Module 15 · Improve

Variation reduction and design

Module 7 introduced Cp and Cpk as a way to check whether a process, as built, meets a spec. This module runs the same idea in the other direction: given a spec, what does it take from the design — the individual part tolerances, the target each process aims for — to make the assembled result capable? Tolerance stack-up connects directly to capability, robust design connects capability data back to the drawing before parts are ever cut, and the Taguchi loss function makes the case that "within spec" was never really the right question.

Learning objectives

Compute a worst-case and an RSS (root-sum-square) predicted tolerance for a linear dimensional stack, and explain when each is the right tool.
State the independence and centring assumptions behind RSS, and what happens to a stack's real capability when centring fails.
Use a Monte Carlo simulation to see how a real, quietly off-target process degrades a stack's actual Cpk relative to RSS's centred prediction.
Reallocate a stack's tolerance budget across parts using the same RSS relationship, favouring the parts that are cheaper to hold tight.
Apply the Taguchi loss function to decompose average loss into a variance component and an off-centre component.
Explain robust design's core idea (reducing sensitivity to noise, not just tightening tolerances) and connect capability data back to a design decision.

Why this matters

The account below is a widely cited illustration, presented as reported, not independently re-verified in this course.

The story most often used to introduce the Taguchi loss function concerns two Sony plants making the same television, one in Japan and one in the United States, both meeting the same colour-density specification.^[6] As reported by Taguchi and Clausing and by Phadke, the two plants' output looked identical by a pass/fail count against the spec, but their distributions did not: the Japanese plant's units clustered tightly around the target value, while the American plant's were spread more evenly across the full tolerance band.^[6]^[7] Taguchi's argument was that this difference is not invisible to the customer just because both plants pass the same inspection: a unit sitting near a spec limit degrades picture quality more than one sitting near target, even though both are "in spec." A traditional pass/fail view of tolerance cannot see this difference at all; the loss function, built to rise continuously with distance from target rather than jump at the spec limit, is built specifically to see it. This module's underlying data are the original 1979 newspaper account cited by later authors, not independently re-verified here, but the argument does not depend on the exact historical figures: it is the same argument, worked on fully traceable constructed numbers, in Worked example 2 below.

Tolerance stack-up revisited: worst case vs RSS

When several toleranced dimensions add up to form one assembled result — a gap, a clearance, an overall length — the assembly's own tolerance depends on how the individual tolerances are assumed to combine. The traditional, conservative answer is worst case: assume every part lands at its own tolerance limit simultaneously, in the direction that makes things worst.

worst-case stack tolerance = Σ |tolerance_i| Guarantees the assembly meets its spec even in the single worst combination of parts, but that combination is rare when tolerances are independent, which makes worst-case tolerancing needlessly expensive to hold in production.

If each part's dimension varies independently and (approximately) normally around its own nominal, the statistical alternative combines the tolerances in quadrature instead of by simple addition, following the same variance-addition rule behind every "sum of independent random variables" result in this course.^[1]

RSS stack tolerance = √(Σ tolerance_i²) Always smaller than the worst-case sum (equal only in the degenerate case of a single part). This is the entire appeal of statistical tolerancing: it predicts a materially tighter, cheaper-to-achieve assembly tolerance than worst case, for the same part tolerances.

RSS's saving is not free. It assumes the part dimensions are independent (no shared cause pulling two of them the same way at once) and that each part's own process is centred on its nominal. Evans' three-part survey of statistical tolerancing methods covers exactly this trade-off, including what happens under shifts and drifts away from the centred assumption; Chase and Parkinson survey the wider tolerance-analysis literature the method sits in.^[2]^[1] Worked example 1 below shows a case where the centring assumption is the one that actually matters in practice.

Allocating tolerance using capability data

RSS combines in quadrature, so a part's tolerance contributes to the stack total in proportion to its square, not linearly. Practically: loosening a large tolerance by a given amount costs the stack relatively little once other tolerances are already smaller, while tightening the smallest tolerance in a stack buys back very little RSS margin. This gives a designer real room to negotiate — loosen the tolerance on whichever part is hardest or most expensive to hold, and recover the RSS budget by tightening a part that is cheap to hold tighter, typically a purchased or standard component rather than a machined feature with its own process to control. Worked example 3 reallocates Worked example 1's own stack this way, on real numbers.

Robust design

Taguchi's robust design approach starts from a different question than tolerancing does. Instead of asking how tightly a dimension must be held to control a known noise source, it asks whether the design itself can be changed so the response is less sensitive to that noise in the first place — parameter design before tolerance design, in Taguchi's own framing.^[4]^[5] A signal-to-noise ratio is used to identify a combination of design (control) factors that keeps the response on target across the range of noise the product will actually see, using the same factorial methods Module 13 already covers to find that combination; only after the design is made robust does tightening any remaining tolerances become worthwhile. The specific signal-to-noise ratio statistics Taguchi proposed have drawn a real, published statistical critique — that they can confound location and dispersion effects in ways an analysis of the mean and log-variance separately does not.^[8] This course teaches the robustness idea and the loss function as Taguchi presented them, and teaches that a standard factorial analysis of mean and variance separately is the more defensible way to carry out the analysis, consistent with the published statistical response to the SN-ratio critique.

The Taguchi loss function

The loss function formalizes the "distance from target matters, not just the spec limits" argument from the Sony story above into a single quadratic curve, calibrated so the loss equals a stated cost A₀ exactly at the spec limit (a distance δ₀ from target).

k = A₀ / δ₀² average loss = k × (s² + (mean − target)²) s² here is the population (n, not n−1) mean square about the sample mean; the identity mean((x−target)²) = variance + (mean−target)² only holds exactly with that convention. The two terms split total average loss into a piece from spread around the mean and a piece from the mean itself sitting off target — two separately actionable causes.

The formula's shape is what carries Taguchi's point: loss rises continuously and smoothly with distance from target, with no discontinuity at the spec limit the way a pass/fail view has one. A part sitting just inside the limit costs almost as much, in this view, as one sitting just outside it; a part sitting exactly on target costs (in principle) nothing. Worked example 2 applies this to real capability data already used elsewhere in this course.

Feeding capability data back into DFM

Every tool in this module runs in the direction production engineers usually do not get to run it: from a spec, backward, into what the design and the tolerances need to be. In practice the flow goes both ways. A supplier's demonstrated process capability (Module 7's Cpk, an MSA study's %R&R, Module 9's multi-vari decomposition of where the variation actually comes from) is exactly the input a stack-up or a loss-function calculation needs to be realistic rather than aspirational, and a design-for-manufacturing review that ignores what a process can actually hold is choosing, by omission, either an expensive over-tight tolerance or an assembly that will not be capable once it is built. Capability data belongs in the tolerance decision, not just in the inspection report that follows it.

Worked examples

Worked example 1: a three-part assembly gap, worst case vs RSS vs a real centring bias

The data below is a constructed example, not a real production assembly.

An assembly gap is formed by three stacked dimensions: a housing depth (A), a spacer thickness (B) subtracted from it, and a shaft-shoulder position (C) also subtracted, gap = A − B − C. The spec on the gap is 2.000 mm ± 0.100 mm.

Table 1. Three-part gap stack, nominal and tolerance (constructed data).
Part	Nominal (mm)	Tolerance (±mm)	Sign in the stack
A, housing depth	25.000	0.050	+
B, spacer thickness	8.000	0.025	−
C, shaft-shoulder position	15.000	0.040	−

Nominal gap = 25.000 − 8.000 − 15.000 = 2.000 mm, matching the spec's centre. Worst case = 0.050 + 0.025 + 0.040 = 0.115 mm — this exceeds the 0.100 mm spec half-width, so worst-case tolerancing says this stack cannot be guaranteed to pass. RSS = √(0.050² + 0.025² + 0.040²) = 0.0687 mm — comfortably within spec, assuming independence and centring. Assuming each part's own process runs at Cpk = 1.0 (a fairly ordinary, "just adequate" process, not an exceptional one), each part's own sigma is tolerance / (3 × 1.0): 0.0167, 0.0083, and 0.0133 mm for A, B, and C. Combining those in quadrature gives a predicted stack sigma of 0.02291 mm and a predicted stack Cpk, assuming everything stays centred, of 1.455 — a comfortable, apparently capable result.

Now drop the centring assumption. Suppose each of the three processes is quietly running one full sigma off its own target — not a large or unusual shift for an unmonitored process, and all three biased in the direction that closes the gap (a plausible scenario: shared tooling wear, a fixture that consistently locates slightly off, or simply three ordinary processes that each happen to drift the same way). A Monte Carlo simulation of 200,000 assemblies built from these three biased-but-otherwise-identical processes gives a simulated mean of 1.9617 mm (spread, sd = 0.02283 mm, essentially the same spread RSS predicted — the bias moves the centre, not the width) and a simulated Cpk of only 0.901, with about 3530 assemblies per million falling outside the 2.000 ± 0.100 mm spec. RSS's centred prediction (Cpk 1.455, negligible reject rate) and the simulated reality (Cpk 0.901, roughly 1 assembly in 280 outside spec) describe the same nominal design; the entire difference is three ordinary, individually unremarkable centring biases combining in the same direction. RSS's spread prediction was essentially exact; its implicit assumption that every part stays centred was the one that cost real capability.

Worked example 2: Taguchi loss for the Module 7 bore data

The data below reuses the 125 bore-diameter readings from Module 7's Worked example (Ø12.000 ± 0.025 mm, Cpk about 1.1), with an illustrative cost at the spec limit added for this module's purpose. Cost figures below are illustrative cost units, not a real currency or a claimed real cost.

With target = 12.000 mm, a cost of A₀ = 40 illustrative cost units at the spec limit (δ₀ = 0.025 mm), k = 40 / 0.025² = 64000 cost units per mm². The 125 readings have mean 12.0039 mm and s = 0.00676 mm. Average loss splits into a variance component of 2.903 and an off-centre component of 0.9754, for a total average loss of 3.879 cost units per part — about 25.15 % of it from the process being slightly off target (mean 12.0039 vs. target 12.000), the rest from ordinary piece-to-piece spread. Summed over the full 125-part sample, total loss is 484.86 cost units. The single worst reading in the sample carries a loss of exactly 40.0000 — exactly A₀, because that reading landed exactly at the spec limit, the boundary the loss function is calibrated against by construction. Every reading closer to target than that one costs less, continuously, all the way down to a reading exactly on target, which is precisely the argument the Sony story above is making on real, traceable numbers: two processes can pass the same inspection and still carry very different real cost, and the loss function is what makes that difference visible.

Worked example 3: reallocating the stack's tolerance budget

Constructed example, continuing Worked example 1's stack.

Part A (the housing depth) is a machined feature on an expensive setup; tightening it further would mean a new fixture. Part B (the spacer) is a purchased shim, available in finer increments at essentially the same unit cost. Loosen A from ±0.050 to ±0.060 mm and tighten B from ±0.025 to ±0.015 mm, leaving C unchanged, and assume a more typical Cpk of 1.33 per part (rather than Worked example 1's deliberately ordinary 1.0, to show a second, independent starting point) with the same 0.5σ centring bias used in the exercises below.

Table 2. The same stack, tolerance reallocated (constructed data).
Part	Original tolerance	Reallocated tolerance
A, housing depth	±0.050 mm	±0.060 mm (loosened)
B, spacer thickness	±0.025 mm	±0.015 mm (tightened)
C, shaft-shoulder position	±0.040 mm	±0.040 mm (unchanged)

Worst case is essentially unchanged (0.060 + 0.015 + 0.040 = 0.115 mm, the same as before to three decimals). RSS rises only from 0.0687 to 0.0737 mm — a real increase, but still comfortably inside the 0.100 mm spec half-width — even though A was loosened by 20 % (10% more room for the hard-to-hold machined feature). Squaring a large tolerance and squaring a small one do not cost the same: loosening the part that already carried the largest tolerance costs relatively little RSS budget, while tightening the part with the smallest tolerance buys back a smaller absolute amount than the same percentage change would on a larger tolerance. With the more typical Cpk = 1.33 assumption, predicted stack Cpk here is 1.806; simulated with the same kind of 0.5σ centring bias used in Exercise 1 below, it comes in at 1.544 (about 5 per million outside spec) — still comfortably capable. The reallocation bought real relief on the expensive part without meaningfully hurting the assembly.

Common mistakes

Using RSS tolerancing without checking that each part's process is actually independent and centred. Consequence: Worked example 1's exact failure mode — a stack that looks comfortably capable on paper and is not, once real (even modest) centring bias is accounted for. Fix: state the centring assumption explicitly whenever RSS is used, and check it against actual process data, not just a nominal drawing.
Treating "passes RSS" as a guarantee rather than a probabilistic prediction. Consequence: RSS was designed to replace an overly conservative worst-case guarantee with a realistic prediction; treating its output as a new guarantee reintroduces the same overconfidence in a different place. Fix: report RSS results with the assumptions attached (independence, centring, approximate normality), not as a bare pass/fail.
Reading a capability index in isolation from the spec limit's cost structure. Consequence: two processes with the same Cpk can carry very different real cost if one is centred and the other is not, exactly Worked example 2's point. Fix: pair a capability index with a target-sensitive view (Cpm, or the loss function directly) whenever centring is in question.
Tightening every tolerance in a stack equally instead of reallocating. Consequence: wastes money tightening parts that were already cheap to hold, while the genuinely expensive part stays tight too. Fix: use RSS's quadratic relationship to loosen the expensive-to-hold part and tighten the cheap-to-hold one, as in Worked example 3.
Treating robust design as "just do a DOE." Consequence: a standard factorial finds which factors move the mean and which move the variance, but robust design specifically asks whether a design (control) factor can be set to make the response insensitive to noise factors the customer cannot control — a different question than optimizing the mean alone. Fix: separate control factors from noise factors explicitly before designing the experiment.
Using Taguchi's original signal-to-noise ratio statistics without knowing they have been challenged. Consequence: presenting a contested statistical method as settled practice. Fix: analyze the mean and the log-variance as separate responses, the more defensible route this course teaches, and cite the SN-ratio critique if the original ratios come up.
Presenting the Sony story's specific numbers as independently verified. Consequence: repeating a widely told but not fully re-sourced anecdote as if it were audited data. Fix: label it as reported (this page does), and use Worked example 2's fully traceable numbers to make the same point when a verified figure is needed.

Exercises

Exercise 1: a two-part shaft-in-bore clearance

Constructed example, not a real production assembly.

A bore (D) and a shaft (E) form a clearance, clearance = D − E, spec 0.050 mm ± 0.045 mm.

Table 3. Two-part clearance stack (constructed data).
Part	Nominal (mm)	Tolerance (±mm)
D, bore	20.000	0.030
E, shaft	19.950	0.020

Tasks. (a) Compute the worst-case and RSS stack tolerances, and check each against the 0.045 mm spec half-width. (b) Assuming each part runs at Cpk = 1.33, compute the predicted stack Cpk assuming centring. (c) A Monte Carlo simulation with a 0.5σ centring bias on each part gives a simulated Cpk. Would you expect it higher or lower than (b)'s prediction, and why?

Show the worked solution

(a) Worst case = 0.030 + 0.020 = 0.050 mm, which exceeds the 0.045 mm spec half-width. RSS = √(0.030² + 0.020²) = 0.03606 mm, which is within spec.

(b) Predicted stack Cpk (centred) = 1.660.

(c) Lower, for the same structural reason as Worked example 1: any consistent off-centre bias uses up some of the margin RSS's centred assumption implicitly claimed. The simulated Cpk here is 1.428, about 20 per million outside spec — still comfortably capable, because a 0.5σ bias at a starting Cpk of 1.33 leaves more margin than Worked example 1's harder case (a full 1σ bias starting from Cpk 1.0). The direction of the effect is the lesson: centring bias always pulls the real Cpk below the centred prediction; how much depends on how large the bias is and how much margin the centred prediction started with.

Exercise 2: Taguchi loss for a fill-weight process

Constructed example, not a real production run. Setting: packaged fill weight (g), target 250.0 g, spec ±3.0 g, 25 packages, each weighed to 0.01 g. Cost figures are illustrative cost units, not a real currency.

Table 4. Fill weight, 25 packages (constructed data).
#	g	#	g	#	g	#	g	#	g
1	249.46	6	249.77	11	248.49	16	248.35	21	249.13
2	249.13	7	251.00	12	249.35	17	248.59	22	251.08
3	250.13	8	250.53	13	246.28	18	248.89	23	248.44
4	249.44	9	248.39	14	249.02	19	249.84	24	249.76
5	248.60	10	247.65	15	247.68	20	250.66	25	250.47

Tasks. With A₀ = 20 cost units at the spec limit (δ₀ = 3.0 g): (a) compute k. (b) Compute the mean and the average loss, split into its variance and off-centre components. (c) What fraction of the average loss comes from the process running below target, rather than from spread?

Show the worked solution

(a) k = 20 / 3.0² = 2.222 cost units per g².

(b) Mean = 249.205 g (s = 1.135 g), against a target of 250.0 g. Variance component = 2.748; off-centre component = 1.404; average loss = 4.152 cost units per package. Total loss over the 25 packages: 103.80.

(c) 33.81 % of the average loss comes from the process running under target (249.205 g vs. 250.0 g), a materially larger off-centre share than Worked example 2's bore data (about 25 %). Unlike a pass/fail spec check, which would treat any package inside 247.0-253.0 g identically, the loss function is already saying this process has real, quantifiable room to save cost simply by shifting its target up, before touching its spread at all.

Quiz

Ten questions. Score 70 % or more to mark the module complete on this device.

1. Worst-case tolerancing assumes every part is perfectly centred on nominal parts are never independent every part lands at its own tolerance limit at once, in the worst direction the assembly tolerance is always smaller than any single part's

2. RSS combines individual tolerances by simple addition the square root of the sum of squares taking the largest single tolerance averaging the tolerances

3. In Worked example 1, RSS predicted a comfortable Cpk of 1.455, but a Monte Carlo with a 1σ centring bias per part gave a simulated Cpk of only 0.901, because the Monte Carlo used a different spec RSS's spread prediction was wrong the parts were not independent RSS's prediction assumed every part stays centred, and the simulation deliberately violated that assumption

4. Reallocating a stack's tolerance budget (Worked example 3) works because RSS combines tolerances in quadrature, so loosening an already-large tolerance costs relatively little of the total worst-case tolerancing allows it and RSS does not the nominal stack value changes it only works when all tolerances are exactly equal

5. Robust design's core idea is to tighten every tolerance until the process is capable replace tolerancing with inspection find design settings that make the response insensitive to noise the customer cannot control, before tolerancing only apply to electronic products

6. The Taguchi loss function's key structural feature, versus a plain pass/fail spec check, is that it only applies at the spec limit rises continuously with distance from target, with no discontinuity at the spec limit is identical to a Cpk calculation ignores the process mean entirely

7. In Worked example 2, the average loss's two components (variance and off-centre) require which convention for s² in the variance term? the sample standard deviation squared, dividing by n − 1 the range divided by d₂ it does not matter which convention is used the population (biased, dividing by n) mean square, for the decomposition identity to hold exactly

8. In Worked example 2, the single worst bore reading carried a loss of exactly A₀ because that reading landed exactly at the spec limit, which is what A₀ is calibrated against A₀ is always the maximum possible loss that reading was outside the spec limit it was a coincidence unrelated to the spec

9. The published critique of Taguchi's original signal-to-noise ratios argues that they are mathematically impossible to compute apply only to attribute data can confound location and dispersion effects in ways a separate analysis of mean and log-variance does not were never actually used in industry

10. The Sony television story in this module is presented as independently verified, audited data as reported by Taguchi and Clausing and by Phadke, with the underlying figures not independently re-verified here as a fabricated illustration invented for this course as a case this course's author worked on directly

Answer key

c. Every part at its own limit at once, worst direction.
b. Square root of the sum of squares.
d. RSS assumed centring, which the simulation violated.
a. Quadrature combination makes loosening a large tolerance cheap.
c. Insensitivity to uncontrollable noise, before tolerancing.
b. Rises continuously, no discontinuity at the spec limit.
d. Population (biased, n-divisor) mean square.
a. That reading landed exactly at the spec limit.
c. Can confound location and dispersion effects.
b. As reported, not independently re-verified here.

Key takeaways

Worst-case tolerancing (Σ tolerances) guarantees an assembly even in the rare simultaneous-limit case; RSS (√Σ tolerance²) predicts a realistic, usually much tighter, assembly tolerance assuming independence and centring.
RSS's centring assumption is often the one that actually matters: Worked example 1 showed a stack RSS predicted as comfortably capable (Cpk 1.455) turn out much less capable (Cpk 0.901) once a realistic per-part centring bias was simulated, while the predicted spread stayed almost exactly right.
Because RSS combines in quadrature, a stack's tolerance budget can be reallocated: loosen the part that is expensive to hold tight, tighten the part that is cheap to hold tight, and the total RSS moves only modestly.
Robust design (parameter design before tolerance design) asks whether a design can be made insensitive to noise before tightening any tolerance; Taguchi's original SN-ratio statistics have a real, published critique, so this course prefers analyzing mean and log-variance separately.
The Taguchi loss function is calibrated so loss equals a stated cost A₀ exactly at the spec limit, and splits cleanly into a variance component and an off-centre component — two separately actionable causes of the same total cost.
A pass/fail spec check cannot distinguish a part near target from one near the spec limit; the loss function, and the Sony story it is built to explain, both make the case that this distinction has real, continuous cost.

References

All web sources accessed 2026-09-09 or 2026-09-10 (Phase A) unless noted.

Evans, D. H. (1974–1975). Statistical tolerancing: the state of the art. Parts I–III. Journal of Quality Technology, 6(4), 188–195; 7(1), 1–12; 7(2), 72–76. https://www.tandfonline.com/doi/abs/10.1080/00224065.1975.11980657 (abstract-level: linear propagation/RSS, non-linear methods, and Part III's treatment of shifts and drifts confirmed at this depth)
Chase, K. W., & Parkinson, A. R. (1991). A survey of research in the application of tolerance analysis to the design of mechanical assemblies. Research in Engineering Design, 3, 23–37. https://link.springer.com/article/10.1007/BF01580066 (abstract-level)
ASME Y14.5-2018. Dimensioning and Tolerancing. ASME. https://www.asme.org/codes-standards/find-codes-standards/y14-5-dimensioning-tolerancing/2018 (catalogue-level: existence and edition; statistical-tolerancing symbol details are not read at this depth, so no single claim on this page rests on it)
Taguchi, G. (1986). Introduction to Quality Engineering: Designing Quality into Products and Processes. Asian Productivity Organization. https://archive.org/details/introductiontoqu0000tagu (catalogue-level; the quadratic loss function, parameter and tolerance design)
Creveling, C. M. (1997). Tolerance Design: A Handbook for Developing Optimal Specifications. Addison-Wesley. https://searchworks.stanford.edu/view/3399329 (catalogue-level; links the loss function and capability)
Simpson, T. W. "IE 466: Concurrent Engineering, 32.3 Taguchi's robust design method" (Penn State course handout). https://www.me.psu.edu/simpson/courses/ie466/ie466.robust.handout.pdf (the Sony USA vs Sony Japan colour-density story, citing Phadke 1989; the most directly readable source found for this story)
Taguchi, G., & Clausing, D. (1990). Robust quality. Harvard Business Review, 68(1), 65–75. https://hbr.org/1990/01/robust-quality (secondary, paywalled beyond title and lead; the Sony comparison and loss-function argument, presented as reported per this course's standing decision, underlying 1979 data not independently verified)
Box, G. E. P. (1988). Signal-to-noise ratios, performance criteria, and transformations (with discussion). Technometrics, 30(1), 1–40. https://www.tandfonline.com/doi/abs/10.1080/00401706.1988.10488313 (abstract-level; the published statistical critique of Taguchi's SN-ratio statistics — a log transformation achieves the same result more simply)