A learning curve is a graphical representation of the relationship between how proficient people are at a task and the amount of experience they have. Proficiency (measured on the vertical axis) usually increases with increased experience (the horizontal axis), that is to say, the more someone, groups, companies or industries perform a task, the better their performance at the task. The common expression "a steep learning curve" is a misnomer suggesting that an activity is difficult to learn and that expending much effort does not increase proficiency by much, although a learning curve with a steep start actually represents rapid progress. In fact, the gradient of the curve has nothing to do with the overall difficulty of an activity, but expresses the expected rate of change of learning speed over time. An activity that it is easy to learn the basics of, but difficult to gain proficiency in, may be described as having "a steep learning curve". The learning curve may refer to a specific task or a body of knowledge. Hermann Ebbinghaus first described the learning curve in 1885 in the field of the psychology of learning, although the name did not come into use until 1903. In 1936 Theodore Paul Wright described the effect of learning on production costs in the aircraft industry. This form, in which unit cost is plotted against total production, is sometimes called an experience curve, or Wright's law.
In psychology
Hermann Ebbinghaus' memory tests, published in 1885, involved memorizing series of nonsense syllables, and recording the success over a number of trials. The translation does not use the term 'learning curve' — but he presents diagrams of learning against trial number. He also notes that the score can decrease, or even oscillate. The first known use of the term 'learning curve' is from 1903: "Bryan and Harter (6) found in their study of the acquisition of the telegraphic language a learning curve which had the rapid rise at the beginning followed by a period of slower learning, and was thus convex to the vertical axis." Psychologist Arthur Bills gave a more detailed description of learning curves in 1934. He also discussed the properties of different types of learning curves, such as negative acceleration, positive acceleration, plateaus, and ogive curves.
In economics
History
In 1936, Theodore Paul Wright described the effect of learning on production costs in the aircraft industry and proposed a mathematical model of the learning curve. In 1952, the US Air Force published data on the learning curve in the airframe industry from 1940 to mid-1945. Specifically, they tabulated and plotted the direct man-hour cost of various products as a function of cumulative production. This formed the basis of many studies on learning curves in the 1950s. In 1968 Bruce Henderson of the Boston Consulting Group (BCG) generalized the Unit Cost model pioneered by Wright, and specifically used a Power Law, which is sometimes called Henderson's Law. He named this particular version the experience curve. Research by BCG in the 1970s observed experience curve effects for various industries that ranged from 10 to 25 percent.
Models
The main statistical models for learning curves are as follows:
Wright's model ("log-linear"): y = K x n {\displaystyle y=Kx^{n}} , where
y {\displaystyle y} is the cost of the x {\displaystyle x} -th unit,
x {\displaystyle x} is the total number of units made,
K {\displaystyle K} is the cost of the first unit made,
n {\displaystyle n} is the exponent measuring the strength of learning. Plateau model: y = max ( K x n , K 0 ) {\displaystyle y=\max(Kx^{n},K_{0})} , where K 0 {\displaystyle K_{0}} models the minimal cost achievable. In other words, the learning ceases after cost reaches a sufficiently low level. Stanford-B model: y = K ( x + B ) n {\displaystyle y=K(x+B)^{n}} , where B {\displaystyle B} models worker's prior experience. DeJong's model: y = K ( M + ( 1 − M ) x n ) {\displaystyle y=K(M+(1-M)x^{n})} , where M {\displaystyle M} models the fraction of production done by machines (assumed to be unable to learn, unlike a human worker). S-curve model: y = K ( M + ( 1 − M ) ( x + B ) n ) {\displaystyle y=K(M+(1-M)(x+B)^{n})} , a combination of Stanford-B model and DeJong's model. The key variable is the exponent n {\displaystyle n} measuring the strength of learning. It is usually expressed as n = log ( ϕ ) / log ( 2 ) {\displaystyle n=\log(\phi )/\log(2)} , where ϕ {\displaystyle \phi } is the "learning rate". In words, it means that the unit cost decreases by 1 − ϕ {\displaystyle 1-\phi } , for every doubling of total units made. Wright found that ϕ ≈ 80 % {\displaystyle \phi \approx 80\%} in aircraft manufacturing, meaning that the unit cost decreases by 20% for every doubling of total units made.
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