Lecture 5

1Lecture #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


Step 2: Subtract the (average) response of the blank samplesfrom each measured standard to obtain the corrected value.

Corrected = Measured - Blank


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.

2Calibration Curves

Concentration of Standard

Ana

lytic

al R

espo

nse

= Known Amount/Concentration of Standard

Measure of Unknown

Amount/Concentrationof Unknown

Method of Least Squaresto 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) = yx

yx

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


(xiyi) xiSlope: m = D

yi n

(xi2) (xiyi)Intercept: b = D

xi yi

(xi2) xiD =

xi n

Determinants

A B

C D AD - BC

3Method of Least Squares

m = n(xiyi) - xiyi n (xi2) - (xi)2

b = (xi2)yi - (xiyi)xi n (xi2) - (xi)2


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)



xi yi xiyi xi20 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

75 1.225 22.07 1375n = 6 6 data points

m = n(xiyi) - xiyin (xi2) - (xi)2

= (6)(22.07) - (75)(1.225) (6)(1375) - (75)2

m = 0.015445714



b = (xi2)yi - (xiyi)xin (xi2) - (xi)2

= (1375)(1.225) - (22.07)(75) (6)(1375) - (75)2

b = 0.01109524

xi yi xiyi xi20 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

75 1.225 22.07 1375n = 6

4Method of Least Squares


m = 0.015445714

b = 0.01109524

y = (0.015445714)x + (0.01109524)

Method of Least Squaresto draw the best straight line through experimentaldata points that have some scatter and do not lieperfectly on a straight line

x

y

y = mx + by (xi,yi)

Vertical Deviation (di)= yi - y

di = yi - y= yi - (mxi + b)

(di)2 = (yi - mxi - b)2

Uncertainty and Least Squares

y sy = (d1 - d)2(degrees of freedom)

sy = (d1)2(degrees of freedom)

sy = (d1)2 n-2

5xi yi xiyi xi2 di (=yi - mx - b) di20 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 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.




sy = (d1)2 n-2

= (0.00065442)/(6-2)

= 0.0001636

= 0.012790808


sm2 = sy2n D

sb2 = sy2(xi2) D



(xi2) xiD =

xi n

= (1375 x 6) - (75 x 75)

1375 75D =

75 6

= 2625

xi yi xiyi xi2 di2 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 0.00017525 75.0 1.225 22.07 1375 0.00065442n = 6sy = 0.012790808



sm2 = sy2n D

= 0.000000373954

= (0.012790808)2 (6) (2625)

sm = 0.000611518

xi yi xiyi xi2 di2 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 0.00017525 75.0 1.225 22.07 1375 0.00065442n = 6sy = 0.012790808, D=2625



= 0.0000856977

sb2 = sy2 (xi2) D

= (0.012790808)2 (1375) (2625)

sb = 0.009257307

xi yi xiyi xi2 di2 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 0.00017525 75.0 1.225 22.07 1375 0.00065442n = 6sy = 0.012790808, D=2625

6Example: 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.


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 = [(xi - x)(yi - y)]2

(xi - x)2 (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 (

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Lecture 5