For a movement completed within a fixed time, error grows in proportion to speed

What it is

In a movement that must be completed within a fixed time and has no well-defined target width to land inside — swinging an arm toward a rough direction within 200 milliseconds, say, where the endpoint is a heading or an approximate distance rather than a bounded target — the spatial error of the endpoint doesn't grow slowly like a logarithm of speed; it grows linearly with average movement speed: double the speed, roughly double the error. This linear relationship is Schmidt's Law, also known as impulse-variability theory, and it describes the most basic quantitative relationship between speed and endpoint variability in time-constrained movements.

Why it happens

The muscular torque driving a fast movement is generated by a single impulse — force applied over a duration — and in the human motor system, the variability of the neural drive signal during a muscle contraction scales in proportion to the size of the torque being produced: going faster requires exerting more force in less time, and the noise carried by that force scales up right along with it. The spatial variability of the endpoint is ultimately the accumulated effect of this force/timing variability over the trajectory, and that accumulated quantity is mathematically proportional to average speed — which is exactly why the observed relationship is linear, unlike the logarithmic form seen in aimed movements toward a defined target width where in-flight correction is possible.

Studying it

The standard paradigm is a timed throw or timed swing: participants are given a fixed total time to complete the movement (set by a metronome or a visual/auditory cue) and asked to move as far and as straight as possible within that window; the deviation of the endpoint from the intended direction or distance is measured. The independent variable is the required movement speed (distance divided by the allotted time); the dependent variable is the standard deviation of the endpoint deviation.

Methodological note: this paradigm requires making sure participants genuinely have no time to look and correct mid-movement — if the movement is too slow or the allotted time too generous, participants will unconsciously insert visually guided corrections, and the measured relationship then mixes in other motor-control components and is no longer a pure linear relationship.

Where it stops holding

This linear relationship holds reliably only when the movement is genuinely brief, driven by a single impulse, and offers no chance for mid-course correction; once the movement is allowed to run long enough for visual feedback to intervene with a secondary correction, the error-versus-speed relationship is no longer purely linear. This relationship also describes the distribution of endpoints for the same person repeating the same timed movement — it doesn't mean any two different movements can be compared on the same line; the proportionality constant varies by person, by movement type, and by which muscle groups are involved.

Related

• Same group: A8.10.2 This relationship applies to fast movements with no well-defined target width · A8.10.3 Not to be confused with the logarithmic pointing law — the applicable conditions differ
• Nearby: A8.09 Speed-accuracy trade-off · A8.06 Open-loop control
• Search terms: Schmidt's Law · impulse-variability theory · motor timing