Distance and width affect time logarithmically, not linearly

What it is

Fitts's-law index of difficulty is commonly written log₂(1 + D/W): the ratio of distance D to effective width W enters logarithmically rather than directly. Doubling distance or halving width adds roughly one comparable step of difficulty. It does not mean every additional pixel adds the same amount of time.

Why it happens

People do not pay a separate control cost for every pixel. They progressively reduce uncertainty about the endpoint. A farther target with a narrower tolerance requires more discriminations of position near the end of the movement. The logarithm expresses this proportional demand in bit-like units: equal multiplicative changes yield comparable increments at different absolute scales.

Studying it

Sample many D/W combinations, then fit movement time against index of difficulty rather than comparing only a few sizes. Report slope, intercept, misses, and residuals to distinguish a fixed start-up cost from degraded high-difficulty performance. Estimate effective width from the observed endpoint distribution; geometric width alone can make conditions with different error strategies look comparable when they are not.

Where it stops holding

The logarithmic form is a model, not a pixel-to-milliseconds converter. Very small targets expose resolution, tremor, and sampling limits; very large ones can be dominated by reaction and visual-confirmation time. Parameters from different formulae or devices cannot simply be exchanged. With few conditions, high error rates, or substantial target search, claiming a logarithmic relation is not informative.

Applying it

• Compare layouts proportionally, not only in pixels: doubling travel or halving hit width should both be treated as meaningful increases in difficulty.
• Target unusually high D/W combinations, such as small icons reached from far away, rather than distributing minor changes across every control.
• In prototype tests log hit area, start point, and trajectory, then compare time and misses by difficulty band instead of relying on one overall mean.

Related

• Same group: B1.01.1 Pointing time is jointly determined by target distance and width · B1.01.3 Width is the effective size along the movement direction · B1.01.4 The law describes an average trend, not one operation · B1.01.5 Enlarging a target and shortening travel have different costs
• Nearby: B1.02 Fitts's law on touchscreens · B1.05 Steering law
• Search terms: index of difficulty · Shannon formulation · effective width