Your question doesn't seem clear.

My expectations are that you found this curve doing least squares approximation (normal equations). So somehow you have concentration as a function of some variable. It seems to me that each of your 30 samples are dependent on this independent variable, plug each one into your approximation to find the concentration for each sample. 

check your standardized control first. 

You need to find the inverse function, i.e. express

y(x|a,b,c,d) = ax³ + bx² + cx + d

as a function for x of y:

x(y|a,b,c,d)

The solution for a cubic equation has several possible solutions (and looks quite complicated for the general case):

http://www.wolframalpha.com/input/?i=Solve[y%3D%28ax^3%2Bbx^2%2Bcx%2Bd%29,x]

so you will need to consider restrictions on x.

A pragmatic solution is to numerically find the value of x for a given y by "searching" (define an interval on x and apply some more or less smart search algorithm to find the x-value for which y(x) will be close enough to the measured value; for ideas see https://en.wikipedia.org/wiki/Root-finding_algorithm).

In general, I doubt that using a 3rd-order polynomial as a calibration function is smart or sensible. I doubt that there is any scientific rationale behind this curve, I think that this is simply some (not-so-well-thought-of) selection of some kind of function that is able to fit reasonably well to the data. I may be wrong, but anyway, it might be worth to re-consider the selection of the calibration function.

Dear Rudra,

I guess

a) you are dealing with experimental data.

b) You "measured" (Y) some "knowns" (X) and then fitted a model to your data Y=f(X)

Now, the questions:

c) why did you chose a 3rd degree polynomial as your model? It is the true (theorectical) relationship between Y and X or was the model that originated the highest (R2)? (you tried the simplest Y=a+bx, then the y=a+bx+cx2 and finally the y=a+bx+cx2+dx3)

d) are the 3rd and 2nd degree terms significant, or in other words, is the 3rd degree model the "right" model?

Can you disclose some additional information (what are you measuring, etc) and/or share the raw data?

Thanks Luis

Yes I am dealing with the experimental data and choose a 3rd degree poynomial because it originated the highest R2. I can send you the raw data. Please provide me your email.

Regards

You are most probably overfitting your model. You may send the data (in a spreadsheet or csv format) to lgouveia(at)campus.ul.pt

In the meantime, google for "regression overfitting" and/or check http://blog.minitab.com/blog/adventures-in-statistics/the-danger-of-overfitting-regression-models

Regards

Luis

Are you looking at the concentration of a chemical as time goes on?

A lot of chemicals follow some sort of an exponential distribution. such as, Conc = A*exp(b*x). Does that seem reasonable? I highly doubt a cubic model is the best.

Can you show a plot of the data for X vs Conc? You might have some non-linear regression model to work with.