I used lagrangian interpolation and for some equations the average value of the function is astronomical despite the polynomial and the points it interpolates. For example Polynomial A: $$(X_0, Y_0) = (62,181)\\ (X_1, Y_1) = (40,210)\\ (X_2, Y_2) = (67,185)\\ (X_3, Y_3) = (19, 201)\\ (X_4, Y_4) = (41, 199) $$ Lagrangian Polynomial: $$f(x)=-3893/4980690 x^4+5847/39130x^3-353200229/34864830x^2+121372073/430430x-8446030382/3486483$$ The average value of the function within $19\le x\le 67: 5248520$ Polynomial B: $$(X_0, Y_0) = (1,200)\\ (X_1, Y_1) = (81,250)\\ (X_2, Y_2) = (32,212)\\ (X_3, Y_3) = (77, 248)\\ (X_4, Y_4) = (93, 245) $$ $$f(x)=-1173601/155489214720 x^4+28814081/22212744960 x^3-1451597501/22212744960 x^2+3639509257/2549003520 x+1482612997/7463960$$ $$dx=32773432105907/145771138800≈224.8$$ Am I doing something wrong in polynomial A. What is mathematically the cause for this? Thanks!
Given $n+1$ points in the plane, Lagrangian interpolation gives the unique polynomial of degree less or equal to $n$.  From the examples I've seen so far, interpolation polynomials all have degree exactly $n$ but I don't know if there might be a set of points for which the interpolation gives a polynomial of degree strictly less than $n$.  Many thanks for your answers.
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