Error Analysis Procedures:
Written by Ashok Maliakal, Zhiqiang Liu, and Kirsten Ostberg
1) Reporting Single Values:
All single value measurements must be reported with an estimate on the
uncertainty in the measurement. Uncertainty may be measured as following:
a) Average Error: Report mean of 2 or more measurements and ± average error.
Ex. Measured quantity (9, 11) Reported value 10 ± 1
b) Standard Deviation: Report mean of 3 or more measurements and standard
deviation.
Ex. Measured quantity (9, 11, 14) Reported value 11 ± 3
Very important to match significant figures with the size of the error. Error typically
should only have 1, possibly two significant figure. Value should be reported to only upto
1 significant figure more than error term.
Ex. 10.53 ± 1 is unacceptable; 10.5 ± 1 is reasonable; 11 ± 1 is also ok.
2) Curve Fitting:
In fitting curves it is important to observe the following error analysis procedures
to prevent overinterpretting data and/or misleading oneself and others. Also make sure
that the number of data points is greater than the number of fitting parameters, otherwise
your fit could be absurd. (Ex. Fitting a line with 2 points, or worse a parbola with two
points).
a) Operate under the multiple hypothesis paradigm, so that no one hypothesis
(ex. Linear fit) is given unfair advantage.
Ex. Fitting data with 0.5, 1, and intermediate curves to test out multiple
hypotheses.
0.25
kobs
0.20
0.15
0.10
0.05
0.00
0
10
20
[PSLi]
30
40x10
-3
b) Make sure you have sufficient points, especially in critical areas. Critical
Areas are where the multiple hypotheses differ the most in their expected
outcomes. See Figure.
Ex. Critical Area for 0.5 and 1 order behavior:
0.25
kobs
0.20
0.15
0.10
Critical Area
0.05
0.00
0
10
20
[PSLi]
30
40x 10
-3
c) Incorporate error bars on your plots. This can be done using IGOR Pro. See
Steffen or Ashok for assistance. This will clearly help identify whether
alternative hypotheses can be disqualified or not.
0.30
0.25
kobs
0.20
0.15
0.10
0.05
0.00
0
10
20
[PSLi]
30
40x10
-3
3) Propagation of Error:
When performing mathematical functions on data with errors, there are specific
ways to treat the error that take into account the propagation of errors throughout the
calculation. The following table contains a list of the most frequently used mathematical
functions and the specific error calculation that must accompany it.
Function
Error Calculation
y  x1  x2
Ey  (E x1  E x 2 )
y  x1  x2
E 2 E 2 
Ey
x1
x2




 
 






y
 x1   x 2 
y  x1 x
Same as above
2
2
2
y  log( x)
 1  E x 
Ey  
  
2.303   x 
y  10 x
Ey 10 x 2.303E x 
y  ln( x)
Ey 
Ex
x
y  ex
Ey  y  E x
y  xa
E 
Ey  y  a   x 
 x 
** In the equations above E represents the error of the variable indicated in the subscript
For more complex mathematical functions such as the treatment of error when using
partial derivatives, etc. there are other good references available. For quick and easy
references the following websites are recommended:
www.pitt.edu/~jchem/ch25/ERROR_PROP.htm
www.rit.edu/~vwlsps/uncertainties/Uncertaintiespart2.html#propagation
4) Instrument Related Errors:
a) Appropriate calibration and internal standard. Standard samples to quickly
check performance of instruments, sensitivity, drift, etc..
GPC: Polystyrene and PMMA standards, check the retention time and
polydispersity and compare to previous calibration and specs of the samples. If
necessary, use toluene or some other compounds as flow rate marker.
GC: performance mixture for S/N. Baking and flushing of column to fend off
spurious peaks.
GC/MS: performance mixture for S/N. Calibration gas, PFTPA and methanol for
Mass calibration.
b) Integral related data analysis. Check integration protocol, make sure the
selection of baseline, peak (start and stop) are appropriate. For non-resolved
peaks, use of mixture of known concentration of the overlapping components
to establish right choice of baseline, peak, especially useful for GC, GPC. For
noisy peaks, avoid over trusting the data, especially if not reproducible
enough.
Example on the integration parameters,
For the Varian chiral GC, a concentration range between 10-2 to 10-5 gives reasonably
reproducible results. If the slanted baseline is chosen as in the picture shown above, a
50:50 isomer mixture give <1% error throughout the whole dynamic range.
Another way to define the baseline/peaks, shown in the picture below as being
horizontal, gives ca. 5% error at both the high and low end of the dynamic range.
1. Loading related error. Check appropriate loading for best results.
GC/MS. High concentration causes trouble for ion trap detector. Suggest no more
than 10-4M concentration and start with high split ratio (70:1) and reduce it if
necessary.
GPC: <=1mg/ml. Higher loading causes tailing.
2. Reproducibility. Repeat the measurements for distribution of measured results.
3. Random order of data analysis to avoid taking instrument monotonal drift as real
results in a series of data.