Determining the Uncertainty in the Coefficients

   One might wonder what the uncertainty in values of the coefficients in the last blog might have been. Repeated "experiments" shows there is some variation present in the results. Can we get a better measure of it?

  The ith coefficient is equal to the scalar product of a vector, λ_i, and the y value. For the polynomial fit, 

  Since only the squared terms of (λ_i^Tδy)² contribute to σ_ai², the average of each δy_k being approximately zero and each δy_k² approximately σ², one can show,

  So for the 9 experiments we then get,

  Note that errors of a are bounded by 3σ. With this information we can properly design an experiment.

Supplemental (May 14): I failed to mention that the σ_y used for the y averages was σ/√9 with σ being the value for a single measurement of y so the values in the table above are σ_a =|λ|σ/3.