Hello Isogeochemers,

this is an additional question on the error propagation issue a few weeks ago:

When using the Gaussian error propagation formula (using partial derivatives multiplied by the uncertainties of the variables) on the normalization equation to estimate combined uncertainty of the final sample delta value, is it allowed to use standard errors of the mean as the uncertainty component (e.g. of the frequently analyzed lab standards) instead of the standard deviations used in the formula?

Calculating the normalized sample value out of mean values, and knowing that scatter of the mean values of a population is of course better determined then the scatter of the poulation itself, this sounds quite obvious and legitimate to me. But i am absolutely not sure about it. Where are the statistical pitfalls? Or is it legitimate to replace SD by SE in the components of the uncertainty calculation?

I am wandering in the dark...is there anybody with a torch?

Thank you very much in advance,

Marc Ruppenthal
Mainz University



_________________________________
Von: Stable Isotope Geochemistry [[log in to unmask]] im Auftrag von Manfred Groening [[log in to unmask]]
Gesendet: Montag, 17. November 2008 15:06
An: [log in to unmask]
Betreff: Re: [ISOGEOCHEM] error propagation in the normalization of isotope data

Dear Marc,
For stable isotope analyses, basically the last approach stated in your
email is the best one. There are even more sources of uncertainty, since
you normalize your data and therefore use two international measurement
standards (VSMOW2 and SLAP2) for the calibration of your internal
laboratory standards, and so in daily work you also use two internal lab
standards for combined calibration and normalization.

In September 2008 at the JESIUM2008 conference in Giens, France, I
presented some examples to illustrate the basic process. Two years ago
an Excel macro based software was developed at our laboratory to
automate as much as possible the process of data correction,
calibration/normalization and calculation of combined uncertainty for
several raw data formats and instruments (dual inlet IRMS with
equilibration, Zn reduction, CF-IRMS pyrolysis, laser). You can contact
me separately if you wish.

Here just some details towards an answer to your question in public:

You can break up the uncertainty calculation in two separate steps:

1. Uncertainty calculation for calibration of your internal laboratory
standards here called ISL1 and ISL2:
A) VSMOW2 assigned uncertainty
B) SLAP2 assigned uncertainty
C) VSMOW2 measured uncertainty during your calibration
D) SLAP2 measured uncertainty during your calibration
E) ISL1 measured uncertainty during your calibration
F) ISL2 measured uncertainty during your calibration
From these contributions A-F you can derive the combined uncertainty of
the reference values for your internal laboratory standards ISL1 (G) and
ISL2 (H) using partial derivatives.

2. Daily measurements
G) ISL1 assigned uncertainty as in step 1
H) ISL2 assigned uncertainty as in step 1
I) ISL1 measured uncertainty
J) ISL2 measured uncertainty
K) Sample measured uncertainty: This one is not directly assessable, as
the sample is often only measured once. You need to include a proxy for
it (normally the long-term reproducibility (L) of an additional quality
control sample analysed daily in your system, or its standard deviation
during the measuring day). Be aware to think properly how to incorporate
the uncertainty in a proper manner not to over- or underestimate its
contribution. Too long for now...
From these contributions G-L you can derive the combined uncertainty of
the unknown sample.

There are some more details like repeatability component for readings of
one individual measurement (often negligible), other error components as
bias, drift, memory and of course ensuring enough repetitions of
standard measurements to get significant statistics.

So be aware there are quite a number of components. Finally only few
will be of significance, but that depends on your setting.

If you would just use the simple error propagation formula on all, you
somewhat overestimate the combined uncertainty (e.g. consider a sample
having the same composition as VSMOW2, then the uncertainty contribution
by SLAP2 uncertainty for the normalisation process is insignificant).
For a proper error propagation, the use of partial derivatives is
recommended (sounds terrible, but is done easily, as it needs only once
the calculation of the partial derivatives of your basic calibration
formula, which can then be used automatically in an Excel spreadsheet).
At IAEA, we use such calculations in daily praxis to be able to merge
data from different instruments and different precision into one final
mean value, which is practically only possible by weighing with the
uncertainties.

Best regards,
Manfred Groening
======================================
Manfred Groening
Head, Isotope Hydrology Laboratory
International Atomic Energy Agency
P.O.Box 100   A-1400 Vienna   AUSTRIA
Tel: +43-1-2600-21740/21766 Fax: +43-1-2600-7
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http://www-naweb.iaea.org/NAALIHL/
======================================


-----Original Message-----
From: Stable Isotope Geochemistry [mailto:[log in to unmask]] On
Behalf Of Ruppenthal, Marc
Sent: Thursday, 13 November 2008 09:36
To: [log in to unmask]
Subject: error propagation in the normalization of isotope data

Dear Isogeochemers,

i have got a question concerning error propagation in the normalization
of isotope data to the respective reference scales. For example: I
measure my samples versus two certified standards (e. g. VSMOW and
SLAP), that come with a predefined uncertainty of e.g. 0.3 permill (the
uncertainty of the interlaboratory comparison data set eveluated by the
IAEA).

When normalizing my data to the reference scale, i of course have to
include this error. Assuming an additive relationship between this error
and my measurement standard deviation (both are absolute errors), the
easiest error propagation formula d(x+y) = [(dx)2 + (dy)2]1/2
(d(x+y)=total error; (dx)=IAEA uncertainty; (dy)=measurement standard
deviation) should be applicable and yield my total error of the
normalized data.

Here are my questions:

First: Am i right?

Second: In the formula above, do i have to include the standard
deviation of my measurement of the certified standards, too? This would
mean that my total sample error is composed of three fractional errors
(1. IAEA uncertainty, 2. standard deviation of my measurement of the
standards and 3. standard deviation of my sample measurement). Right or
wrong?

And third: What if i use laboratory standards to normalize my data? Then
i would have five error sources: 1. IAEA uncertainty, 2. standard
deviation of my measurement of the standards, 3. uncertainty of my
calibration of the lab standards, 4. standard deviation of my
measurement of the lab standards and 5. standard deviation of my sample
measurement. Right or wrong? And if right, which error propagation
calculation is the right one to apply? The one mentioned above? Or a
more complicated one...?

As i am quite new in the field of isotope geochemistry, i hope that some
of the very experienced scientists on the list are willing to help me a
little bit...

Thanks a lot in advance!

Marc Ruppenthal
Mainz University

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