# Interval dispersion in SuperMemo

This text is part of: "*History of spaced repetition*" by Piotr Wozniak (June 2018)

## Definition

**Interval dispersion**, in spaced repetition, is the practice of distributing intervals around the optimum interval value. For example, if the Algorithm SM-18 computes that the optimum interval for a given item is 365 days, **interval dispersion** may result in scheduling a review in 400 days instead. **Interval dispersion** is employed for two reasons: (1) it speeds up the optimization process used by the spaced repetition algorithm, and (2) it prevents clustering of items on particular days.

## SuperMemo 1.0 for DOS (1987)

The first version of SuperMemo software was SuperMemo 1.0 for DOS in 1987. It used a spaced repetition algorithm that did not change much for the next two years (described as Algorithm SM-2). In early SuperMemos, there was no **interval dispersion**. As a result, adding a great deal of items on a single day, e.g. on the weekend, would result in clusters of items reviewed at the same time. If 70-80% of items scored a good grade, they would often reoccur on the same day in the future. As early SuperMemos did not have any rescheduling tools, this item clustering was often mentioned in the criticism of the program.

## SuperMemo 4.0 for DOS (1989)

In 1989, a new spaced repetition algorithm (Algorithm SM-4) would attempt to dynamically and incrementally modify the matrix of optimum intervals during the learning process. In essence, the algorithm was running a live version of the first spaced repetition experiment that lead to determining the outline of the function of optimum intervals (1985). A serious flaw of the Algorithm SM-4 was its agonizingly slow convergence. This is why it was promptly replaced with the Algorithm SM-5 that included **interval dispersion** for the first time.

## SuperMemo 5.0 for DOS (1989)

Later in 1989, SuperMemo replaced optimum intervals with O-factors. In addition, it included a simple **interval dispersion** procedure that survived in SuperMemo for 30 years. See: Random interval dispersion in SuperMemo 5.0.

## Algorithm SM-18 (2019)

Algorithm SM-18 introduced first improvements to **interval dispersion** in 30 years. The two-argument procedure was replaced with one-argument procedure. Instead of `Interval`

and `Modifier`

arguments, the procedure only requires `Interval`

on input. In addition, a symmetric distribution was replaced with a distribution that favors longer intervals (to account for the spacing effect).

The dispersion brackets determined heuristically defines the maximum and minimum values of actual intervals for a given optimum interval:

Figure:Interval dispersion bracket:In Algorithm SM-18, intervals are distributed randomly around the optimum interval value. The brackets of dispersion are shown in red and blue in the graph. For example, for optimum interval of 10 days (horizonal line), in the process of interval dispersal, the minimum interval used cannot fall below 5 days (50% shown by the blue line). This corresponds with a spread quotient of 0.5

For example, for optimum interval of 365 days, the distribution of items around the optimum value may look as follows:

Figure:Exemplary simulation of interval dispersion in SuperMemo 18. The graph shows the distribution of intervals around the optimum value of 365 days.Item counton the vertical axis shows how many items in a sample of n=100,000 draw intervals of a given length (Intervalon the horizonal axis, in days)

This text is part of: "*History of spaced repetition*" by Piotr Wozniak (June 2018)