9.4 Summary (EMCJT)

• Curve fitting is the process of fitting functions to data.

• Intuitive curve fitting is performed by visually interpreting if the points on the scatter plot conform to a linear, exponential, quadratic or some other function.

• The line of best fit or trend line is a straight line through the data which best approximates the available data points. This allows for the estimation of missing data values.

• Interpolation is the technique used to predict values that fall within the range of the available data.
• Extrapolation is the technique used to predict the value of variables beyond the range of the available data.

• Linear regression analysis is a statistical technique of finding out exactly which linear function best fits a given set of data.

• The least squares method is an algebraic method of finding the linear regression equation. The linear regression equation is written $\hat{y}=a+bx$, where $$\begin{align*} b & = \frac{n{\sum }_{i=1}^{n}{x}_{i}{y}_{i}-{\sum }_{i=1}^{n}{x}_{i}{\sum }_{i=1}^{n}{y}_{i}}{n{\sum }_{i=1}^{n}{\left({x}_{i}\right)}^{2}-{\left({\sum }_{i=1}^{n}{x}_{i}\right)}^{2}} \\ a & = \frac{1}{n}\sum _{i=1}^{n}{y}_{i}-\frac{b}{n}\sum _{i=1}^{n}{x}_{i}=\bar{y}-b\bar{x} \end{align*}$$

• The linear correlation coefficient, $r$, is a measure which tells us the strength and direction of a relationship between two variables, determined using the equation:

  $$r = b(\frac{\sigma_{x}}{\sigma_{y}})$$

• The correlation coefficient $r\in \left[-1;1\right]$. When $r=-1$, there is perfect negative correlation, when $r=0$, there is no correlation and when $r=1$, there is perfect positive correlation.