Lecture #5 - Overview Calibration Curves - FIU
7
1 Lecture #5 - Overview Statistics - Part 3 Statistical Tools in Quantitative Analysis • The Method of Least Squares • Calibration Curves • Using a Spreadsheet for Least Squares Calibration Curves Concentration of Standard “Analytical Response” = Known Amount/Concentration of Standard Measure of Unknown Amount/Concentration of Unknown Construction of Calibration Curves Standard Solutions = “Solutions containing known concentrations of analyte(s)” Blank Solutions = “Solutions containing all the reagents and solvents used in the analysis, but no deliberately added analyte” 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. e.g. 1x 1/5x 1/25x 1/125x 1/625x Serial Dilution Blank Measure response with analytical procedure 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 2: Subtract the (average) response of the blank samples from each measured standard to obtain the corrected value. Corrected = Measured - Blank 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 2: Subtract the average response of the blank samples from each measured standard to obtain the corrected value. 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 .
Transcript of Lecture #5 - Overview Calibration Curves - FIU
1
Lecture #5 - OverviewStatistics - Part 3
Statistical Tools in Quantitative Analysis• The Method of Least Squares• Calibration Curves• Using a Spreadsheet for Least Squares
Calibration Curves
Concentration of Standard
“Ana
lytic
al R
espo
nse”
= Known Amount/Concentration of Standard
Measure of Unknown
Amount/Concentrationof Unknown
Construction of Calibration Curves
Standard Solutions = “Solutions containing knownconcentrations of analyte(s)”
Blank Solutions = “Solutions containing all the reagentsand solvents used in the analysis, but no deliberatelyadded analyte”
Construction of Calibration CurvesStep 1: Prepare known samples of analyte covering a rangeof concentrations expected for unknowns. Measure theresponse of the analytical procedure for these standards.
e.g.
1x 1/5x 1/25x 1/125x 1/625x
Serial Dilution
Blank
Measure response with analytical procedure
Construction of Calibration CurvesStep 1: Prepare known samples of analyte covering a rangeof concentrations expected for unknowns. Measure theresponse of the analytical procedure for these standards.
Step 2: Subtract the (average) response of the blank samplesfrom each measured standard to obtain the corrected value.
Corrected = Measured - Blank
Construction of Calibration CurvesStep 1: Prepare known samples of analyte covering a rangeof concentrations expected for unknowns. Measure theresponse of the analytical procedure for these standards.
Step 2: Subtract the average response of the blank samplesfrom each measured standard to obtain the corrected value.
Step 3: Make a graph of corrected versus concentration ofstandard, and use the “method of least squares” procedure tofind 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 theblank to obtain the corrected value and use the correctedvalue to determine the concentration based on yourcalibration curve.
2
Calibration Curves
Concentration of Standard
“Ana
lytic
al R
espo
nse”
= Known Amount/Concentration of Standard
Measure of Unknown
Amount/Concentrationof Unknown
“Method of Least Squares”“to draw the ‘best’ straight line through experimentaldata points that have some scatter and do not lieperfectly on a straight line”
x
y
y-intercept (b)
y = mx + b
Slope (m) = ΔyΔx
Δy
Δx
Vertical Deviation = di = yi - y = yi - (mxi + b)
di2 = (yi - y)2 = (yi - mxi - b)2
We wish to minimize to minimize the magnitude ofthe deviations (regardless of sign) so we square the terms. This is where “Method of least Squares” takes its name.
Method of Least Squares
“Method of Least Squares”
Σ(xiyi) ΣxiSlope: m = ÷ D
Σyi n
Σ(xi2) Σ(xiyi)
Intercept: b = ÷ DΣxi Σyi
Σ(xi2) Σxi
D = Σxi n
Determinants
A B
C D⇒ AD - BC
3
“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 tomeasure a colored product which results from chemical reaction withprotein. To construct a calibration curve, you make the followingmeasurements of absorbance (of the colored product) for several knownamounts of protein. Use the “method of least squares” to determine thebest fit line.
AmountProtein (mg) Absorbance0 0.0995.0 0.18510.0 0.28215.0 0.34520.0 0.42525.0 0.483
Corrected*0.0000.0860.183 0.246 0.3260.384
* Absorbance - Average Blank (=0.0993)
“Method of Least Squares”
Example: To analyze protein levels, you use a spectrophotometer tomeasure a colored product which results from chemical reaction withprotein. To construct a calibration curve, you make the followingmeasurements of absorbance (of the colored product) for several knownamounts of protein. Use the “method of least squares” to determine thebest fit line.
xi yi xiyi xi2
0 0 0 05.0 0.086 0.43 2510.0 0.183 1.83 10015.0 0.246 3.69 22520.0 0.326 6.52 40025.0 0.384 9.60 625
Σ 75 1.225 22.07 1375
n = 6 ⇒ 6 data points
m = nΣ(xiyi) - ΣxiΣyi
nΣ (xi2) - (Σxi)2
= (6)(22.07) - (75)(1.225) (6)(1375) - (75)2
m = 0.015445714
“Method of Least Squares”
Example: To analyze protein levels, you use a spectrophotometer tomeasure a colored product which results from chemical reaction withprotein. To construct a calibration curve, you make the followingmeasurements of absorbance (of the colored product) for several knownamounts of protein. Use the “method of least squares” to determine thebest fit line.
b = Σ(xi2)Σyi - (Σxiyi)Σxi
nΣ (xi2) - (Σxi)2
= (1375)(1.225) - (22.07)(75) (6)(1375) - (75)2
b = 0.01109524
xi yi xiyi xi2
0 0 0 05.0 0.086 0.43 2510.0 0.183 1.83 10015.0 0.246 3.69 22520.0 0.326 6.52 40025.0 0.384 9.60 625
Σ 75 1.225 22.07 1375
n = 6
4
“Method of Least Squares”
Example: To analyze protein levels, you use a spectrophotometer tomeasure a colored product which results from chemical reaction withprotein. To construct a calibration curve, you make the followingmeasurements of absorbance (of the colored product) for several knownamounts of protein. Use the “method of least squares” to determine thebest 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 experimentaldata points that have some scatter and do not lieperfectly on a straight line”
x
y
y = mx + bσy (xi,yi)
Vertical Deviation (di)= yi - y
di = yi - y= yi - (mxi + b)
(di)2 = (yi - mxi - b)2
Uncertainty and Least Squares
σy ≈ sy = Σ(d1 - d)2
(degrees of freedom)
sy = Σ(d1)2
(degrees of freedom)
sy = Σ(d1)2
n-2
5
xi yi xiyi xi2 di (=yi - mx - b) di
2
0 0 0 0 -0.0111 0.000123215.0 0.086 0.43 25 -0.0022 0.0000054010.0 0.183 1.83 100 0.0174 0.0003044215.0 0.246 3.69 225 0.0032 0.0000103620.0 0.326 6.52 400 0.0060 0.0000358925.0 0.384 9.60 625 -0.0132 0.00017525
Σ 75.0 1.225 22.07 1375 0.00065442
n = 6
Example: To analyze protein levels, you use a spectrophotometer tomeasure a colored product which results from chemical reaction withprotein. To construct a calibration curve, you make the followingmeasurements of absorbance (of the colored product) for several knownamounts of protein. Use the “method of least squares” to determine thebest fit line. Calculate the uncertainty associated with this line.
Uncertainty and Least Squares
Example: To analyze protein levels, you use a spectrophotometer tomeasure a colored product which results from chemical reaction withprotein. To construct a calibration curve, you make the followingmeasurements of absorbance (of the colored product) for several knownamounts of protein. Use the “method of least squares” to determine thebest fit line. Calculate the uncertainty associated with this line.
Uncertainty and Least Squares
sy = Σ(d1)2
n-2
= (0.00065442)/(6-2)
= 0.0001636
= 0.012790808
Uncertainty and Least Squares
sm2 = sy
2n D
sb2 = sy
2Σ(xi2)
D
Example: To analyze protein levels, you use a spectrophotometer tomeasure a colored product which results from chemical reaction withprotein. To construct a calibration curve, you make the followingmeasurements of absorbance (of the colored product) for several knownamounts of protein. Use the “method of least squares” to determine thebest fit line. Calculate the uncertainty associated with this line.
Uncertainty and Least Squares
Σ(xi2) Σxi
D = Σxi n
= (1375 x 6) - (75 x 75)
1375 75D =
75 6
= 2625
xi yi xiyi xi2 di
2
0 0 0 0 0.00012321 5.0 0.086 0.43 25 0.00000540 10.0 0.183 1.83 100 0.00030442 15.0 0.246 3.69 225 0.00001036 20.0 0.326 6.52 400 0.00003589 25.0 0.384 9.60 625 0.00017525Σ 75.0 1.225 22.07 1375 0.00065442
n = 6sy = 0.012790808
Example: To analyze protein levels, you use a spectrophotometer tomeasure a colored product which results from chemical reaction withprotein. To construct a calibration curve, you make the followingmeasurements of absorbance (of the colored product) for several knownamounts of protein. Use the “method of least squares” to determine thebest fit line. Calculate the uncertainty associated with this line.
Uncertainty and Least Squares
sm2 = sy
2n
D
= 0.000000373954
= (0.012790808)2 (6) (2625)
sm = 0.000611518
xi yi xiyi xi2 di
2
0 0 0 0 0.00012321 5.0 0.086 0.43 25 0.00000540 10.0 0.183 1.83 100 0.00030442 15.0 0.246 3.69 225 0.00001036 20.0 0.326 6.52 400 0.00003589 25.0 0.384 9.60 625 0.00017525Σ 75.0 1.225 22.07 1375 0.00065442
n = 6sy = 0.012790808, D=2625
Example: To analyze protein levels, you use a spectrophotometer tomeasure a colored product which results from chemical reaction withprotein. To construct a calibration curve, you make the followingmeasurements of absorbance (of the colored product) for several knownamounts of protein. Use the “method of least squares” to determine thebest fit line. Calculate the uncertainty associated with this line.
Uncertainty and Least Squares
= 0.0000856977
sb2 = sy
2 Σ(xi2)
D
= (0.012790808)2 (1375) (2625)
sb = 0.009257307
xi yi xiyi xi2 di
2
0 0 0 0 0.00012321 5.0 0.086 0.43 25 0.00000540 10.0 0.183 1.83 100 0.00030442 15.0 0.246 3.69 225 0.00001036 20.0 0.326 6.52 400 0.00003589 25.0 0.384 9.60 625 0.00017525Σ 75.0 1.225 22.07 1375 0.00065442
n = 6sy = 0.012790808, D=2625
6
Example: To analyze protein levels, you use a spectrophotometer tomeasure a colored product which results from chemical reaction withprotein. To construct a calibration curve, you make the followingmeasurements of absorbance (of the colored product) for several knownamounts of protein. Use the “method of least squares” to determine thebest fit line. Calculate the uncertainty associated with this line.
Uncertainty and Least Squares
m = 0.015445714 ± 0.000611518
b = 0.01109524 ± 0.009257307
= 0.0154 ± 0.0006
= 0.011 ± 0.009
Linearity
Linear Range vs. Dynamic Range
Linear Range
Dynamic Range
Determining Linearity
R2 = [Σ(xi - x)(yi - y)]2
Σ(xi - x)2 Σ(yi - y)2
Square of Correlation Coefficient
R2 close to 1 (e.g. ≥ 0.99, 0.98, 0.95)
R2 High (>0.95)
R2 Low (<<0.95)
7
