Evidence for a normal distribution of normalized gains

Recently Burkholder et al. argued that class normalized gains over the entire population of courses is approximated by a Cauchy distribution, not by a normal distribution, and therefore should not be used to compare different classes because means and standard deviations cannot be calculated. They argued that multiple linear regression should be used instead. Burkholder et al. provided very little data in support of their belief, but instead provided a theoretical argument with certain assumptions. We shall show that these assumptions are unreasonable. We also show explicitly how normalized gains from 114 clases taught by six different instructors are indeed normally distributed and how two other even larger samples are roughly normally distributed and are not remotely like a Cauchy distribution. The continued use of normalized gains by physics instructors is therefore justified.

Figure 1

Comparison of the distribution of class normalized gains for 114 IE classes with similar Lawson scores, median of 0.46, and a standard deviation of 0.10 (blue) with a normal distribution having the same median and SD (red).

Figure 2

Comparison of distribution of class normalized gains for 382 IE classes from Physport (blue) with a normal distribution (red) having the same mean (0.30) and standard deviation (0.14).

Figure 3

Comparison of the distribution of class normalized gains for 189 IE classes reported by Von Korff (blue) with a normal distribution (red) having the same median (0.40) and standard deviation (0.14).

Figure 4

Comparison of a normal distribution (dashed line) and a Cauchy distribution (solid line).

Figure 5

Average FCI normalized gain for each Lawson score from a sample of 1432 individual $G$’s and Lawson scores; standard error for each $\text{average}=0.02-0.04$.