The Method of Least Squares
• used to find the best straight line through a set of experimental points
• appropriate for most analytical techniques
Finding the slope
• linear relationship between y (signal) and x (analyte concentration): y = mx + b
◦ where b=intercept and m=slope of calibration curve (aka sensitivity)
• any deviation of individual points from the straight line results from error in the measurement
◦ i.e. negligible error in x values of the points
◦ usually true if standards are prepared with care: their concentration is known exactly Origin of the method of least squares
• the vertical deviation of each point from the straight line = residual d = y – y i i
= y i (mx +bi
◦ where y=ordinate of the calculated straight line corresponding to x=x
• d'i can be positive and/or negative
◦ to minimize their magnitude irrespective of their sign → use di = (y - y) = (y2- mx -i) 2 i i 2
• minimizing the squares of the residuals = Method of Least squares
“Best” straight line
• Slope = m = = sensitivity
• intercept = b =
• standard deviation of the vertical deviations:
• standard deviation of the slope = s = m
• standard deviation of the intercept: = s = b
• Reagent blank: contains all the reagents and solvents used in the analysis
• Method blank: contains all the reagents and solvents used in the analysis and is taken through
all steps of the analytical procedure
• Field blank: method blank that has also been exposed to the sampling site
Q: Which of the following is untrue regarding a blank solution?
a) Solutions containing known concentrations of analyte are called blank solutions.
b) A solution containing all of the reagents and solvents used in the analysis and no deliberately
added analyte is called a blank solution.
c) Ablank solution measures the response of the analytical procedure to impurities or interfering
species in the reagents. Which of the following is true regarding a blank in an analysis?
a) Amethod blank is a sample containing all components except the analyte and is taken
through all steps of the analytical procedure.
b) Afield blank is a sample containing all components except the analyte and is taken through all
steps of the analytical procedure.
c) Areagent blank is the same as a method blank; that is, it has been taken through all of the steps
of the analytical procedure.
Building and using calibration curves
• Analyse blank solutions.
• Analyse a series of standard solutions whose concentrations encompass those expected in the
samples. → 0.5-1.5 x expected analyte concentration
• Analyse the sample solutions (with calibration check).
• Subtract the blank signal from that of all samples and standards.
• Apply the method of least squares to find the equation of the best straight line for the standards.
• Apply the equation to the samples.
Propagation of uncertain