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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.