Calibration
Many Things Are Not Absolute
Calibration Graph
• line through center
of distribution
• dN/N - normal
distribution eqn.
• fraction of universe
between
x and (x + dx) is the
probability that x will
lie between
x and (x + dx)
Method of Least Squares
• It can be shown:
• best straight line through a series of
experimental points:
• the line for which the sum of the squares
of the deviations of the points from the
line is a minimum
• quadratic summation allows for sign of
deviation to be ignored
Sum of Squares of Differences
• equation of line y = mx + b
• square of sum of differences, S is
• assumes no error in x - independent variable
• yl is the value on the line
Minimize S by Differentiation
• Best straight line occurs when S goes
through a minium
• Differentiate and set the derivatives of S
wrt m and b equal to zero and solve for m
and b
• The result is:
• where x is the mean of all the values of xi
and y is the mean of all the values of yi
Easier Form of Least Squares
Equations
• n is the number of data points
Least Squares Example
• riboflavin (vitamin B2) determined in
cereal by fluorescence in 5% acid soln.
• calibration curve made with standards
• data are on next slide
• use least squares method to obtain
straight line and calculate riboflavin
concentration
• sample fluorescence intensity was 15.4
Data for Riboflavin
Riboflavin least squares calc.
• Use equations for slope, m, and intercept, b.
•
•
•
•
•
retained max. sig. figs. for calculation
experimental values given to 1st decimal
therefore, round m and b to 1st decimal
y = 53.8x + 0.6
sample conc. 15.4 = 53.8x + 0.6, x = 0.275 µg/mL
Homework
• Plot the Riboflavin data with a
spreadsheet - use the spreadsheet to do
the least squares calculations
• FTP spreadsheet to your account on Auld
Reekie
• Name spreadsheet Ribo[your initials].xls
• Due Wednesday Feb. 28
Standard Deviation of Deviations
• used to obtain s of slope and intercept
• deviation of each yi from line is:
yi - yl = yi - (mx + b)
• require - standard deviation sy
• standard deviation of the deviations:
• one less degree of freedom, as two used to
define slope and intercept
Uncertainty in the slope
• sy can be used to calculate sm from:
2
Uncertainty in the intercept
• sy can also be used to calculate sb from:
• in calculations of concentration,
uncertainties in y, m, and b obtained by
propagation of error
Example calculation
• uncertainty in m, b and y for riboflavin e.g.
need values for:
and m2
from the example:
and m2 = (53.75)2 = 2.889
(yi)2 values: (0.0)2, (5.8)2, (12.2)2, (22.3)2, (43.3)2
= 0.0, 33.6, 148.8, 497.3 and 1,874.9 and
Riboflavin calculations sy & sm
• from equation for sy:
• from equation for sm:
Riboflavin calculation of sb
• from which m = 53.8 ± 1 and b = 0.6 ± 4
• unknown riboflavin calculated from:
propagation of error gives x - 0.27 ± 0.01
Homework
• Repeat the above uncertainty calculation
on the spreadsheet from the prior
homework.
• Do problem 5.3 in Harris in a
spreadsheet
• FTP the two spreadsheets to your folder
on Auld Reekie by Feb. 28
– Filenames: Ribo[your initials].xls
– and LS5_3[your initials].xls
Standard Addition
• known quantities of analyte added to
unknown sample
• requires linear calibration curve
• appropriate if component of matrix
affects signal from analyte
• signal from standards would be wrong if
the standard not put in sample
Basic Standard Addition Eqn
Standard
Addition
Flasks
Standard Addition