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Appendix C: Uncertainty Formulae

For a function F of multiple variables x, each with uncertainty
σx, 
 the square of the uncertainty in F is given as
( 1 )
σF2 = Σi
δ F
δ xi
 σxi
2
 

Addition rule:
F(x1,x2) = x1 + x2 

( 2 )
σF2 =
δF
δx1
 σx1
2
+
δF
δx2
 σx2
2
 
= σx12 + σx22
 
Example: Assume you must determine a total length L consisting of two measurements x1 and x2.
( 3 )
L = x1 + x2 
( 4 )
σL2 =
δF
δx1
 σx1
2
 +
δF
δx2
 σx2
2
 
If
x1 
 = (1.23 ± 0.02) m and
x2 
 = (4.17 ± 0.01) m, then
( 5 )
σL2 = ((0.02)2 + (0.01)2) m2
 
= (4 × 10−4 + 1 × 10−4) m2
 
= 5 × 10−4 m2
 
σL = 2.2 × 10−2 m, and
 
L = (5.40 ± 0.02) m
 

Product rule: F(A,B,C) = kAmBnCp

( 6 )
σF2 =
δF
δA
 σA
2
 +
δF
δB
 σB
2
 +
δF
δC
 σC
2
 
=
mF
A
 σA
2
 +
nF
B
 σB
2
 +
pF
C
 σC
2
 
= F2
mσA
A
2
 +
nσB
B
2
 +
pσC
C
2
 
σF = F
mσA
A
2
 +
nσB
B
2
 +
pσC
C
2
 
Example:
F = k
Q1Q2
r2
 
( 7 )
σF = F
σQ1
Q1
2
 +
σQ2
Q2
2
 +
–2σr
r
2
 
Assume
Q1 = (6.1 ± 0.4) 
 µC,
Q2 = (4.7 ± 0.3) 
 µC,
r = (0.025 ± 0.003) 
 m.
F = 8.99 × 109
(6.1 × 10−6)(4.7 × 10−6)
0.0252
= 4.1 × 102 N 
( 8 )
σF = 4.1 × 102
0.4
6.1
2
 +
0.3
4.7
2
 +
(−2)(0.003)
0.025
2
 
= 4.1 × 102
4.30 × 10−3 + 4.07 × 10−3 + 5.76 × 10−2
 
= (4.1 × 102) (0.26)
 
= 1.07 × 102 N
 
F = (410 ± 100) N 

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