# Transforming a given distribution into a normal distribution

The grading scheme in the my university is based on a 0-100 range. In some courses, some kind of normalization is applied to the course score. The simplest such normalization is simply adding some fixed amount of points to every grade (to shift the mean upwards to some desired value). I’ve recently encountered these two:

$$\text{Normalized grade}=83\left(\frac{\text{Grade}}{83}\right)^{0.64}+10$$ $$\text{Normalized grade}=4.16\cdot\min\left(\text{Grade}+10,100\right)^{0.69}$$

I assume the goal is to make the transformed grades appear as though they have a Gaussian distribution, which got me thinking - how does one come up with such a transformation? Say for some desired mean $\mu$ and variance $\sigma^{2}$? The two examples above do not look like they were derived from the same closed form.

A Gaussian distribution should provide probabilities for arguments over the entire real number line, $(- \infty, \infty)$. Both transformations you cite in your question map non-negative grades into necessarily positive transformed values, so strictly they aren't producing normal distributions. These seem to be ad hoc attempts chosen by trial and error to look good when you plot histograms of the transformed grades, and perhaps also to meet a university's idea of what an average grade and the distribution of grades among students taking a course should be.