Do we compare digits and dot clouds with the same mental system?
A Weber's-law 'approximate number' model described how people compare dot clouds well, but it systematically mispredicted how they compare written digits, suggesting digits use a different system.
Source
Symbolic Number Comparison Is Not Processed by the Analog Number System: Different Symbolic and Non-symbolic Numerical Distance and Size Effects
Study at a glance
- Design
- Human experiment — Within-subject number comparison task: each participant chose the larger of two digits (1-9) and of two dot arrays (ratio-matched 5-45) over many trials; error rates, reaction times and diffusion-model drift rates were fitted against Analog Number System predictions.
- N
- N=20 · 24 university students tested; 4 excluded for high error rates, leaving 20.
- Population
- Hungarian university students aged 19-24
- Outcome
- Error rates, reaction times and EZ-diffusion drift rates for every number pair in symbolic (digit) and non-symbolic (dot) comparison
Structured fields used in claim comparison tables when every cited study has a complete layer.
What they did
20 students chose the larger of two numbers in two conditions: Arabic digits from 1 to 9, and dot arrays using the same ratios scaled up to 5-45 dots to avoid instant counting. Every pair was shown many times, giving 720 trials per condition. The authors fitted Analog Number System (ANS) predictions to error rates, compared reaction-time patterns across notations, and used a diffusion model to recover drift rates (the quality of evidence) for each pair.
What they found
For dots, one Weber ratio (about 0.19) predicted both the mean error rate and the shape of errors across pairs, and drift rates fell toward zero as pairs became indistinguishable, as the ANS predicts. For digits, the Weber ratio implied by mean errors (about 0.09) did not produce the best-fitting error pattern: the model predicted too many errors for close pairs and too few for medium pairs. Digit drift rates stayed well above zero even for the hardest pairs, and digit and dot reaction-time patterns were not simple rescalings of each other.
The limits
What it doesn't show
The sample was small and the analysis rests on group-mean patterns across number pairs rather than a formal comparison against a fitted alternative model. The authors themselves say the reaction-time analysis follows older reasoning and is not a reliable test, and the simple EZ diffusion method has known limits. The dot stimuli did not control all visual cues such as density and area. The study argues digits are not processed by the ANS but does not identify which system does process them.
Key terms
- Analog Number System (ANS)
- A proposed noisy, approximate representation of quantity in which discriminability depends on the ratio between numbers.
- Weber's law / Weber ratio
- The principle that the just-noticeable difference grows with magnitude; the Weber ratio indexes how precise a representation is.
- Numerical distance effect
- Comparing two numbers is slower and more error-prone when they are close together.
- Numerical size effect
- At a fixed distance, comparing larger numbers is harder than comparing smaller ones.
- Drift rate
- In diffusion models of decision-making, the rate at which evidence accumulates toward a response, reflecting information quality.
- Subitizing
- Fast, accurate enumeration of very small sets of about four or fewer items.
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Quiz yourself
What did participants compare in the non-symbolic condition?
Common questions
Why can't good model fit alone support the ANS?
An alternative account, the discrete semantic system model, makes very similar predictions, so a high correlation with ANS predictions could arise even if the ANS is wrong.
Why were dot numbers multiplied by five?
Small sets of up to about four are subitized rather than estimated; scaling by five kept the same ratios while avoiding that range.
What is the single most damaging finding for the ANS account of digits?
Digit drift rates did not approach zero for near-equal pairs, which no noisy analog system obeying Weber's law should allow.
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