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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.
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