ERROR PROPAGATION
1. Measurement of Physical Properties
The value of a physical property often depends on one or more measured quantities
u  f ( x , y , z , )
Example: Volume of a cylinder
V  f ( h, r )
r = 6 cm
2
 hr
 (10 cm)  (6 cm) 2
 1131 cm 3
h = 10 cm
2. Systematic Errors
A systematic error in the measurement of x, y, or z leads to an error in the determination of u.
 u 
 u 
 u 
du    dx    dy    dz
 x  y ,z
 z  x , y
 y 
x ,z
This is simply the multi-dimensional definition of slope. It describes how changes in u depend
on changes in x, y, and z. [Note: Sime refers to this expression as the “maximum error.”]
Example: A miscalibrated ruler results in a systematic error in length measurements. The
values of r and h must be changed by +0.1 cm.
 V 
 V 
dV  
 dr  
 dh
 r  h
 h  r
 (h  2 r )dr  (  r 2 )dh
 [(10 cm)  2 (6 cm)](01
. cm)  [  (6 cm) 2 ](01
. cm)
 38 cm 3  11 cm 3
 49 cm 3
3. Random Errors
Random errors in the measurement of x, y, or z also lead to error in the determination of u.
However, since random errors can be both positive and negative, one should examine (du)2 rather
than du.
2
 u 
 u 
 u 
(du )    dx 2    dy 2    dz 2
 x  y ,z
 z  x , y
 y  x ,z
2
2
2
 u   u 
 u   u 
 u   u 
 2    dxdy  2    dydz  2    dxdz
 x  y,z  y  x ,z
 x  y ,z  z  x , y
 y  x ,z  z  x , y
If the measured variables are independent (non-correlated), then the cross-terms average to zero
dx dy  0, dy dz  0, and dx dz  0
as dx, dy, and dz each take on both positive and negative values.
Thus,
du  (du) 2
2
 u 
 u 
 u 
   dx 2    dy 2    dz 2
 x  y ,z
 z  x , y
 y  x ,z
2
2
[Note: Sime refers to this expression as the “most probable error.”]
Equating standard deviation with differential, i.e.,
su  du, sx  dx, s y  dy, and sz  dz
results in the famous error propagation formula
2
 u 
 u 
 u 
su    s x 2    s y 2    sz 2
 x  y ,z
 z  x , y
 y  x ,z
2
2
This expression will be used in the Uncertainty Analysis section of every Physical Chemistry
laboratory report!
Example: There is 0.1 cm uncertainty in the ruler used to measure r and h.
 V  2  V  2
sV  
 s    sh
 r  h r
 h  r
2
2
 (h  2 r ) 2 sr 2  (  r 2 ) 2 sh 2
 [(10 cm)  2 (6 cm)]2 (01
. cm) 2  [  (6 cm) 2 ]2 (01
. cm) 2
 1,421 cm 6  128 cm 6
 39 cm 3
Thus, the expected uncertainty in V is 39 cm3.
4. Purpose of Error Propagation
 Quantifies precision of results
Example: V = 1131  39 cm3
 Identifies principle source of error and suggests improvement
Example: Determine r better (not h!)
 Justifies observed standard deviation
If sobserved  scalculated then the observed standard deviation is accounted for
If sobserved differs significantly from scalculated then perhaps unrealistic values were
chosen for sx, sy, and sz.
 Identifies type of error
If uobserrved  uliterature  scalculated then error is random error
If uobserrved  uliterature  scalculated then error is systematic error
5. Calculating and Reporting Values when using Error Propagation
Use full precision (keep extra significant figures and do not round) until the end of a calculation.
Then keep two significant figures for the uncertainty and match precision for the value.
Example: V = 1131  39 cm3
6. Comparison of Error Propagation to Significant Figures
Use of significant figures in calculations is a rough estimate of error propagation.
Example:
V  h  r 2  (10.0 cm)  (6.0 cm) 2  1100 cm3
Keeping two significant figures in this example implies a result of V = 1100  100 cm3, which is
much less precise than the result of V = 1131  39 cm3 derived by error propagation.
7. Common Applications of the Error Propagation Formula
Several applications of the error propagation formula are regularly used in Analytical Chemistry.
u  xy
Example:
2
 u 
 u 
su    s x 2    s y 2
 x  y
 y  x
2
2
2
 1 s x 2  1 s y 2
 sx 2  s y 2
Example:
u  xy
2
 u 
 u 
su    s x 2    s y 2
 x  y
 y  x
2

 y

y 2 sx 2  x 2 s y 2
2
sx 2   x s y 2
2
2
sx 2 s y
u 2  2
x
y
Analytical chemists tend to remember these common error propagation results, as they encounter
them frequently during repetitive measurements. Physical chemists tend to remember the one
general formula that can be applied to any case, as they encounter widely varying applications of
error propagation. (Or perhaps analytical chemists take a more utilitarian approach, whereas
physical chemists take a more "from first principles" approach.)