Data analysis for Geochemists
9/25/12
Lecture outline:
1) designing an analytical strategy
2)
error analysis in isotope
geochemistry
3)
clean rooms &
chemical separation
Contents of the sample canister from NASA’s Genesis mission,
in a “cleanroom”.
Designing an analytical strategy for isotopic analysis
1.
How much material do you have available for analysis?
- often set by external factors (no sample is unlimited)
2.
What is the expected concentration of the isotopes of interest?
3.
What is the error on the isotope ratio expected from counting statistics?
4.
What are the other sources of error?
- blanks (know the sources of contamination and their isotopic signatures)
5.
Is the expected/desired isotopic signal larger than the sum of all expected errors?
yes? proceed
no? back to square one – can you use more sample? limit blanks? etc
6.
What instrument will deliver you the required precision?
7.
What particular sources of error are associated with this analysis technique?
- poor yield from sample injection to detection (lowers N)
- mass fractionation, abundance sensitivity, etc
8.
Is the expected/desired isotopic signal larger than the sum of all expected errors?
A review of terms
accuracy: how close the measurement is to a true value
precision: how well we can measure something analytically
Good science: quote values that are accurate within the precision
Systematic error: cannot be assessed by repeated measurements (ex?)
Random error: can be assessed by repeated measurements (ex?)
Internal error: measure ratio repeatedly, assess scatter (aka precision)
External error: compare measurements of standards with internal
errors to truth (aka accuracy)
Systematic error
Examples:
detector gain (only counts a fraction of signal, usually close to 1)
uncorrected blank or “memory”
wrong mass discrimination law assumed
spike calibration not accurate
Reducing systematic errors:
1) minimize systematic errors, add them to random errors
2) make sure systematic errors are small compared to random errors (<10%)
3) measure unknowns relative to a standard so systematic errors cancel out
* Different applications require different approaches (2 & 3 most popular in mass spec work)
How do you hunt for
systematic errors?
Random error
Counting Statistics:
problem of counting subset of a large set (sometimes the subset will reflect the
large set, sometimes over-estimate, sometimes under-estimate)
theoretical limit: 68% chance of being within 1/ n of measured values
so need 10,000 realizations to get 1% error (at 68% confidence, or 1s)
Internal error:
derives from imperfect measurement (collector noise, electronic noise, etc)
measure ratio repeatedly and use scatter to assess uncertainty
External error:
the ability to reproduce standards over many runs (why might this change long-term?)
measure standard repeatedly, over a very long time
cite as 2 s.d. and mention how many standards based on
Ex.: “External reproducibility was assessed with repeated measurements
of the NBS-19 carbonate standard, and is reported as ±0.05‰ (2 s.d., N=550).”
Example – U isotope ratios in single run
238U/235U ratios over typical multi-collector ICPMS run
238U/235U
1.40E+02
1.40E+02
1.40E+02
1.40E+02
1.40E+02
1.40E+02
0
50
100
150
200
Scan number
Statistic
1) mean = 140.0833
What is the 238U/235U
ratio in nature?
What sources of
error are implicit
in this plot?
x


i
N
2) standard deviation (1s) = 0.038
* variance = (s)2
1
2
(
x

x
)
 i
N 1
s
3) standard error (1s) = 0.0027 s.e. 
4) relative standard error (1s) = 1.93 x
s
N
10-5
r.s.e. 
s.e.

The Gaussian, or “normal” distribution
 1  x   2 
1
PG 
exp   
 
s 2
 2  s  
Frequency
+2s
+1s
MEAN
-1s
-2s
Probability density equation:
range
CI
0.6826895
0.9544997
0.9973002
0.9999366
0.9999994
238U/235U ratio
How could you test whether
a process is Gaussian?
Would the shape of this
distribution change with
more measurements?
238U/235U ratio
What’s limiting the precision
of this measurement?
+1s
+2s
Frequency
Would the distribution
for many standard runs
be greater or smaller than
for individual runs?
-1s
-2s
How many measurements
should fall outside the 2s
boundaries?
MEAN
Fun with Gaussian statistics
A note on error propagation
Addition and subtraction: square root of sum of squared absolute errors
Example: subtracting a blank
blank = 230 ± 20 pg 230Th
measurement = 3532 ± 50 pg 230Th
blank-corrected 230Th = 3302 ± 70 pg 230Th
multiplication and division: square root of sum of squared relative errors
Example: correcting value for mass discrimination
by normalizing to standard value
mass discrimination = std ratio(meas) / std ratio(true) = 1.003322 ± 0.01%
unknown ratio = 1932 ± 10 or 1932 ± 0.52%
m.d.-corrected ratio = 1926 ± 0.53%
Usually you are dealing with multiple sources of error
errors from….
mass spectrometry
2 s.d. external
weighing
2 s.d. of repeat measurements on balance
spike calibration
2 s.d. of repeat spike calibration attempts
blank correction
2 s.d. of blank variability
IF errors are unrelated (orthogonal) – no error correlation (examples?)
then combine errors quadratically
A reminder about significant figures
Number of significant figures:
1. leftmost nonzero digit is the most significant digit
2. if there is no decimal point, rightmost nonzero digit is the least significant digit
3. if there is a decimal point, the rightmost digit is the least significant digit (0 included)
4. all digits between the least and most significant digits are counted as significant digits
Example:
How many significant figures in the following numbers?
1234
123,400
123.4
1001
10.10
0.0001010
100.0
NOTE: When performing calculations with data, the number of significant figures in the answer
must be equal to the smallest number of significant figures in the input data
Rules for reporting data:
Do not report data as more precise than the error (internal or external).
Example: mass spec printout reads -5.322‰, external error = ±0.05‰, report as -5.32‰
Linear regression, R-value, and slope uncertainty
Fitting your data to a linear model y=ax+b
Questions to ask yourself:
1) which is my dependent (x) and independent (y) variable?
- if error in both variables, then use
total least squares (minimize residuals
in both x and y) versus
ordinary least squares (y only)
2) How do I interpret my R value?
-R2 represents the amount of variance
that are you explaining with the
relationship, in %
Ex: if R=0.6, then you are explaining
36% of the variance in your variables
3) What is the uncertainty in my slope?
-can use LINEST in Excel
-if the slope is used in a calculation, the
uncertainty in slope must be propagated through
Assessing the significance of an R-value
Is the relationship between x and y significant?
Note: you can have a very high R-value that is meaningless!
Can use several approaches, most common is
the Student’s T-test, where you are testing
whether your R value is significantly different
from ‘0’, given your degrees of freedom.
Table at right uses a Pearson’s Product-Moment
Correlation Coefficient.
Ex: For 7 sample pairs of x and y,
I have 5 degrees of freedom.
I need an R-value of ≥0.754 for a 95% significance.
≥0.874 for a 99% significance.
http://www.gifted.uconn.edu/siegle/research/correlation/corrchrt.htm
Determining your need for clean
Every geochemical measurement must be corrected for a blank.
In general, the blank correction should be at least an order of magnitude less than
the analytical error of the measurement.
Your relative blank contribution depends on:
- the concentration of the element of interest and the amount of sample available
- the amount of contaminants that will be incorporated into the sample
(depends on analytical technique)
(Nspl*Rspl + Ncont*Rcont)/Ntotal = Rfinal
Three scenarios
14C/12C
in carbonates
-introduction of modern CO2 gas
during conversion of sample CaCO3
to CO2 is fatal
-blanks are extremely important
-conversion performed in evacuated
vacuum line
-blanks carefully monitored
18O/16O
in carbonates
-oxygen is 48% of CaCO3 by
weight
-but we can measure R of small
samples precisely (~0.005%)
-atmospheric CO2, water vapor
must be limited during analysis
U-Th isotopes in carbonates
-U is 3ppm in carbonates
Th is ultra-trace
-not easy to contaminate U,
but very easy to contaminate Th
How clean is clean?
A cleanroom is a fully-enclosed laboratory that is:
1. maintained at positive pressure
2. fed by filtered air
3. inhabited by scientists in white labcoats, booties, gloves, and hats
4. constructed entirely of white plastic (usually)
Cleanroom specifications range from
class 10,000 to class 1.
Cleanliness costs time
and money (~$500,000 to $2M).
NOTE: they only prevent
particulate contaminants.
Which analyses need to be conducted in a clean room?
14C/12C
in carbonates
18O/16O
in carbonates
-introduction of modern CO2 gas
during conversion of sample CaCO3
to CO2 is fatal
-blanks are extremely important
-conversion performed in evacuated
vacuum line
-blanks carefully monitored
-oxygen is 48% of CaCO3 by
Maybe.
You cannot prevent CO2
from entering a cleanroom.
but
Particulates may contain
modern CO2 (esp. if
biologists using tracer 14C
are nearby), so limiting
sample exposure
to particulates can be
important.
No.
You cannot prevent H2O(g)
from entering a cleanroom.
weight
-but we can measure R of small
samples precisely (~0.005%)
-atmospheric CO2, water vapor
must be limited during analysis
U-Th isotopes in carbonates
-U is 3ppm in carbonates
Th is ultra-trace
-not easy to contaminate U,
but very easy to contaminate Th
Definitely.
Your Th blank would render
your analysis useless.
Chemical separation
If the isotope of interest is present in low concentrations (ppm and lower), a chemical
separation is often required prior to analysis.
benefits: -reduces mass interferences
-improves conversion of atoms to ions (ionization efficiency)
Usually achieved by column chromatography
Principle: -it’s a competition between sample ions and acid ions
for the binding sites on the resin
-ions have different binding strengths to resin beads,
can be pulled off with specific acids
example of cation exchange reaction:
M+ + H-R  M-R + H+
-this reaction is associated with a K-value that is specific
to the acid and ion in question
Column Chromatography
Procedure:
1) sample pre-concentrated, converted to solution form
2) anion or cation exchange resin slurry loaded into column
3) sample loaded
4) progress through sequence of eluants (acids of different
strengths)
5) collect fractions of interest