Error Propagation
The analysis of uncertainties (errors) in measurements and calculations is essential in the
physics laboratory. We will start with a review of differentials from calculus, introduce
partial derivatives, and then derive the general equation for error propagation.
Review From Calculus
If y = f(x), and x changes from some initial value xo to a final value x1, then there is
corresponding change in y from yo = f(xo) to y1 = f(x1). Thus, the increment
∆x = x1 – xo
(1)
produces a corresponding increment
∆y = y1 – yo = f(x1) - f(xo)
(2)
From equation (1) we have x1 = xo + ∆x and thus (2) can be written as
∆y = f(xo + ∆x) - f(xo)
(3)
We will drop the zero subscript on xo and use x to denote some initial position as well as
the variable. Equation (3) now becomes
∆y = f(x + ∆x) - f(x)
(4)
1
The derivative can now be defined in the following equivalent forms:
dy
∆y
= lim
dx ∆x →0 ∆x
dy
f ( x + ∆x ) − f ( x )
= lim
dx ∆x →0
∆x
∆f
∆x →0 ∆x
f ′( x) = lim
Differentials
dy
is the symbol for the derivative. Up to now the symbols dy and dx, called
dx
differentials, have no meaning by themselves. However, we shall now define them so
that they do have a meaning.
The symbol
Let’s suppose that x is fixed and define dx to be an independent variable than can be
assigned any value. If f(x) is differentiable at x, we define by the formula
dy = f ′( x)dx
(5)
If dx ≠ 0 , then we can divide both sides to obtain
dy
= f ′( x)
dx
(6)
dy
is equal to the slope of the tangent line to f(x) at x. Therefore, dy and dx
dx
correspond to the rise and run of this tangent line.
But f ′( x) =
In general, the increment ∆y and the differential dy are different. To see the difference
let’s consider the case when dx = ∆x and look at the tangent line to y = f(x) at x.
2
a) ∆y is the change in y that occurs if we start at x and move along the curve until
we have moved a distance ∆x.
b) dy is the change in y that occurs if we start at x and move along the tangent line
until we have moved a distance ∆x.
Consider the function f(x) evaluated at the point xo. The figure below shows that for x
close to xo, the tangent line to the curve at xo is a good approximation to the curve f(x).
The tangent line goes through the point (xo, f(xo)) and has slope f ′( xo ) . Using the pointslope form of a line for the tangent line,
y - yo = m(x – xo)
or
y = yo + m(x – xo)
Since we are considering x close to xo, then the height y of this tangent line will closely
approximate the height f(x) of the curve and then y ≈ f(x). Thus,
f(x) ≈ f(xo) + f ′( xo ) ∆x
(7)
But ∆x = x - xo implies that x = ∆x + xo, thus
f(xo+ ∆x) ≈ f(xo) + f ′( xo ) ∆x
(8)
This is called the linear approximation of f(x) near x when ∆x is small.
3
Example - Use equation (8) to approximate
9.1
f ( x) = x
x = 9.1
xo = 9.0
∆x = x − xo = 0.1
f (9.1) ≈ f (9.0) + f ′(9.0)(0.1)
f (9.0) = 9.0 = 3
1
f ′( x) =
2 x
1
f ′(9) =
= 0.167
2 9
f (9.1) ≈ 3.0167
This gives a 0.0026% error when comparing to the true value of 9.1 = 3.0166 .
Going back to Eq. (8) and rearranging gives the following,
f(xo+ ∆x) ≈ f(xo) + f ′( xo ) ∆x
f ( xo + ∆x) − f ( xo ) ≈ f ′( x0 )∆x
∆y ≈ f ′( x0 )∆x
If we drop the subscript on xo
∆y ≈ f ′( x)∆x
(9)
Now recall that,
dy = f ′( x)dx
(10)
Now comparing Eq. (9) and Eq. (10) we see that if ∆x = dx, then
∆y = dy
(11)
Thus, if ∆x is relative “small” and we ensure that ∆x = dx, then to a very good
approximation
dy = f ′( x)dx
4
 df 
can be written in the following form ∆y =   ∆x . Using f instead of y :
 dx 
 df 
∆f =   ∆x
 dx 
(12)
This equation allows us to determine the error (uncertainty) in the function f associated
with the error (uncertainty) ∆x in the physical measurement x. In this way we say that the
error ∆x propagates to produce an error ∆f in the calculated value of f. We refer to this as
propagation of error. Equation (12) is valid only for functions of one variable. Later we
will generalize this equation to include functions of more than one variable.
Example – A sphere has radius r = 50.00 cm ± 0.02 cm. Calculate the uncertainty in the
volume.
 dV 
∆V = 
 ∆r
 dr 
4
V = π r3
3
dV
= 4π r 2
dr
(
)
∆V = 4π r 2 ∆r
∆V = 4π (50.00cm) 2 (0.02cm)
∆V = 600cm3
V = 5.236 × 105 ± 600 cm3
Partial Derivatives
There are many formulas in which a function depends on two or more variables. For
example, the volume of a box of lengths b, w, and h
V = bwh
Likewise, the function z = f(x,y) is a function of two independent variable x and y. As an
example
z = f(x,y) = x2 + 3xy
is a function of the independent variable x and y.
5
Let f = f(x,y). If we hold y constant, then f(x,y) is a function of x alone. The derivative of
∂f
f(x,y) with respect to x is denoted by and is called the partial derivative of f with
∂x
respect to x. Likewise, we could have fixed x and taken the partial derivative of f with
∂f
. Using the above function
respect to y. This would be denoted by
∂y
z = x2 + 3xy
∂z
= 2x + 3y
∂x
∂z
= 3x
∂y
Example
Let f = 4xyz + x2 +3yz3
∂f
= 4 yz + 2 x
∂x
∂f
= 4 xz + 3z 3
∂y
∂f
= 4 xy + 9 yz 2
∂z
Propagation of Errors
For a function of one variable we’ve shown that
 df 
∆f =   ∆x .
 dx 
Extending this equation to a function of 3 variables f = f(x,y,z) the result is
 ∂f 
 ∂f 
 ∂f 
∆f =   ∆x +   ∆y +   ∆z
 ∂x 
 ∂z 
 ∂y 
where the derivatives are partial derivatives. I have stated this result without proof but
 df 
the proof is similar to the derivation of ∆f =   ∆x .
 dx 
Suppose you make a series of measurements determining sets of values for the variables
x, y, and z with uncertainties σx, σy, and σz. We will now show how to calculate the
uncertainty in f = f(x,y,z) produced by such uncertainties.
6
If the uncertainties of the measurements are random and independent (uncertainties must
also be “small”), then after repeated measurements the measurements will be distributed
(centered) about the mean. The distribution that results is called a Gaussian or Normal
Distribution (bell-shaped curve) which is given by the following formula:
( x − x )2
−
1
2
f ( x) =
e 2σ
σ 2π
σ = standard deviation
x = mean
The following are graphs of the Normal Distribution equation with different values of the
standard deviation σ.
This function has very important properties in probability and statistics. For our purpose
we will focus on the representation of the quantity σ which is called the standard
deviation. The standard deviation is defined by the following statistical expression
n
σ=
∑(X
i =1
i
− X )2
n
Some of the properties of the standard deviation are the following:
1.
2.
3.
4.
Measure of the average deviation from the mean.
Measure of the dispersion of a set of data from the mean.
68% of data falls within one standard deviation.
68% probability that a measurement falls within one standard deviation.
7
5. Small σ corresponds to a sharp Gaussian Distribution and a large σ corresponds to
a broad distribution.
The standard deviation σ can also be written as
σ=
∑ ( ∆x )
2
n
where ∆x = xi − x is the deviation of a single measurement from the mean. We shall now
define the standard deviation σ to be equal to the uncertainty of a single measurement.
The uncertainty in the function f = f(x,y,z) is given by
σf =
∑ ( ∆f )
2
n
(13)
where ∆f is given by
 ∂f 
 ∂f 
 ∂f 
∆f =   ∆x +   ∆y +   ∆z
 ∂x 
 ∂z 
 ∂y 
(14)
Substituting Eq. (14) into Eq. (13) gives
8
σ 2f = ∑
( ∆f )
n
 ∂f 
1  ∂f 
 ∂f  
σ = ∑   ∆x +   ∆y +   ∆z 
n  ∂x 
 ∂z  
 ∂y 
2
2
f
σ 2f =
 ∂f 
 ∂f 
1   ∂f 
 ∂f    ∂f 
 ∂f   
   ∆x +   ∆y +   ∆z    ∆x +   ∆y +   ∆z  
∑
n   ∂x 
 ∂z    ∂x 
 ∂z   
 ∂y 
 ∂y 
2
2
 ∂f  2

∂f 
 ∂f 
2
2
2

 ∂f   ∂f 
 ∂f   ∂f 
  ( ∆x ) +   ( ∆y ) +   ( ∆z ) + 2     ∆x∆y + 2     ∆x∆z + 
1
 ∂x 
 ∂z 
 ∂x   ∂y 
 ∂x   ∂z 
 ∂y 

σ 2f = ∑ 

n   ∂f   ∂f 
 2     ∆y∆z

  ∂y   ∂z 

 ∂f 
( ∆y )  ∂f  ( ∆z )
 ∂f   ∂f  1
∑ n +  ∂y  ∑ n +  ∂z  ∑ n + 2  ∂x   ∂y  n ∑ ∆x∆y
 
 
 ∂f   ∂f  1
 ∂f   ∂f  1
+2     ∑ ∆x∆z + 2     ∑ ∆y∆z
 ∂x   ∂z  n
 ∂y   ∂z  n
 ∂f 
σ = 
 ∂x 
2
f
2
( ∆x )
2
2
2
2
2
Since the uncertainties in the measurement x, y, and z are random and independent, then
Σ∆x∆y = 0
Σ∆x∆z = 0
Σ∆y∆z = 0
Therefore,
2
2
2
 ∂f 
 ∂f 
 ∂f 
σ =   σ x2 +   σ y2 +   σ z2
 ∂x 
 ∂z 
 ∂y 
2
f
2
2
2
 ∂f 
 ∂f 
 ∂f 
σ f =   σ x2 +   σ y2 +   σ z2
 ∂x 
 ∂z 
 ∂y 
General Error Propagation Equation
Example
f=x +y
9
∂f
=1
∂x
∂f
=1
∂y
σ f = σ x2 + σ y2
Example
f = xy
∂f
=y
∂x
∂f
=x
∂y
σ f = y 2σ x2 + x 2σ y2
σf
xy
=
2 2
y 2σ x2 x σ y
+
x2 y 2 x2 y 2
σ  σ 
σ f = f  x  + y 
 x   y 
2
Example
2
Calculate the value of z and the uncertainty if z = wx + y
w = 4.52 ± 0.02 cm
x = 2.0 ± 0.2 cm
y = 3.0 ± 0.6 cm
2
10
z = zbest ± σ z
2
2
2
 ∂z 
 ∂z 
 ∂z 
σ z =   σ x2 +   σ y2 +   σ w2
 ∂x 
 ∂w 
 ∂y 
∂z
= w = 4.52cm
∂x
∂z
= 2 y = 6.0cm
∂y
∂z
= x = 2.0cm
∂w
σ z = (4.52cm) 2 (0.2cm) 2 + (6.0cm) 2 (0.6cm) 2 + (2.0cm) 2 (0.02cm) 2
σ z = 3.7cm 2 ≈ 4cm 2
2
zbest = wbest xbest + ybest
zbest = (4.52cm)(2.0cm) + (3.0cm) 2 = 18.0400cm 2 ≈ 18cm 2
zbest = 18 ± 4cm 2
11