I have some physical experiments done at various locations. The locations produces a set of observations y for one value of x, the independent variable. In the end across a set of locations I have values in the following form

[y11, y21, y31, y41, y51, y61...] for one value of x, say x1

Then I repeat the experiment and get a new set of values

[y21, y22, y32, y42, y52, y62...] for a different value of x say x2

Ans so on.

In the end I have readings for y for 5 distinct values of x, [x1, x2, x3, x4, x5]

I wish to fit a quadratic to this data, and my main goal is to find the value of x, for which y is maximum. One way to do this is to define an average y for each x1 and fit a quadratic with the averaged out values. I know that fitting a quadratic to just 5 points is not a good idea. I am open to other ideas, that can help me solve the problem without directly fitting a functional relationship.

Some non-parametric idea for instance. One idea I have is to do some distribution analysis on values of y for x1, vs values of y for x2 and so on. This enables me to take all the y values for one x1, without averaging them. I would be open to other ideas and suggestions.