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Lecture 2, PHYS1140
Propagation of errors
Given a function f = f(x,y), where x and y have uncorrelated uncertainties δx and δy, the
uncertainty in f is
2
2
⎛ ∂f ⎞⎟ ⎛⎜ ∂f ⎞⎟
⎜
δf = ⎜ δx ⎟⎟ + ⎜ δy⎟⎟
⎜⎝ ∂x ⎠ ⎝⎜ ∂y ⎠⎟
This is a general case formula that works for any function f(x,y). (We will derive this
equation later.)
There are many special cases:
Special Case: Function of one variable only: f = f(x)
2
⎛ df ⎞
df
δf = ⎜⎜ δx ⎟⎟⎟ =
δx
⎜⎝ dx ⎠
dx
(δf > 0, by definition)
f(x)
f ± δf
f(xo)
x ± δx
x
xo
Special Case: f = A x, A = constant
df
= A ⇒ δf = A δx,
dx
δf
δx
=
f
x
Special Case: f = A xn , A, n = constants
df
= A n x n−1 ⇒ δf = A n x n−1 δx,
dx
M. Dubson
δf
δx
=n
f
x
©University of Colorado at Boulder
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Special case: addition or subtraction f(x) = x + y or f(x) = x – y
∂f
∂f
=1,
=1
∂x
∂y
δf = δx 2 + δy2
⇒
“Addition in quadrature”
Regular addition = A + B
δx 2 + δ y 2
Addition in quadrature =
A2 + B2
δy
δx
Why addition is quadrature?
Have two numbers: x ± δx, y ± δy that we add to get f = x + y.
If we add the numbers, the errors add.
We might be unlucky and the errors might both be in the same direction (both numbers
too high or both too low), leading to the
δf
δx
WORST CASE: δf = δx + δy
δy
Or we might get lucky and the errors cancel (one number too high and the other too low),
leading to
δy
δf
BEST CASE: δf = δx −δy
M. Dubson
δx
©University of Colorado at Boulder
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But neither of these is very likely, the MOST LIKELY ERROR is something in between:
MOST LIKELY CASE turns out to be δf = δx 2 + δy2
δf
δy
δx
The most likely error is the average error we would get if we measured x and y
repeatedly and each time we got random errors in the windows x ± δx, y ± δy.
Reminder: this all assumes that the errors in x and y are uncorrelated.
Special case: multiplication or division f(x) = x y or f(x) = x / y
Multiplication: f = x y
∂f
∂f
=y,
=x
∂x
∂y
⇒
2
2
δf = ( y δx ) + ( x δy)
This can be written as..
δf
f
2
=
2
⎛ ⎞
⎛ δx ⎟⎞
⎜⎜ ⎟ + ⎜⎜ δy ⎟⎟
⎜⎝ y ⎠⎟⎟
⎝⎜ x ⎟⎠
You get exactly the same formula for division f = x/y.
Summary:
For multiplication or division, fractional uncertainties add in quadrature.
For addition or subtraction, absolute uncertainties add in quadrature.
M. Dubson
©University of Colorado at Boulder
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Example: A cylinder of radius R and height H has volume V = π R2 H
The radius and height are measured to be
R = 10.0 ± 0.2 cm, H = 22.5± 0.1 cm
R
H
What is the volume and its uncertainty?
Answer:
The volume is V = π R2 H = π (10.0)2 (22.5) = 7068 cm3
(This answer is way too precise but we won’t be sure how to round until we compute the
uncertainty.)
∂V
∂V
V(R, H) = π R2 H
so
= 2π R H,
= π R2
∂R
∂H
2
2
⎛ ∂V ⎞⎟ ⎛ ∂V ⎞⎟
δV = ⎜⎜
δR + ⎜
δH
⎜⎝ ∂R ⎠⎟⎟ ⎝⎜⎜ ∂H ⎠⎟⎟
=
=
(2π R H δR ) + (π R 2 δH)
2
⎡2π (10.0) (22.5) (0.2)⎤ + ⎡π (10.0)2 (0.1)⎤
⎢⎣
⎥⎦
⎣
⎦
2
2
2
=
2
2
(282) + (31) = 283 ≈ 300 cm3
Notice that the largest error term usually dominates the uncertainty due to the squaring
when adding in quadrature — especially since we always round the uncertainty to 1 (or at
most 2) sig.figs.
Now that we have the uncertainty, we can write the properly-rounded final answer in
“standard format”:
V = 7100 ± 300 cm3
M. Dubson
©University of Colorado at Boulder