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Accuracy of linear function parameters for the least squares method

Dependence type y=a+b*x

Average square error defining the scattering of the values measured y against the

calculated ones from Y measurements for n values of measurements, it's expressed as:

Sy y i el Y i el - (2^i 1 n sum n 2 - / sqrt :

In equation of a line there are parameters a i b, so number of freedon degrees is:

k n 2 - :

The average square errors with which the parameters a and b are intercepted are:

Sa Sy x i el (2^i 1 n sum n x i el (2^x i el i 1 n sum (2^- i 1 n sum * / sqrt * :

Sb Sy n n x i el (2^x i el i 1 n sum (2^- i 1 n sum * / sqrt * :

Confidence intervals for a and b are found using the coefficient of Student-distribution

t=t(α,k), for cofidence level eg. α=0.95

Δa t Sa * :

Δb t Sb * :

Dependence type y=b*x

In this case a=0,

k n 1 - :

Sy y i el Y i el - (2^i 1 n sum n 1 - / sqrt :

Sb Sy x i el 2^i 1 n sum sqrt / :

Δb t Sb * :

As an example may be the measurement of the elasticity coefficient p for Hook's law where

the stress is proportional to deformation:

p E ε * :

where E is a Younga module, ε is a deformation depending on the deformation method.

For elastic deformations, an elongation Δl has a linear relationship to the tensile force

of a bar of length lo and of cross section S:

Δl b F * :

where

b lo E S * / :

Therefore, the measurement concern to a parameter b and to determine its error.

Having been experimentally specified parametr b we can calculate Young module.