1. V. Rouillard  2003 1 Introduction to measurement and statistical analysis CURVE FITTING In graphical form, drawing a line (curve) of best fit through the data can be useful to illustrate the best relationship between the variables. The most common and useful curve fit is the straight line. In graphical form, drawing a line (curve) of best fit through the data can be useful to illustrate the best relationship between the variables. Apparent line x bias y bias True line Y precision x precision y x

2. V. Rouillard  2003 2 Introduction to measurement and statistical analysis CURVE FITTING The gradient and y-intercept can be easily determined graphically  x (run)  y (rise) Y axis intercept : C Gradient: m =  y/  x y = m  x + C

3. V. Rouillard  2003 3 Introduction to measurement and statistical analysis CURVE FITTING Method of least squares: Identifies the slope m and the intercept C so that the sum of the squared deviations of the data from the fitted line is minimised: Where for various values x i, y i is the measured value, y(x i ) = mx i + C is the corresponding value for the fitted line and n is the total number of observations used.

4. V. Rouillard  2003 4 Introduction to measurement and statistical analysis CURVE FITTING Method of least squares: Identifies the slope m and the intercept C so that the sum of the squared deviations of the data from the fitted line is minimised: Where for various values x i, y i is the measured value, y(x i ) = mx i + C is the corresponding value for the fitted line and n is the total number of observations used. The gradient, m is calculated as follows: And the y intercept, C is calculated from:

5. V. Rouillard  2003 5 Introduction to measurement and statistical analysis CURVE FITTING The method of least squares addresses precision (random) errors in y and assumes that the x measurements are error-free. The method should only be used if the precision errors in x are small Most software packages also compute the Pearson product moment correlation coefficient, r: If the errors are solely produced from precision errors, then S 2  0, r 2  1 and a “perfect” fit is obtained. Unfortunately, the correlation coefficient is not a good indicator of the precision of the data. Instead, the fit quality is best assessed with the quantity which is proportional to the precision error of the data.

6. V. Rouillard  2003 6 Introduction to measurement and statistical analysis CURVE FITTING If the data is non linear, it is useful to plot the data in such a manner that a straight line will be produced. This is of course easier if the fundamental mathematical relationship between the variables is known: