Learning curves

In one line

The time per unit is not constant at the start: it falls at a regular rate each time the total produced doubles, until it flattens out once the operator masters the task.

What it is

Niebel and Freivalds explain that learning a task takes time: even the simplest operation can take hours, and complicated work days or weeks, until the operator achieves the mental and physical coordination to move from one element to the next without hesitation or delays. That period and the associated level of learning make up the learning curve. Once the operator reaches the flat part, rating their performance becomes simpler, but it is not always practical to wait that long to set a standard: the analyst sometimes needs to set it in the initial part, where the slope is steeper. Having representative curves for each type of work helps there.

The theory: when the total number of units produced doubles, the time per unit decreases by a constant percentage. With a learning rate of 80%, doubling production cuts the time by 20%. The book's table starts from 12 minutes for the first unit:

Unit number	1	2	4	8	16	32
Time (min)	12.00	9.60	7.68	6.14	4.92	3.93

Mathematically it is a power curve: y = k · xⁿ, where y is the time of unit number x, k is the time of the first unit and n is the slope, which is calculated as the logarithm of the learning percentage divided by the logarithm of 2. For 80%, n = −0.322. The book gives the equivalence between percentage and slope: 70% is −0.514, 75% is −0.415, 80% is −0.322, 85% is −0.234, 90% is −0.152 and 95% is −0.074.

Two clarifications from the book. First, counterintuitively, 70% is the highest learning rate (the lower the percentage, the faster the learning) and 100% represents no learning at all, as in a completely automated operation. Second, the typical rates: large-scale or non-assembly work (such as aircraft), 70 to 80%; welding, 80 to 90%; machining, 90 to 95%.

Two models

The book distinguishes the Crawford unit model (1944), in which the improvement corresponds to a particular unit, which is the one it develops, and the Wright model (1936, aviation industry), also called the cumulative average model, where the improvement corresponds to the cumulative average unit. The latter gives cumulative average values higher than the unit cost of the nth unit, although the models converge as the quantity grows. Some analysts prefer Crawford because it does not mask individual variability.

How to apply it

Niebel and Freivalds mention three uses: adjusting standards for new workers, adjusting standard data for small batches, and knowing how many repetitions it takes to reach a target time. A practical point: standard data almost always reflect performance once learning has leveled off, so for small quantities you have to adjust them upward. In the book, a complex assembly whose standard data give 1.47 hours at unit 300 and with a rate of 95% (n = −0.074) has a first unit of 2.24 hours: that is the basis for estimating the cost of the first assemblies, not the 1.47 hours.

Calculate your curve

Real example

Example 18.1 from the book. It takes 20 minutes to produce unit 50 and 15 minutes for unit 100. The slope is n = (log 20 − log 15) ÷ (log 50 − log 100) = −0.4152, which means learning of 2^−0.4152 = 75%. Solving for k with the point (50, 20 minutes) gives 101.5 minutes for the first unit, not the 10 minutes that came out of the standard data. To reach a standard time of 10 minutes, about 265 cycles are needed, and the total time spent up to that point is about 4,424 minutes (73.7 hours), an average of 16.7 minutes per unit. The two sections of the calculator reproduce these numbers (the book approximates the total by integrating and the calculator adds up unit by unit, so they can differ slightly, and the rounding of the exponent moves the 265 to 265 or 266).

Backsliding. If the operator stops (for example, vacation), they forget part of what they learned. The book calls this backsliding and approximates it with a straight line between the first-cycle time and the standard time. In example 18.2, the operator from the previous case takes two weeks of vacation after 50 cycles: their cycle 51 takes 84.17 minutes instead of the 19.84 it would have taken without interruption, a backslide of 64.33 minutes, and they start a new curve from there.

Template to use

Unit number	Observed time (min)	Time projected by the curve (min)	Difference
1
2
4
8
16

Time units 1, 2, 4, 8, 16... with a stopwatch (each time the count doubles) and compare them with the projection. The actual learning percentage comes from two points, using the second section of the calculator.

So you do not have to build it by hand, there is a learning curve spreadsheet in Excel with the formulas already in place: it compares observed times with projected ones and calculates the learning percentage from two measurements. The values it comes with are examples and you need to delete them.

Benefits

• It lets you estimate the time of the first units in small batches or new products, without waiting for the operator to stabilize.
• It is useful for setting standards for new workers and for planning training.
• It helps you understand why a newly launched operation does not meet the standard and when it should.

Limitations to keep in mind

• Many factors affect human learning: the book mentions the complexity of the work (the longer the cycle, the more uncertainty in the movements; the more simultaneous movements, the more training) and individual characteristics such as age. The percentage is an average rate, not a prediction for each person.
• The curve assumes continuous repetition: interruptions cause backsliding and the curve has to be restarted.
• With rates close to 100% (automated operations) the curve barely exists and the calculation of "units to reach the target" makes no sense.

In summary

Every time the total produced doubles, the time per unit drops by a constant percentage: y = k · xⁿ. With the task's typical learning rate you can project the first times, decide when to set a standard and estimate the effect of an interruption. It complements the time study and stopwatch timing.