Let be \(\displaystyle m\) the slope and \(\displaystyle b\) the intercept. We are under the assumption that the uncertainties on the values of [MATH]y[/MATH] are much larger than the uncertainties on the values of [MATH]x[/MATH]
With the Method of Least Squares these parameters are found to be
\(\displaystyle m = \dfrac{nS_{xy} - S_xS_y}{D}\)
\(\displaystyle b = \dfrac{S_yS_{xx} - S_xS_{xy}}{D}\)
with
\(\displaystyle S_x = \sum^n_{i=1} x_i\)
\(\displaystyle S_{xy} = \sum^n_{i=1} x_iy_i\)
\(\displaystyle S_{y} = \sum^n_{i=1} y_i\)
\(\displaystyle S_{xx} = \sum^n_{i=1} x^2_i\)
\(\displaystyle D =
\begin{vmatrix}
S_{xx} & S_{x}\\
S_{x} & n
\end{vmatrix}
\)
The parameters \(\displaystyle m\) and \(\displaystyle b\) are affected by uncertainty since the \(\displaystyle y_i\) values are. \(\displaystyle m\) and \(\displaystyle b\) uncertainty are given by
\(\displaystyle \sigma^2_m = \sum^n_{i=1} \left( \dfrac{\partial m}{\partial y_i} \right)^2 \sigma^2_{y_i}\)
\(\displaystyle \sigma^2_b = \sum^n_{i=1} \left( \dfrac{\partial b}{\partial y_i} \right)^2 \sigma^2_{y_i}\)
So, let's make these partial derivatives
\(\displaystyle \sigma^2_m = \left [ \left( \dfrac{1}{D} \right) \dfrac{\partial m}{\partial y_i} (nS_{xy} - S_xSy) \right]^2 \sigma^2_{y_i} = \left[ \dfrac{1}{D} (nS_{x} - S_x) \right]^2 \sigma^2_{y_i} = \dfrac{n^2 \sigma^2_{y_i}}{D^2} \)
The result is the same as given by the text, except for the value of [MATH]n[/MATH] and [MATH]D[/MATH], which should not be squared.
Did I make any mistakes?
With the Method of Least Squares these parameters are found to be
\(\displaystyle m = \dfrac{nS_{xy} - S_xS_y}{D}\)
\(\displaystyle b = \dfrac{S_yS_{xx} - S_xS_{xy}}{D}\)
with
\(\displaystyle S_x = \sum^n_{i=1} x_i\)
\(\displaystyle S_{xy} = \sum^n_{i=1} x_iy_i\)
\(\displaystyle S_{y} = \sum^n_{i=1} y_i\)
\(\displaystyle S_{xx} = \sum^n_{i=1} x^2_i\)
\(\displaystyle D =
\begin{vmatrix}
S_{xx} & S_{x}\\
S_{x} & n
\end{vmatrix}
\)
The parameters \(\displaystyle m\) and \(\displaystyle b\) are affected by uncertainty since the \(\displaystyle y_i\) values are. \(\displaystyle m\) and \(\displaystyle b\) uncertainty are given by
\(\displaystyle \sigma^2_m = \sum^n_{i=1} \left( \dfrac{\partial m}{\partial y_i} \right)^2 \sigma^2_{y_i}\)
\(\displaystyle \sigma^2_b = \sum^n_{i=1} \left( \dfrac{\partial b}{\partial y_i} \right)^2 \sigma^2_{y_i}\)
So, let's make these partial derivatives
\(\displaystyle \sigma^2_m = \left [ \left( \dfrac{1}{D} \right) \dfrac{\partial m}{\partial y_i} (nS_{xy} - S_xSy) \right]^2 \sigma^2_{y_i} = \left[ \dfrac{1}{D} (nS_{x} - S_x) \right]^2 \sigma^2_{y_i} = \dfrac{n^2 \sigma^2_{y_i}}{D^2} \)
The result is the same as given by the text, except for the value of [MATH]n[/MATH] and [MATH]D[/MATH], which should not be squared.
Did I make any mistakes?