Not to be confused with the logarithmic pointing law — the applicable conditions differ

What it is

The linear growth of error with speed and the better-known logarithmic law — where movement time grows with the logarithm of the distance-to-width ratio — answer questions posed by two different task structures, and neither can substitute for or be tested against the other. Their independent variables, dependent variables, and task preconditions are entirely different; both sound, on the surface, like "going faster makes you less accurate," but they are two independent quantitative relationships.

Why it happens

Telling the two apart comes down to the task structure each one presupposes. The logarithmic law holds when: target width and position are fixed, and time is a free variable — the user decides how long to take, the dependent variable of interest is "how long does completion take," and error is held at a roughly constant level (typically maintained through mid-flight visual correction). The linear relationship holds under the opposite structure: the time allowed is externally fixed — there is no room to negotiate it — and the task itself has no bounded target to hit; the dependent variable of interest becomes "how far off does the endpoint land," and this deviation is no longer suppressed by mid-course correction, because the movement happens too fast for correction to occur at all.

The two relationships can even apply to the very same physical movement, viewed from two different angles: for one fast swing, asking "how quickly can this be done while still landing inside some range" invokes the logarithmic law; asking "given that it had to be done this fast, how far off did it actually land" invokes the linear relationship described here. The classic mistake is applying the logarithmic law's conclusions to predict error in a time-constrained, targetless task — because the logarithmic relationship grows very slowly with increasing difficulty, it badly underestimates how steeply error actually climbs with speed under a genuine time constraint. The reverse mistake — applying the linear model to a discrete pointing task with a well-defined target and unconstrained time — ignores that the user could simply slow down and use mid-flight visual correction to hold error near a roughly constant level, wrongly predicting that error keeps climbing linearly with the required speed with no ceiling.

Studying it

Determining which model applies to a specific task requires answering two prior questions rather than reaching straight for a formula: is the movement's duration chosen by the participant, or is it externally imposed? Does the movement have a bounded target that must be hit or missed? The combination of answers determines whether "completion time" or "endpoint deviation" should be the dependent variable measured, and whether the expected functional form is logarithmic or linear. Mixed designs — a target of moderate width paired with a vague time cap, say — need particular caution: the measured relationship may be a blend of both mechanisms, and neither pure-form conclusion can simply be applied wholesale.

Where it stops holding

This distinction depends on correctly reading the task structure in the first place. Many real interface actions sit between the two extremes — a target that isn't especially narrow, paired with a time cue that's present but not especially tight — and the error-versus-speed relationship in that middle ground may be neither purely logarithmic nor purely linear. Such cases need to be measured directly for the specific scenario rather than assumed to follow either formula on theoretical grounds alone.

Related

• Same group: A8.10.1 For a movement completed within a fixed time, error grows in proportion to speed · A8.10.2 This relationship applies to fast movements with no well-defined target width
• Nearby: A8.09 Speed-accuracy trade-off
• Search terms: Schmidt's Law · Fitts' Law · impulse-variability theory · index of difficulty