# The Distance Effect

In whatever way they are presented, it is easier to compare two magnitudes that are quantitatively further apart, than it is to compare two magnitudes that are quantitatively closer together. This is the distance effect.

Figure 1: Distance Effect: 2 versus 8 and 8 versus 9

It is easier to differentiate 2 and 8 dots (on the left) to decide which is smaller or larger, than it is to differentiate 8 and 9 dots (on the right). This is because these two comparisons have different separation distances of 6 (8 – 2) and 1 (9 – 8) respectively and it is easier to recognise a difference of 6 than a difference of 1.

# The Size Effect

If the numerical distance between two magnitudes remains the same, then it is easier to compare two small magnitudes than it is to compare two larger magnitudes. This is the size effect.

Figure 2: Size Effect: 3 versus 4 and 8 versus 9

It is easier to differentiate 3 and 4 dots (on the left) than it is to differentiate 8 and 9 dots (on the right), where both comparisons have a separation distance of 1 (4 – 3) and (9 – 8) respectively.

The distance and size effects follow Weber’s Law, are present in different modalities and are present with two-digit numbers. For instance, when comparing two-digit numbers against a fixed reference such as 65, comparing 79 with 65 (difference 14) takes less time than comparing 71 and 65 (difference 6) an effect that is not subject to decade boundaries.

Figure 3: Numerical Distance Effect (from Dehaene 2011, p.64)

# Origin of Size and Distance Effect

According to Gelman and Gallistel, in *Language and the Origin of Numerical Concepts* (Gelman and Gallistel, 2004) one can imagine that repeated perceptions of a given numerosity give rise to normal signal distributions like those shown below. The wider spread the distribution is, the less precise and noisy the representation. The extent to which two signal distributions overlap, determines the likelihood of confusion as to which distribution an individual signal belongs to and the more processing time is required to resolve the situation. So, no matter how the numerosities are presented, it is much easier (takes less time) to discriminate 10 from 2 and is much harder (takes longer) to discriminate 2 from 3.

Figure 4: Mental Magnitude Signal Distributions (after Gelman and Galliestal 2004, p.442 A)

This is the generally accepted explanation of the size and distance effects. It ties basic arithmetic reasoning with numerical symbols (order judgements) to an imprecise non-verbal representation of number.

Gelman and Galliestal 2004, p.441