The article's own example: 85% curve, T1 = 10,000 h, b = ln(0.85) ÷ ln(2) = −0.2345, so unit X takes 10,000 × X^−0.2345. The doubling ladder is 1 → 10,000 h, 2 → 8,500, 4 → 7,225, 8 → 6,141, 16 → 5,220, 32 → 4,437, 64 → 3,771 — each 85% of the one before, while the hours saved fall 1,500, 1,275, 1,084, 921, 783, 666. Both panels plot those seven points and nothing else; the left has linear axes with an untruncated 0–10,000 h scale, the right has both axes logarithmic, where a constant ratio per doubling is a straight line of slope b. Continuing the curve: unit 100 = 10,000 × 100^−0.2345 = 3,397 h, or 34% of T1.
The Observation That Changed Production Planning
In 1936, T.P. Wright observed that labor hours per aircraft at Curtis-Wright decreased by a consistent percentage each time cumulative production doubled. Unit 1 took 100,000 hours. Unit 2 took 80,000. Unit 4 took 64,000. Unit 8 took 51,200. The ratio between doublings was constant: 80%. This was not coincidence — it was a fundamental property of repetitive human work.
The reason is straightforward: as workers repeat a task, they develop more efficient methods, better tool handling, fewer errors, and smoother coordination. Tooling and fixtures improve. Processes are refined. Supply chains stabilize. Each of these improvements contributes a small reduction in hours per unit, and the cumulative effect follows a predictable mathematical pattern.
The Crawford (Unit) Model
The Crawford model predicts the hours for any individual unit:
Yx = T1 × Xb
Where:
- Yx = hours for unit X
- T1 = theoretical first unit hours
- X = cumulative unit number
- b = ln(learning rate) ÷ ln(2)
Example: 85% learning curve, T1 = 10,000 hours
b = ln(0.85) ÷ ln(2) = –0.1625 ÷ 0.6931 = –0.2345
| Unit | Calculation | Hours | % of T1 |
|---|---|---|---|
| 1 | 10,000 × 1–0.2345 | 10,000 | 100% |
| 2 | 10,000 × 2–0.2345 | 8,500 | 85% |
| 4 | 10,000 × 4–0.2345 | 7,225 | 72% |
| 10 | 10,000 × 10–0.2345 | 5,828 | 58% |
| 50 | 10,000 × 50–0.2345 | 3,996 | 40% |
| 100 | 10,000 × 100–0.2345 | 3,397 | 34% |
By unit 100, the hours have dropped to 34% of the first unit. This is the power of the learning curve — and why it is critical for program planning, pricing, and forecasting.
Crawford vs. Wright
| Model | Predicts | Formula | When to Use |
|---|---|---|---|
| Crawford (Unit) | Hours for each specific unit | Yx = T1 × Xb | Estimating specific unit costs, production planning, make-or-buy analysis |
| Wright (Cumulative Average) | Average hours for all units 1 through X | ĀYx = T1 × Xb (where ĀY is the cumulative average) | Total program cost estimation, lot pricing, cumulative budget forecasts |
⚠️ The Same Percentage Means Different Things
An “85% learning curve” produces different unit hours depending on whether it is Crawford or Wright. Always specify which model you are using. In aerospace, Crawford (unit) is more common for production planning. Wright (cumulative average) is more common for pricing and total cost estimation. Using the wrong model can produce 10–15% estimation errors on large programs.
T1: The Theoretical First Unit
T1 is not the actual hours for unit 1 — it is the theoretical first unit hours derived from the learning curve regression. In practice, actual unit 1 hours are often higher than T1 because unit 1 includes one-time setup, first-article inefficiencies, and process debugging that are not part of the repeatable learning pattern.
| T1 Source | Method | Accuracy |
|---|---|---|
| Engineering estimate | Bottom-up estimate of first-unit labor by operation | ±20–30% (before production data exists) |
| Analogous program | Scale T1 from a similar product using weight, complexity, or feature ratios | ±15–25% |
| Regression from actuals | Plot actual units on log-log paper, fit curve, extrapolate to X=1 | ±5–10% (best, requires production data) |
Typical Learning Rates
| Product Type | Typical Rate | Why |
|---|---|---|
| Airframe assembly | 80–85% | High labor content, complex manual operations, significant method improvement opportunity |
| Systems integration | 85–90% | Mix of labor and testing, some tasks are labor-intensive but test procedures stabilize |
| Machined components | 90–95% | Machine-paced operations, less labor variability, learning mostly in setup and handling |
| Electronics assembly | 85–92% | Repetitive but precise, learning in component placement and rework reduction |
| Composite fabrication | 82–88% | Manual layup processes, cure cycle learning, tooling refinement |
🎯 The Bottom Line
Learning curves are not optional — they are a mathematical reality of repetitive production. The Crawford model (Y = T1 × Xb) predicts unit hours. The Wright model predicts cumulative averages. Typical aerospace assembly rates are 80–85%. T1 estimation accuracy improves dramatically once you have actual production data to regress. Every program plan, bid, and EAC that does not account for learning curves is wrong by definition. Next: Aerospace Learning Curve Application — applying these models to real programs with lot midpoints, rate adjustments, and multi-shop curves.
Interactive Demo
Move the learning rate and watch what a few points are worth across a whole lot. Then flip to log-log, where the curve becomes the straight line you can judge a fit against.
Stop reading, start modeling
Model your process flow, run simulations, optimize staffing with TOC math, and test your knowledge with 107 interactive checks — all in one platform.