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Error Correction in Context 281

an error. In this scenario, change in collective performance over time ought to
be derivable from the learning of the participating individuals. If a collective S
consists of a population of N individual learners s s , s , … s , and if each of the
N
1, 2
3
latter is accurately described by a learning curve with a gradually and smoothly
declining learning curve, then what is the resulting learning curve for S?
The answer is that the aggregated learning curve is of the same shape as
the individual curves. For example, Figure 8.6 shows the result of aggregating
over 50 independent, simulated agents who were assumed to learn in accor-
dance with an exponential equation. The simulated agents were assumed to
have no relevant prior experience (i.e., the E parameter was set to zero) but
to vary in cognitive ability. The B parameter – performance on the first trial –
was assumed to be normally distributed with a mean of 100 and a standard
deviation of 10, while the a parameter – the learning rate – was assumed to be
normally distributed with a mean of 1 and a standard deviation of 0.50. The B
distribution was chosen arbitrarily but the distribution of the a parameter was
guided by empirically found values. The resulting curve is negatively acceler-
ated. Although the individual curves are exponential, the aggregated curve
exhibits nearly perfect fit to a power law equation.
Some social systems approximate populations with multiple, nearly inde-
pendent operators. Transportation industries provide a stock of examples.
The occurrence of an error and a subsequent accident in a vehicle moving on
land, on the sea or in the air frequently leaves the probability of error in other
vehicles of the same type unaffected. Consider the airline system: All pilots (or
cockpit crews, if each crew seen as a single agent or operator) who fly com-
mercial airplanes form a population. It is not too gross a simplification to say
that each crew executes its flight independently of the other crews and that its
errors are its own. The commercial airline system is error free in a given period
of time only if every crew flies every flight without error in that period of time.
It is sufficient for one crew to make an error – flying too close to another air-
plane, for example – for the overall system to exhibit at least one error.
How do error rates change over time in air traffic and other transportation
industries? Romney B. Duffey and co-workers have made an extensive study of
this issue in their 2003 book Know the Risk: Learning from Errors and Accidents.
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They argue that in the case of airlines, the appropriate measure of prior experi-
ence is not elapsed time – how many years has a particular airline been in busi-
ness – but the total number of pilot flying hours. Plotting the number of near
misses, a type of error for which extensive quantitative data are available, as a
function of experience measured by total flying hours yields the error reduction
curve in Figure 8.7. As the figure shows, the curve follows a negatively accelerated