# 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.

The word "normalization" can mean many different things, as this page and this page nicely illustrate. It doesn't necessarily mean making the transformed values "appear as though they have a Gaussian distribution," although it often does.

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.

If you want to force the transformed grades to match a chosen or estimated Gaussian distribution, you could place the grades in order to determine their quantiles (percentiles), then use the inverse of the Gaussian distribution to get the values along the real line to which those quantiles would correspond. This is called the "rankit," which when plotted with the actual quantiles in a normal probability plot provides a nice graphical display of how well the original data match what would be expected from a Gaussian distribution. If the original data don't follow a Gaussian distribution, however, grades close to each other in the original scale could be quite far from each other in this type of transformed scale.

Power transforms like the Box-Cox transformation can be used to make the distribution of data points seem more like those expected of a Gaussian distribution while being less likely to lead to wide discrepancies of transformed values between close neighbors on the original scale.

• Thank you for the correction. I suppose that instead of "Gaussian distribution" I should have used "Bell shaped histogram". So using rankit you would project the grades on the "normal" linear function and then use the projection as the transformation? – D.M. Aug 30 '18 at 16:24
• @D.M. yes, if I understand you correctly, although it's not a "'normal' linear function." Take each percentile and figure out what x-axis value has that percentile for the chosen Gaussian mean and SD. For example, to force a Gaussian with mean of 80 and SD of 10 onto a set of grades, the grade at the 16th percentile has a transformed grade of 70, the median grade has a transformed grade of 80, and the grade at the 84th percentile has a transformed grade of 90. – EdM Aug 30 '18 at 19:15