Fitts' law

Shannon formulation

In 1954, Paul Fitts proposed a mathematical function to estimate the mean time required to reach a target from its distance and width. In human–computer interaction, it is often introduced as

where MT is the mean time to reach the target, D is the target's distance, W is its width, and a and b are empirically determined constants.

The index of task difficulty (ID) is the equation without the leading constants:

If the distance D increases, the ID increases, but if the width W increases, the ID decreases. However, if both D and W increase or decrease proportionally, the ID does not change.

For example, Target A and Target B have the same ID, although Target B's distance and width are double those of Target A. This property is called scale independence.

The red line is ID if D varies with W set to 1 cm, but the blue line is ID if W varies with D set to 10 cm:

The Shannon formulation of ID includes + 1. This addition improves on Fitts' original ID because it better fits observations, always results in a positive value for ID, and follows Shannon's information theorem precisely (McKenzie & Buxton, 1992).

Extending to two dimensions

Fitts' law models a lateral movement to a target along one dimension. The target's height is not considered. The question arises of how to apply Fitts' law to human–computer interaction where the position of the target varies in two dimensions and its height and width also vary. One of the most straightforward solutions is to measure the target's width along the direction of approach W'.

This solution, however, requires considering the angle of approach θ. Several other alternatives have been proposed. One that is simple and comparable in effectiveness for rectangular targets is to use the height or width, depending on which has the smaller value: 

Determining the constants a and b

The mean time is a linear function of the index of difficulty, MT = a × b ID, so a is just the y-intercept of the line, and b is its slope. Thus, these constants can be determined by fitting a line to user data, typically collected during an experiment.

In the experiment, D and W are independent variables. D has |D| levels, and W has |W| levels, yielding |D| × |W| conditions. Each condition comprises a block of trials presenting the user with the task of clicking on a target of width W at a distance D. ID is determined from D and W, and MT is the time the user takes to complete the trial. ID and MT are recorded, and a line is fit to these values by linear least squares to determine the y-intercept a and the slope b. The constants a and b for the above plot are listed below:

Pearson's correlation r = 0.89. Typical values range from 0.85 to 0.95, where a higher value indicates the equation better fits the data.

Linear least-squares regression minimizes the sum of the squared differences between the recorded values and the line.

References