# Spurious Power Laws of Learning and Forgetting: Mathematical and Computational Analyses of Averaging Artifacts

- Jaap Murre,
*University of Amsterdam*
- Antonio Chessa,
*Statistics Netherlands (CBS)*

## Abstract

It has frequently been claimed that learning performance improves
with practice according to the so-called “Power Law of Learning”.
Similarly, forgetting may follow a Power Law. It has been shown on the basis of
extensive simulations that such Power Laws may emerge as artifacts through
averaging functions with other shapes. Here, we present a mathematical analysis
that power functions will indeed emerge as a result of averaging over exponential
functions, if the distribution of learning rates follows a gamma distribution.
Power Laws may, thus, arise as a result of data aggregation over subjects or
items. Through a number of simulations we further investigate to what extent
these findings may affect empirical results in practice. We conclude that
spurious Power Laws will be more likely with large numbers of subjects and
shorter time scales and with gamma distributions with much probability mass close
to zero and with a not too low variance.

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