default improvement method in a job shop is OFAT — one factor at a time. Vary speed, hold everything else, pick the better level, then vary feed, and so on. It is slow, it cannot detect interactions, and it converges to a local optimum that depends on where you started. A factorial design varies all factors together and estimates every main effect and interaction with the same number of runs.

A replicated 2³

Surface roughness on a turned EN8 shaft, three factors at two levels: cutting speed A (60 / 120 m/min), feed B (0.08 / 0.15 mm/rev), depth of cut C (0.5 / 1.5 mm). Eight runs, two replicates, randomised order. The model in coded units (\pm1):

With 16 observations and 8 parameters, the residual has 8 degrees of freedom and — because the design is replicated — those degrees of freedom are pure error, not model lack-of-fit. The pooled standard deviation was s = 0.0539 µm, giving a coefficient standard error of:

and a 95% confidence half-width of t_{0.025,8} \times SE = 2.306 \times 0.0135 = \mathbf{0.031\ \mu m}.

Term · Coefficient (µm) · t · Significant at 95%?

Intercept · 1.5823 · 117.5 · —

A: speed · −0.1609 · −11.9 · yes

B: feed · +0.4274 · +31.7 · yes

C: depth · +0.0412 · +3.1 · yes (marginal)

AB · +0.1877 · +13.9 · yes

AC · +0.0120 · +0.9 · no

BC · −0.0016 · −0.1 · no

ABC · +0.0086 · +0.6 · no

R^2 = 0.994. Three things come out of this that OFAT would have missed:

  1. Feed dominates. Its coefficient is 2.7× the next largest main effect (speed) and 2.3× the largest interaction. In coded terms, going from 0.08 to 0.15 mm/rev adds 2 \times 0.427 = 0.85 µm of Ra. The theoretical floor for a turned surface, Ra \approx f^2/(32 r_\varepsilon), predicts the same direction and roughly the same magnitude for a 0.8 mm nose radius — which is why feed is the first knob for finish, and speed is the first knob for tool life.
  2. The AB interaction is real. At low feed, raising speed improves finish by reducing built-up edge. At high feed, raising speed makes it worse. A one-factor-at-a-time study run at low feed would conclude "more speed is better" and then produce scrap when the operator later runs high feed. Interactions are the reason OFAT conclusions do not transfer.
  3. Depth of cut barely matters — 0.04 µm against a 0.031 µm confidence half-width. It is statistically significant and practically irrelevant. Knowing that lets a planner take heavier roughing passes for cycle time without touching finish, which is where the money is.

The L9 trap

Taguchi's orthogonal arrays are attractive because they are small: an L_9 screens four 3-level factors in nine runs. The signal-to-noise ratio for smaller-is-better is:

and for nominal-is-best, S/N = -10\log_{10}(\bar{y}^2/s^2), for larger-is-better, S/N = -10\log_{10}\left(\frac{1}{n}\sum 1/y_i^2\right).

Run the L_9 with four factors at three levels and the level averages come out clean — feed's range across levels was 5.47 dB against 0.35 dB for speed, which is a decisive result. But count the degrees of freedom: nine runs give 8 df; four main effects at three levels consume 4 \times 2 = 8 df. Zero degrees of freedom remain for error or interactions. The array cannot estimate an interaction, cannot produce an F-test, and cannot distinguish a real effect from noise. It is a screening tool for a first pass, not a confirmation experiment. If interactions matter — and in machining they almost always do — use a factorial or a response-surface design.

The same caution applies to the S/N ratio computed from a single part per run: with n=1, -10\log_{10}(y^2) conflates the mean and the variance and cannot separate "this setting is off-target" from "this setting is unstable." Replicate, or do not report S/N.

Sample size and power

A design that cannot detect the effect you care about is a waste of machine time. For a two-level factorial, the observations needed per factor level to detect a difference \Delta at power 1-\beta is approximately:

In a 2^3 design each level of any factor appears in 2^{k-1} = 4 runs, so with r replicates each level holds 4r observations. For \sigma = 0.054 µm, \alpha = 0.05 and power 0.80, n_{level} \approx 2(1.96+0.84)^2(0.0029)/0.01 \approx 4.6. With two replicates that level holds 8 observations and the design resolves

which is why the depth-of-cut coefficient at 0.041 µm should be read as "above the noise floor, below the design's comfortable resolution" rather than acted on. Drop to a single replicate and two things break at once: the 2^3 has seven degrees of freedom consumed by its seven terms, leaving zero for error, and the detection threshold rises to 0.107 µm — the feed effect would still show, the depth effect and the interaction would not.

7. The economics: loss, not scrap

The scrap count is a bad cost model, because it counts only the parts that crossed a limit. Taguchi's loss function counts the cost of being away from target, which is what the customer experiences:

where A is the loss at the specification limit \Delta. For the shaft, if a part at the limit costs ₹450 in rework, scrap, or assembly rework downstream, and \Delta = 0.10 mm:

Average loss per part is k\sigma^2 for a centred process:

Process state · \sigma (mm) · Average loss per part · At 5,000 parts/month

Within-subgroup only (no drift) · 0.0174 · ₹13.62 · ₹68,100

Shift-wide drift (actual) · 0.0287 · ₹36.94 · ₹184,700

The gap — ₹116,600 a month — is the cost of the drift, and none of it appears in the scrap report, because the parts were all in tolerance. That is the argument for SPC in one line: the inspection-based quality system sees zero defects while the loss function sees ₹1.4 million a year.

Against that, the investment is trivially small:

The expensive part is not the equipment. It is the discipline to stop a process that is producing good parts because the chart says it is about to stop producing good parts.

8. What to instrument first

A shop with thirty characteristics and no SPC cannot chart thirty. The sequence that returns the most money per hour:

  1. Characteristics that have caused a rejection, rework, or a customer complaint in the last six months. Not the ones on the drawing with the tightest tolerance — the ones that actually failed.
  2. Characteristics that are expensive to inspect late — internal features, features buried under subsequent operations, anything that requires a CMM queue to verify.
  3. Characteristics where the customer has imposed a capability requirement — the special characteristics in the control plan, the KCs in AS9103, anything with a C_{pk} clause in the purchase order.
  4. Then the highest-risk remaining features by severity from the PFMEA.

For each one, before the first chart goes on the wall:

That is the entire method. The statistics are not the hard part — d_2, A_2, \chi^2 and a binomial tail are all in a spreadsheet. The hard part is the sequence: measure the gauge before the process, prove stability before claiming capability, and treat the range chart and the X-bar chart as two different questions rather than one picture.

The numbers to take away

For the measurement side of this, the metrology and dimensional inspection guide covers gauge selection and calibration traceability; the GD&T guide covers how the tolerance in the denominator got there in the first place, and the CNC feeds and speeds guide covers the process parameters that the DOE in section 6 is screening.

The shaft study that opened this article had a passing C_{pk}, a clean range chart, and zero rejects in 125 parts. It also had eleven out-of-control signals, a P_{pk} of 0.93, 731 ppm of expected fallout, and ₹1.4 million a year of invisible loss. All of it was in the same 125 numbers. The only thing missing was the order in which to read them.

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