Applied Statistics
Error propagation
Troels C. Petersen (NBI)
“Statistics is merely a quantization of common sense”
Sunday, August 31, 14
1
Error propagation
Imagine that y is a function of xi , and that we wish to find the error on y
from the errors on xi . Making a Taylor expansion of the function y gives:
n
X
@y
y(x̄) ' y(µ̄) +
(xi
@x
i
i
µi )
In order to get the uncertainty of y as a function of the variables xi we
calculate:
2
x
x̄2 = E[x2 ]
= x2
E[y(x̄)] ' y(µ̄)
2
2
E[y (x̄)] ' y (µ̄) +
Sunday, August 31, 14

n
X
i,j
E 2 [x]
@y @y
Vij
@xi @xj
2
Error propagation formula
Subtracting the two formulae, we obtain:
2
y
=

n
X
i,j
@y @y
@xi @xj
Vij
x̄=ȳ
If there are no correlations, only the diagonal (individual errors) enter:
2
y
=

n
X
i
Sunday, August 31, 14
@y
@xi
2
2
i
x̄=ȳ
3
Specific error propagation formula
Addition
Specific formula:
x=u+v
2
x
General formula:
2
x
=
2
u
+
2
v
+2
2
uv
x = au + bv
= a2
2
u
+ b2
2
v
+ 2ab
2
uv
“When adding numbers, their errors add in quadrature”
4
Sunday, August 31, 14
Specific error propagation formula
Multiplication
x = uv
2
x
= (v
u)
2
+ (u
v)
2
+ 2uv
2
uv
Dividing by x2 to get relative terms, we obtain:
2
x
x2
=
2
u
u2
+
2
v
v2
+2
2
uv
uv
“When multiplying numbers, their RELATIVE errors add in quadrature”
5
Sunday, August 31, 14
Error propagation at work...
Harrison's first sea clock (H1)
John Harrison (24 March 1693 – 24 March 1776)
British clockmaker extraordinaire
“Won” the Longitude Act prize (3 sec/day).
Harrison build H1-H5.
K1 (Copy of H4) was used by James Cook.
6
Sunday, August 31, 14
Error propagation at work...
Harrison’s Gridiron pendulum is
designed to cancel the change in
length (in fact moment of inertia)
with temperature.
Coefficient of thermal expansion:
Iron = 11.8×10-6/C° Zinc = 30.2×10-6/C°
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Sunday, August 31, 14
Error propagation at more work...
Analysis of tiny differences in Uranus’ orbit
from Newtonian prediction led to the
prediction and discovery of Neptune!
Continuing with Mercury...
Urbain Le Verrier (1811-1877)
Sunday, August 31, 14
8
Advanced example of error propagation (Higgs particle mass):
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Sunday, August 31, 14
Reporting uncertainties
The systematic uncertainties of a measurement should be reported in a
table, and if measurements are combined, the correlation needs
consideration.
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Sunday, August 31, 14