Proper method for calculating average visual acuity

TLDR: Once the authors have chosen to compare vision as a ratio using a reference visual angle (20/20), a geometric progression results and a geometric mean must be calculated for a meaningful result.

Abstract: C alculating the average visual acuity and standard deviation on a series of patients is not difficult , but has been done incorrectly in most studies. 1 The basic problem relates to the difference between the arithmetic and geometric mean for a set of numbers. For the correct average visual acuity, the geometric mean must be used, which gives significantly different values than the arithmetic mean. Modern visual acuity charts are designed so that the letter sizes on each line follow a geometric progression (ie, change in a uniform step on a logarithmic scale). 2-4 The accepted step size has been chosen to be 0.1 log unit steps, which is equivalent to letter sizes changing by a factor of 1.2589 between lines. This standard gave rise to the LogMAR (log of the minimum angle of resolution) notation, as shown in Table 1. A geometric progression of lines on the visual acuity chart was chosen because it parallels the way our visual system functions. If patient #1 has a visual acuity of 20/20 and patient #2 has a visual acuity of 20/40, we conclude that patient #1 has two times better visual acuity than patient #2 because he or she can recognize a letter twice as small. Once we have chosen to compare vision as a ratio using a reference visual angle (20/20), a geometric progression results and a geometric mean must be calculated for a meaningful result. Notice in Table 1 that the only values that increase linearly are the line numbers and the LogMar notation. The Snellen acuity, decimal acuity, and visual angle all increase by the geometric factor of 1.2589. Once we decide that equal steps in visual acuity measurement are geometric and not arithmetic , we must use the appropriate geometric mean to compute the correct average (Figure). In Table 1 and the Figure, we see that line 0 is the 20/20 Snellen acuity that corresponds to the LogMAR value zero, since 20/20 is the standard. We also see that line 10 is the 20/200 Snellen visual acuity that corresponds to a LogMAR value of +1.00 (ten times or 1 log unit worse than 20/20). Intuitively, it would appear that halfway between line 0 and line 10 would be line 5, or 20/63. This is the correct average, because geometrically it is halfway between 20/200 and 20/20. The two incorrect methods would be to take the arithmetic …