The Method of Least Squares
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
Assumptions • 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 di = yi – y = yi - (mxi +b) ◦ where y=ordinate of the calculated straight line corresponding to x=x • di's can be positive and/or negative ◦ to minimize their magnitude irrespective of their sign → use di2 = (yi - y)2 = (yi - mxi -b) 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 = sm =
•
standard deviation of the intercept: = sb =
Calculations
Blanks • 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) A blank 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) A method blank is a sample containing all components except the analyte and is taken through all steps of the analytical procedure. b) A field blank is a sample containing all components except the analyte and is taken through all steps of the analytical procedure. c) A reagent 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 uncertainty with a calibration curve • From ysample and equation of the line: concentration xsample = (ysample – b)/m •
sample with standard deviation:
•
◦ m = absolute value of the slope ◦ n= number of data points ◦ k= number of replicate measurements of unknown the confidence interval for x is tsx, with t for n-2 degrees of freedom
Example A spectrometer was used to measure the light absorbed by three standard solutiona containing different concentrations of K2Cr2O7. The following data were obtained: x = [Cr] in ppm y = absorbance. x
y
0.0
0.004
5.1
0.166
10.0 0.324 14.9 0.490
The equation of the straight line generated using the method of least squares is a. y = 0.0331x - 0.0079 b. y = 30.24x + 0.242 c. y = -0.0079x + 0.0331 d. y = 0.0331x + 0.0079
Answer
The eqn of the straight line generated using the method of least squares is a. y = 0.0326x – 0.00233 → Calculate the errors
The eqn of the straight line generated using the method of least squares really is y = 0.0326 (±0.0003) x - 0.002 (±0.003) Spectrophotometric determination of cobalt: The concentration of cobalt in the sample is: (a) 13.11 mg Co/L (b) 2.81 mg Co/L (c) 13.93 mg Co/L (d) 13.9 mg Co/L
The least squares method of linear regression gave the following equation for the regression line of absorbance as a function of Co concentration: y = 0.0196x – 0.008. a) This equation represents the statistical best fit of a line drawn through the data points. b) The amount of unknown in the sample may be determined by substituting the absorbance value of the unknown for x in the equation and solving for the y value. c) The value 0.0196 represents the value on the y axis where the line crosses this axis. exel : LINEST (highlight 5x2 array and Answer press Ctrl+Shift+Enter) x = (y-b)/m = (0.265 – (-0.008))/0.0196 x = (0.265 + 0.008)/0.0196 = 13.9 (d) however the error on the slope and the intercept are sm = 0.00011 and sb = 0.0012 and sy = 0.00079 which leads to sx = 0.048 but DF = 3 - 2 - 1 and t = 12.706 (at 95% level) hence the result is 13.93 =/- 0.61 mg Cr/L or 13.9 +/- 0.6 mg Cr/L (+/-ts)
Linear vs dynamic range
Linear vs non-linear calibration curve • dynamic range = concentration range over which the response to the analyte is measurable ◦ data points may be fitted by quadratic equation y = ax2 + bx +c • linear range = concentration range over which the response to the analyte is proportional to concentration
Assessment of linearity • superficial measure of linearity = square of correlation coefficient: • •
R2 must be very close to 1 to represent a linear fit. good idea to plot the data as a graph to check for outliers (the latter are not shown by the calculator)
used to find the best straight line through a set of experimental points appropriate for most analytical techniques
Finding the slope
Assumptions • 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 di = yi – y = yi - (mxi +b) ◦ where y=ordinate of the calculated straight line corresponding to x=x • di's can be positive and/or negative ◦ to minimize their magnitude irrespective of their sign → use di2 = (yi - y)2 = (yi - mxi -b) 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 = sm =
•
standard deviation of the intercept: = sb =
Calculations
Blanks • 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) A blank 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) A method blank is a sample containing all components except the analyte and is taken through all steps of the analytical procedure. b) A field blank is a sample containing all components except the analyte and is taken through all steps of the analytical procedure. c) A reagent 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 uncertainty with a calibration curve • From ysample and equation of the line: concentration xsample = (ysample – b)/m •
sample with standard deviation:
•
◦ m = absolute value of the slope ◦ n= number of data points ◦ k= number of replicate measurements of unknown the confidence interval for x is tsx, with t for n-2 degrees of freedom
Example A spectrometer was used to measure the light absorbed by three standard solutiona containing different concentrations of K2Cr2O7. The following data were obtained: x = [Cr] in ppm y = absorbance. x
y
0.0
0.004
5.1
0.166
10.0 0.324 14.9 0.490
The equation of the straight line generated using the method of least squares is a. y = 0.0331x - 0.0079 b. y = 30.24x + 0.242 c. y = -0.0079x + 0.0331 d. y = 0.0331x + 0.0079
Answer
The eqn of the straight line generated using the method of least squares is a. y = 0.0326x – 0.00233 → Calculate the errors
The eqn of the straight line generated using the method of least squares really is y = 0.0326 (±0.0003) x - 0.002 (±0.003) Spectrophotometric determination of cobalt: The concentration of cobalt in the sample is: (a) 13.11 mg Co/L (b) 2.81 mg Co/L (c) 13.93 mg Co/L (d) 13.9 mg Co/L
The least squares method of linear regression gave the following equation for the regression line of absorbance as a function of Co concentration: y = 0.0196x – 0.008. a) This equation represents the statistical best fit of a line drawn through the data points. b) The amount of unknown in the sample may be determined by substituting the absorbance value of the unknown for x in the equation and solving for the y value. c) The value 0.0196 represents the value on the y axis where the line crosses this axis. exel : LINEST (highlight 5x2 array and Answer press Ctrl+Shift+Enter) x = (y-b)/m = (0.265 – (-0.008))/0.0196 x = (0.265 + 0.008)/0.0196 = 13.9 (d) however the error on the slope and the intercept are sm = 0.00011 and sb = 0.0012 and sy = 0.00079 which leads to sx = 0.048 but DF = 3 - 2 - 1 and t = 12.706 (at 95% level) hence the result is 13.93 =/- 0.61 mg Cr/L or 13.9 +/- 0.6 mg Cr/L (+/-ts)
Linear vs dynamic range
Linear vs non-linear calibration curve • dynamic range = concentration range over which the response to the analyte is measurable ◦ data points may be fitted by quadratic equation y = ax2 + bx +c • linear range = concentration range over which the response to the analyte is proportional to concentration
Assessment of linearity • superficial measure of linearity = square of correlation coefficient: • •
R2 must be very close to 1 to represent a linear fit. good idea to plot the data as a graph to check for outliers (the latter are not shown by the calculator)
