Quicksort algorithms for data fitting in Gaussian distribution curve

We know that Quicksort algorithm provides the best case runtime when the median is chosen as a pivot. But for any arbitrary data, for finding median you need to sort the array first. But for data which fits in a Gaussian distribution, eg Marks of students in an exam, since we know the probability distribution function if we calculate the mean and standard deviation, it is easier to choose the pivot since we can approximate the median points in each step using the probability distribution function. Write a code to verify this theory, take a dataset that follows Gaussian distribution, and sort it first using conventional quicksort algorithm taking a random pivot, and then sort it by approximating the Pivot using the Probability Distribution Function of Gaussian distribution. Compare the runtime for both cases, do it for several such datasets and compare the runtime of both algorithms as a function of array size. Finally show the two variations of runtime as a function of array size in a graph. Refer to the proposed idea is the presentation.