s p o n s e. ” = Known Amount/Concentration of Standard. Measure of Unknown.
Amount/Concentration of Unknown. Construction of Calibration Curves.
Calibration Curves
Statistics - Part 3
Statistical Tools in Quantitative Analysis • The Method of Least Squares • Calibration Curves • Using a Spreadsheet for Least Squares
“Analytical Response”
Lecture #5 - Overview
Measure of Unknown
Amount/Concentration of Unknown
Concentration of Standard
= Known Amount/Concentration of Standard
Construction of Calibration Curves Standard Solutions = “Solutions containing known concentrations of analyte(s)”
Construction of Calibration Curves Step 1: Prepare known samples of analyte covering a range of concentrations expected for unknowns. Measure the response of the analytical procedure for these standards.
Serial Dilution Blank Solutions = “Solutions containing all the reagents and solvents used in the analysis, but no deliberately added analyte”
e.g.
1x
1/5x
1/25x
1/125x 1/625x
Blank
Measure response with analytical procedure
Construction of Calibration Curves
Construction of Calibration Curves
Step 1: Prepare known samples of analyte covering a range of concentrations expected for unknowns. Measure the response of the analytical procedure for these standards.
Step 1: Prepare known samples of analyte covering a range of concentrations expected for unknowns. Measure the response of the analytical procedure for these standards.
Step 2: Subtract the (average) response of the blank samples from each measured standard to obtain the corrected value.
Step 2: Subtract the average response of the blank samples from each measured standard to obtain the corrected value.
Corrected
=
Measured - Blank
Step 3: Make a graph of corrected versus concentration of standard, and use the “method of least squares” procedure to find the best straight line through the linear portion of the data. Step 4: To determine the concentration of an unknown, analyze the unknown sample along with a blank, subtract the blank to obtain the corrected value and use the corrected value to determine the concentration based on your calibration curve.
1
“Analytical Response”
Calibration Curves
Measure of Unknown
Amount/Concentration of Unknown
Concentration of Standard
= Known Amount/Concentration of Standard
“Method of Least Squares” “to draw the ‘best’ straight line through experimental data points that have some scatter and do not lie perfectly on a straight line”
Method of Least Squares Vertical Deviation = di = yi - y = yi - (mxi + b) di2 = (yi - y)2 = (yi - mxi - b)2
y = mx + b y
Δy Δx
Slope (m) = Δy Δx
y-intercept (b)
We wish to minimize to minimize the magnitude of the deviations (regardless of sign) so we square the terms. This is where “Method of least Squares” takes its name.
x
“Method of Least Squares” Σ(xiyi) Σxi Slope:
m=
Intercept:
b=
D=
Σyi
n
Σ(xi2)
Σ(xiyi)
Σxi
Σ(xi2)
Σxi
Σxi
n
Σyi
÷
Determinants
D
÷
D
A
B
C
D
AD - BC
2
“Method of Least Squares” m
=
nΣ(xiyi) - ΣxiΣyi nΣ (xi2) - (Σxi)2
b
=
Σ(xi2)Σyi - (Σxiyi)Σxi nΣ (xi2) - (Σxi)2
“Method of Least Squares” Example: To analyze protein levels, you use a spectrophotometer to measure a colored product which results from chemical reaction with protein. To construct a calibration curve, you make the following measurements of absorbance (of the colored product) for several known amounts of protein. Use the “method of least squares” to determine the best fit line. Amount Protein (mg) 0 5.0 10.0 15.0 20.0 25.0
Absorbance 0.099 0.185 0.282 0.345 0.425 0.483
Corrected* 0.000 0.086 0.183 0.246 0.326 0.384
* Absorbance - Average Blank (=0.0993)
“Method of Least Squares”
“Method of Least Squares”
Example: To analyze protein levels, you use a spectrophotometer to measure a colored product which results from chemical reaction with protein. To construct a calibration curve, you make the following measurements of absorbance (of the colored product) for several known amounts of protein. Use the “method of least squares” to determine the best fit line.
m=
xi
Σ n=6
0 5.0 10.0 15.0 20.0 25.0 75
yi
0 0.086 0.183 0.246 0.326 0.384 1.225
6 data points
x iy i
0 0.43 1.83 3.69 6.52 9.60 22.07
x i2
0 25 100 225 400 625 1375
Example: To analyze protein levels, you use a spectrophotometer to measure a colored product which results from chemical reaction with protein. To construct a calibration curve, you make the following measurements of absorbance (of the colored product) for several known amounts of protein. Use the “method of least squares” to determine the best fit line.
nΣ(xiyi) - ΣxiΣyi nΣ
(xi2)
-
xi
(Σxi)2
= (6)(22.07) - (75)(1.225) (6)(1375) - (75)2 m = 0.015445714
Σ
0 5.0 10.0 15.0 20.0 25.0 75
yi
0 0.086 0.183 0.246 0.326 0.384 1.225
x iy i
0 0.43 1.83 3.69 6.52 9.60 22.07
x i2
0 25 100 225 400 625 1375
b=
Σ(xi2)Σyi - (Σxiyi)Σxi nΣ (xi2) - (Σxi)2
= (1375)(1.225) - (22.07)(75) (6)(1375) - (75)2 b = 0.01109524
n=6
3
“Method of Least Squares” Example: To analyze protein levels, you use a spectrophotometer to measure a colored product which results from chemical reaction with protein. To construct a calibration curve, you make the following measurements of absorbance (of the colored product) for several known amounts of protein. Use the “method of least squares” to determine the best fit line.
m = 0.015445714 b = 0.01109524 y = (0.015445714)x + (0.01109524)
“Method of Least Squares” “to draw the ‘best’ straight line through experimental data points that have some scatter and do not lie perfectly on a straight line”
y = mx + b σy y
(xi,yi)
Vertical Deviation (di) = yi - y
di = yi - y = yi - (mxi + b) (di)2 = (yi - mxi - b)2 x
Uncertainty and Least Squares σy ≈ sy =
sy =
sy =
Σ(d1 - d)2 (degrees of freedom) Σ(d1)2 (degrees of freedom) Σ(d1)2 n-2
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Uncertainty and Least Squares Example: To analyze protein levels, you use a spectrophotometer to measure a colored product which results from chemical reaction with protein. To construct a calibration curve, you make the following measurements of absorbance (of the colored product) for several known amounts of protein. Use the “method of least squares” to determine the best fit line. Calculate the uncertainty associated with this line.
xi
Σ n=6
0 5.0 10.0 15.0 20.0 25.0 75.0
yi
0 0.086 0.183 0.246 0.326 0.384 1.225
x iy i
0 0.43 1.83 3.69 6.52 9.60 22.07
x i2
0 25 100 225 400 625 1375
di (=yi - mx - b) -0.0111 -0.0022 0.0174 0.0032 0.0060 -0.0132
Uncertainty and Least Squares Example: To analyze protein levels, you use a spectrophotometer to measure a colored product which results from chemical reaction with protein. To construct a calibration curve, you make the following measurements of absorbance (of the colored product) for several known amounts of protein. Use the “method of least squares” to determine the best fit line. Calculate the uncertainty associated with this line.
Σ(d1)2 n-2
sy =
d i2
0.00012321 0.00000540 0.00030442 0.00001036 0.00003589 0.00017525 0.00065442
= =
(0.00065442)/(6-2) 0.0001636
= 0.012790808
Uncertainty and Least Squares
Uncertainty and Least Squares Example: To analyze protein levels, you use a spectrophotometer to measure a colored product which results from chemical reaction with protein. To construct a calibration curve, you make the following measurements of absorbance (of the colored product) for several known amounts of protein. Use the “method of least squares” to determine the best fit line. Calculate the uncertainty associated with this line.
sm 2 =
sy2n D
sb2 =
sy2Σ(xi2) D
xi
0 5.0 10.0 15.0 20.0 25.0 Σ 75.0
yi
0 0.086 0.183 0.246 0.326 0.384 1.225
xi yi
0 0.43 1.83 3.69 6.52 9.60 22.07
xi 2
0 25 100 225 400 625 1375
d i2
0.00012321 0.00000540 0.00030442 0.00001036 0.00003589 0.00017525 0.00065442
n=6 sy = 0.012790808
Uncertainty and Least Squares Example: To analyze protein levels, you use a spectrophotometer to measure a colored product which results from chemical reaction with protein. To construct a calibration curve, you make the following measurements of absorbance (of the colored product) for several known amounts of protein. Use the “method of least squares” to determine the best fit line. Calculate the uncertainty associated with this line.
xi
0 5.0 10.0 15.0 20.0 25.0 Σ 75.0
yi
0 0.086 0.183 0.246 0.326 0.384 1.225
xi yi
0 0.43 1.83 3.69 6.52 9.60 22.07
xi 2
0 25 100 225 400 625 1375
n=6 sy = 0.012790808, D=2625
d i2
0.00012321 0.00000540 0.00030442 0.00001036 0.00003589 0.00017525 0.00065442
sm2 = sy2n D = (0.012790808)2 (6) (2625) = 0.000000373954 sm = 0.000611518
Σ(xi2)
D=
Σxi
Σxi
n
1375
75
75
6
D= = (1375 x 6) - (75 x 75) = 2625
Uncertainty and Least Squares Example: To analyze protein levels, you use a spectrophotometer to measure a colored product which results from chemical reaction with protein. To construct a calibration curve, you make the following measurements of absorbance (of the colored product) for several known amounts of protein. Use the “method of least squares” to determine the best fit line. Calculate the uncertainty associated with this line.
xi
0 5.0 10.0 15.0 20.0 25.0 Σ 75.0
yi
0 0.086 0.183 0.246 0.326 0.384 1.225
xi yi
0 0.43 1.83 3.69 6.52 9.60 22.07
xi 2
0 25 100 225 400 625 1375
n=6 sy = 0.012790808, D=2625
d i2
0.00012321 0.00000540 0.00030442 0.00001036 0.00003589 0.00017525 0.00065442
sb2 =
sy2 Σ(xi2) D
= (0.012790808)2 (1375) (2625) = 0.0000856977 sb = 0.009257307
5
Uncertainty and Least Squares
Linearity
Example: To analyze protein levels, you use a spectrophotometer to measure a colored product which results from chemical reaction with protein. To construct a calibration curve, you make the following measurements of absorbance (of the colored product) for several known amounts of protein. Use the “method of least squares” to determine the best fit line. Calculate the uncertainty associated with this line.
m = 0.015445714 ± 0.000611518 = 0.0154 ± 0.0006 b = 0.01109524 ± 0.009257307 = 0.011 ± 0.009
Linear Range vs. Dynamic Range Dynamic Range
Linear Range
Determining Linearity Square of Correlation Coefficient R2
=
[Σ(xi - x)(yi - y)]2 Σ(xi - x)2 Σ(yi - y)2
R2 close to 1 (e.g. ≥ 0.99, 0.98, 0.95) R2 High (>0.95)
R2 Low (
