White Belt · 13 min
The hidden factory: why every step passes and the process fails
After this you can
- calculate defects per unit and first time yield from inspection data.
Assumes you have done DMAIC: what each phase produces and why the order holds.
The problem
An order fulfilment operation had five steps, a 94% first-time-yield target on each, and five green squares on the monthly dashboard for eleven months running. In the same period, 151 orders in every thousand needed rework somewhere — enough work to occupy four people full time. Nobody had costed those four people, because no step's report showed a problem. Every number on the dashboard was accurate.
The idea
First time yield for a step is the proportion of units that clear it with no rework of any kind. Not "eventually fine" — first time. The distinction is the whole lesson: rework that happens quickly still consumes capacity, and it is invisible to any measure that counts what came out rather than what went through cleanly.
Rolled throughput yield (RTY) is the probability that a single unit clears every step with no rework. And here is the thing everyone gets wrong:
RTY is the PRODUCT of the step yields, not their average.
Five steps at 95% each is not a 95% process. It is 0.95⁵ = 77.4%. Nearly a quarter of everything you make gets touched twice, and every step is meeting its target.
The gap between the average step yield and the RTY has a name: the hidden factory. It is the rework capacity that exists, is staffed, is paid for, and appears in no report — because reporting is organised by step, and no step is failing.
Why the average is so seductive
Averaging step yields gives a number that is always higher than the truth and always feels right. It is also the number a dashboard produces by default, because averaging is what dashboards do to a column.
There is a legitimate average here — normalised yield, which is RTY^(1/steps), the geometric mean. It answers "what per-step yield, applied at every step, would give this RTY?" and it is useful for comparing a five-step process with a twelve-step one. But it is a poor target, because improvement is almost never spread evenly: it belongs at the worst step.
The consequence for improvement work
RTY changes what you do next in a specific way. Adding steps to a process multiplies the yield down even when every added step is excellent. A 99% inspection step added to a 90% process gives 89.1%, not 90% — inspection is a step, and steps have yields too. This is why "add a check" so often fails to improve anything measurable: it catches defects and adds its own.
Worked example
One month, 1,000 orders, five steps.
| Step | Units in | Passed first time | First time yield |
|---|---|---|---|
| Order entry | 1,000 | 960 | 96.0% |
| Credit check | 1,000 | 980 | 98.0% |
| Pick and pack | 1,000 | 940 | 94.0% |
| Despatch | 1,000 | 990 | 99.0% |
| Invoice | 1,000 | 970 | 97.0% |
Every step is at or above the 94% target. The dashboard is entirely green.
The average of the step yields is (96.0 + 98.0 + 94.0 + 99.0 + 97.0) ÷ 5 = 96.80%. That is the number the monthly report quoted.
The rolled throughput yield is the product:
0.96 × 0.98 × 0.94 × 0.99 × 0.97 = 0.849, or 84.9%.
Of 1,000 orders, about 151 were touched twice or more. At any realistic handling cost, that is the four people nobody had counted.
The normalised yield is 0.849^(1/5) = 96.78% — almost exactly the plain average, which is worth noticing: with step yields this similar the arithmetic and geometric means nearly coincide, and neither of them is the RTY. The error is not in which average you pick. It is in averaging at all.
Where to work. Pick and pack is the worst step at 94.0%. Raise it to 99% and nothing else changes:
0.96 × 0.98 × 0.99 × 0.99 × 0.97 = 89.4%
RTY rises 4.5 points — from 151 reworked orders to 106 — from one step. That is the argument you take to a budget meeting, and it is invisible without the multiplication.
What this does not tell you. It does not say why pick and pack fails 6% of the time, and it does not say the rework costs anything in particular until somebody measures the handling time. RTY sizes the problem and points at the step. The cause is still the next question.
Dataset: ds-order-fulfilment-yield — the same data loads in the tool below, so you can reproduce every figure here yourself.
Your turn
The tool opens with the fulfilment data.
- Confirm RTY is 84.9% while the average of the step yields is 96.80%. The tool prints both next to each other on purpose.
- Raise pick and pack from 940 to 990 passing first time. RTY goes to 89.4%. Now instead raise despatch from 990 to 1000 — a perfect step — and watch RTY move by 0.9 points. Effort at the best step buys almost nothing.
- Add a sixth step: an inspection, passing 990 of 1,000. RTY falls to 84.1%. Adding a check to catch defects lowers the rolled yield, because inspection is a step and steps have yields. This is the result that changes how people design processes.
- Set every step to 950 of 1,000 and confirm 77.4% — five steps at 95% each.
Rolled Throughput Yield
practice| Step | Units in | Passed first time | FTY | Remove |
|---|---|---|---|---|
| 96.0% | ||||
| 98.0% | ||||
| 94.0%worst | ||||
| 99.0% | ||||
| 97.0% |
Rolled throughput yield
84.9%
Normalised yield
96.78%
DPMO · one opportunity per step
32,000
Process sigma · includes the 1.5σ shift
3.35
The gapAveraging the step yields gives 96.80%. The real answer — the probability a unit gets through every step with no rework — is 84.9%. Of 1,000 units, about 151 need rework somewhere. That capacity exists, is staffed and is paid for, and it appears in no step’s report.
Normalised yield (96.78%) is the per-step yield you would need at every step to end up with this RTY. It is the geometric mean of the step yields, which is why it lands close to the plain average (96.80%) whenever the steps are similar — the two are near-identical here, and neither of them is the RTY. Normalised yield is for comparing processes with different numbers of steps. It is a poor target, because the fix is nearly always at the worst step rather than spread evenly.
DPMO counts defects, not defective units. 160 step failures against 5,000 opportunities — every unit entering a step is one chance to fail there, so the opportunity count comes from the table rather than from a box you type in. Feeding it the 151 defective units instead would count each unit once and then divide by the number of steps a second time, which raises the sigma level without touching the process. The process sigma includes the conventional 1.5 shift; the long-term Z the defect rate actually implies is 1.85.
How this is calculated
- First time yield per step = units passing first time ÷ units in. “First time” excludes anything reworked, however quickly.
- Rolled throughput yield = the product of the step yields, not their average. It is the probability a single unit clears every step untouched.
- Normalised yield = RTY1/steps — the geometric mean per-step yield.
- DPMO = defects ÷ opportunities × 10⁶, where a defect is a unit failing a step and an opportunity is a unit entering one — so the denominator is the incoming volumes summed across the steps, which is units × steps when the volumes are equal. Defective units in that numerator is the classic error: it counts a unit that failed three steps once, then divides by three opportunities anyway.
- Process sigma = Φ⁻¹(1 − DPMO/10⁶) + 1.5. The 1.5 is a convention — Motorola’s allowance for long-term drift — not a derivation, and it is worth 1.5 sigma of flattery to whoever quotes the number without it.
Source: Montgomery, D.C., Introduction to Statistical Quality Control 7e, Sec. 1.4 and Ch. 6; George, M. et al. (2005), The Lean Six Sigma Pocket Toolbook, rolled throughput yield and the hidden factory.
Check yourself
No hints. Wrong answers are explained, not softened.
A four-step process has first time yields of 99%, 98%, 97% and 96%. What is the rolled throughput yield?
A team proposes adding a 99% inspection step to a three-step process currently at 90% RTY. What happens to RTY, and what does that mean?
Two processes both report 92% RTY. One has 3 steps, the other has 15. Which statistic lets you compare their steps, and what does it say?
Worth remembering
How is rolled throughput yield calculated?
Multiply the first time yields of every step. It is the probability one unit clears the whole process with no rework — never the average of the step yields.
What is the hidden factory?
The rework capacity implied by the gap between the average step yield and the RTY. It is staffed and paid for, and it appears in no report because reporting is organised by step and no step is failing.
What happens to RTY when you add an inspection step?
It falls. Inspection is a step, and every step multiplies the yield down — a 99% check on a 90% process gives 89.1%. Escaped defects may improve, but throughput yield does not.
Can you do this now?
Rate yourself honestly. We compare your rating with how you actually answered — the gap is more useful than either number alone.
I can calculate defects per unit and first time yield from inspection data.
Your rating is recorded alongside your drill results. Neither alone marks the competency as met.