1.17:
Calibration Curves: Linear Least Squares
A calibration curve is a mathematical relationship between the instrument's signal and known analyte concentrations. This curve equation predicts the unknown concentration of a sample.
The experimental data may not lie perfectly on a straight line due to random errors. So, the linear least squares method – a regression analysis is used to obtain a straight line that best fits the different points.
LLS is based on two assumptions. Firstly, a linear relationship exists between the instrument's signal and the analyte concentration. Secondly, the errors are attributed to random error, not human error.
The best-fitting line is drawn by minimizing the sum of the squared differences between the estimated and the actual values.
The recorded plot yields the equation of the line. Here, y is the instrument's signal, x is the analyte concentration, m is the slope of the line, and c is the y-intercept. The unknown concentration of the sample is determined by measuring its instrumental signal and substituting the appropriate values in the equation.
1.17:
Calibration Curves: Linear Least Squares
A calibration curve is a plot of the instrument's response against a series of known concentrations of a substance. This curve is used to set the instrument response levels, using the substance and its concentrations as standards. Alternatively, or additionally, an equation is fitted to the calibration curve plot and subsequently used to calculate the unknown concentrations of other samples reliably.
For data that follow a straight line, the standard method for fitting is the linear least-squares method. This method minimizes the sum of the squared differences between the predicted and actual values.
The linear least square method plots the data points with the concentration on the x-axis and the measured analytical response on the y-axis. The equation of the line that best fits these data points is 'y = mx + c.' Here, y is the instrument's signal, x is the analyte concentration, m is the slope of the line, and c is the y-intercept. Once the best-fit equation has been determined, unknown concentrations can be determined with this equation by solving for x.