Line fitting of y = a + bx: Errors in y only (no errors in x)
\(a\) and \(b\) are obtained by minimizing the quantity, \[ t = \sum_{i=1}^n (y_i - a - bx_i)^2. \] Setting \(\partial t/\partial a = \partial t/\partial b = 0\), we obtain, \begin{eqnarray*} n a + S_x b & = & S_y, \\ S_x a + S_{xx} b & = & S_{xy}, \end{eqnarray*} where \(S_x = \sum_{i=1}^n x_i, S_y = \sum_{i=1}^n y_i, S_{xx} = \sum_{i=1}^n x_i^2, S_{xy} = \sum_{i=1}^n x_i y_i\). Solving these equations, \begin{eqnarray} a & = & \frac{S_{xx} S_y - S_x S_{xy}}{\Delta}, \label{eq01} \\ b & = & \frac{n S_{xy} - S_x S_y}{\Delta}. \label{eq02} \end{eqnarray} where \(\Delta = n S_{xx} - S_x^2\). The correlation coefficient \(r\) is defined by, \[ r = \frac{\sum_{i=1}^n (x_i - \bar x)(y_i - \bar y)} {\sqrt{\sum_{i=1}^n (x_i - \bar x)^2} \sqrt{\sum_{i=1}^n (y_i - \bar y)^2}}. \] where \(\bar x = S_x/n\) and \(\bar y = S_y/n\). Using equation \eqref{eq02}, \(r\) is expressed as, \begin{equation} r = b \frac{\sigma_x}{\sigma_y} \label{eq03} \end{equation} where \(\sigma_x = \sqrt{(S_{xx} - S_x^2/n)/(n-1)}, \sigma_y = \sqrt{(S_{yy} - S_y^2/n)/(n-1)}\). According to the propagation of errors, \[ \sigma_a^2 = \sum_{i=1}^n \sigma_i^2 \left(\frac{\partial a}{\partial y_i}\right)^2, \quad \sigma_b^2 = \sum_{i=1}^n \sigma_i^2 \left(\frac{\partial b}{\partial y_i}\right)^2, \] where \(\sigma_i^2\) is a variance of \(y_i\). Suppose \(\sigma_i = \sigma\) for all \(y_i\), and noting \(\partial S_y/\partial y_i = 1, \partial S_{xy}/\partial y_i = x_i\), \[ \frac{\partial a}{\partial y_i} = \frac{S_{xx} - x_i S_x}{\Delta}, \quad \frac{\partial b}{\partial y_i} = \frac{n x_i - S_x}{\Delta}. \] Hence, we obtain, \begin{eqnarray} \sigma_a^2 & = & \sigma^2 \frac{S_{xx}}{\Delta} \label{eq04}, \\ \sigma_b^2 & = & \sigma^2 \frac{n}{\Delta} \label{eq05}. \end{eqnarray} Here, the best estimate of \(\sigma^2\) is usually given by, \begin{equation} \sigma^2 = \frac{\sum_{i=1}^n (y_i - a - b x_i)^2}{n-2} \label{eq06} \end{equation} which is called the standard error of estimate.