Fitts's Law - Mathematical Details

Mathematical Details

The logarithm in Fitts's law is called the index of difficulty ID for the target, and has units of bits. We can rewrite the law as

where

Thus, the units for b are time/bit; e.g., ms/bit. The constant a can be thought of as incorporating reaction time and/or the time required to click a button.

The values for a and b change as the conditions under which pointing is done are changed. For example, a mouse and stylus may both be used for pointing, but have different constants a and b associated with them.

An index of performance IP (also called throughput TP), in bits/time, can be defined to characterize how quickly pointing can be done, independent of the particular targets involved. There are two conventions for defining IP: one is IP = 1/b (which has the disadvantage of ignoring the effect of a), the other is IP = ID_avg/MT_avg (which has the disadvantage of depending on an arbitrarily chosen "average" ID). For a discussion of these two conventions, see Zhai (2002). Whatever definition is used, measuring the IP of different input devices allows the devices to be compared with respect to their pointing capability.

Slightly different from the Shannon formulation is the original formulation by Fitts:

The factor of 2 here is not particularly important; this form of the ID can be rewritten with the factor of 2 absorbed as changes in the constants a, b. The "+1" in the Shannon form, however, does make it different from Fitts's original form, especially for low values of the ratio D/W. The Shannon form has the advantage that the ID is always non-negative, and has been shown to better fit measured data.