Propagation of Measurement Uncertainties Using Calculus

Uncertainty is intrinsic to measurement. Doing a calculation with measured quantities requires an understanding of how the uncertainties effect or propagate through the calculation.

Uncertainties may be written in absolute or relative terms

a ± δa or a ± a% ,

where α = (δa/a) × 100.

The uncertainty δa/a or α should be given to one significant digit. There is no difference between the two forms. However, you may be required to present your results in one form rather than the other.

We distinguish between uncertain quantities and exact quantities such a π or 2.2 that have no uncertainty. Exact quantities may be represented as a point on a number line whereas uncertain quantities are represented by a line segment. Figure 1 below shows the exact quantity 1.3 and the uncertain quantity 1.3 <± 0.4 .

The key to error propagation is understanding that we are dealing with a range of numbers, and that any calculation will produce a range of values as an answer, from the smallest that the answer could be to the largest. As a simple example consider the difference between adding two exact quantities, for example adding 4.5 to 3.8, compared to adding 4.5 <± 0.2 to 3.8 <± 0.4. The exact result is 4.5 + 3.8 = 8.3 . The smallest answer we can get for the uncertain quantities is (4.5-0.2) + (3.8-0.4) = 7.7 . The largest answer we can get is (4.5+0.2) + (3.8+0.4) = 8.9 . We would state the answer for the range as 8.3 <± 0.6 .

Simple addition with uncertain quantities is easy to do, but throw in other operations like multiplication and functions like cosine, and things get complicated fast. Luckily if one can do simple derivatives, then even the most complicated problems can be tackled in a straightforward systematic way. The only caveat is that we will be making use of approximations and our results will only be accurate if we are dealing with small relative uncertainties in our measurements, say 10% or less.

Notation

There is some special notation that we use when we discuss uncertainties. First we will use uppercase letters to represent measured quantities, e.g. A = 4.5 <± 0.2 and B = 3.8 <± 0.4. We will use lower case letters to represent the principle value and the uncertainty. Thus A = a ± δa and B = b ± δb, where a = P(A), b = P(B), δa = δ(A), and δb = δ(B). We define the operator P( ) to mean the principle value of the measured quantity, i.e. the value without the uncertainty. Thus P(A) = 4.5 and P(B) = 3.8. We define the operator δ( ) to mean the uncertainty of the measurement, e.g. δ(A) = 0.2 and δ(B) = 0.4.

When we do mathematical operations such as addition or multiplication with measured quantities, we are looking for both the principle value of the result as well as the maximum uncertainty. For example, consider

A + B = P(A + B) ± δ(A + B)

P(A+B) is the principle value of the sum A + B, the value found when the uncertainty in neglected. Here P(A+B) is 4.5 + 3.8 = 8.3. δ(A + B) is the maximum uncertainty in the sum A + B, which we already found to be 0.2 + 0.4 = 0.6.

Warning! This notation is not standard. As an example, considered the product A × B.

A × B = P(AB) ± δ(AB)

As we will see later, this is

A × B = P(AB) <± [δ(A) P(B) + δ(B) P(A)]

It is common, if sometimes misleading, to see the above written as

A × B = AB <± (δA B + δB A)

In the above A refers to both the measured value and its principle value and δA refers to the uncertainty in A. Unfortunately, this "sloppy" notation is close to being the standard.

Functions of One Variable and Calculus

Consider some function f which depends on only one variable x. We will assume that the function f(x) that we are investigating will always be well-behaved and always have a derivative which will be written as df/dx or f ' (x). The diagram below is of a typical function. Recall that the derivative of a function at a point a, df/dx|_a or f ' (a), is slope of a line tangent to the function curve at the point a as shown in the diagram below.

In terms of a formula, the definition of the derivative is

If δa is not vanishingly small but just very small the following is true

This can be rearranged to give,

f(a + δa) ≈ f(a) + δa × f ’(a)

Or if we had considered f(a-δa)

f(a - δa) ≈ f(a) - δa × f’(a)

And thus we have the general result that

f(a ± δa) ≈ f(a) ± δa × f ¢(a)

The equation above is an example of a Taylor Expansion

In our proper notation, A = a ± δa is the measurement. Then to first order

δ{f(A)} = δa × f ’(a)

In the "sloppy" notation, we have

δ{f(A)} = δa × f ’(A)

Just to reiterate, this result is very good as long as δa is small compared to a, generally less than 10%.

Examples

(i) Let θ = 1.23 <± 0.02 radians. Evaluate F = 3cos(θ).

First set your calculator to radians and determine the principle value

P(F) = 3cos(1.23) = 1.0027 .

Next, we take the derivative of F,

F' = -3sin(θ).

Hence the uncertainty is

δ(F) = |Δθ × -3sin(θ)| = 0.02 × 3sin(1.23) = 0.0565

Thus the answer is

F = 1.0027 <± 0.0565

We can only keep one figure in the uncertainty, thus

F = 1.00 <± 0.06

(ii) Let θ = 22.2 <± 0.2° . Evaluate F = ½tan(θ).

First set your calculator to degrees and determine the principle value

P(F) = ½tan(22.2°) = 0.2040 .

Next, we take the derivative of F,

F' = -1 / {2cos²(θ)}.

Hence the uncertainty is

δ(F) = Δθ × |-1 / {2cos²(θ)}|. = (0.2° × π/180°) × |-1 / {2cos²(22.2°)}| = 0.0020

Thus the answer is

F = 0.2040 <± 0.0020

We can only keep one figure in the uncertainty, thus

F = 0.204 <± 0.002

Note that we converted Δθ from degrees to radians in the error calculation.

WARNING:For all trigonometric functions, Δθ must be in radians!

(iii) Let R = 0.151 ± 0.005. Evaluate V = (4/3)πR³.

First we determine the principle value

P(V) = (4/3)π(0.151)³ = 0.01442 .

Next, we take the derivative of V,

V' = 4πR².

Hence the uncertainty is

δ(V) = δR × 4πR² = 0.005 × 4π(0.151)² = 0.0014

Thus the answer is

V = 0.01442 ± 0.0014

We can only keep one figure in the uncertainty, thus

V = 0.014 ± 0.001

Functions of Many Variables and Calculus

Taylor expansions also exist for functions of more than one variable and have a form nearly identical to the one variable term. For a function of two variables, say f(x,y), the Taylor expansion to first order about x = a ± δa and y = b ± δb is

f(a±δa,b±δb) @ f(a,b) ± δa´¶f(x,y)/¶x½_x=a,y=b ± δb´¶f(x,y)/¶y½_x=a,y=b .

The quantity ¶f/¶x is the partial derivative of function f with respect to variable x. One calculates a partial derivative with respect to x by treating the other variables as constants independent of x and doing an ordinary derivative. The partial derivative obeys the product, quotient, and chain rules just like ordinary derivatives.

It is easy to generalize to function of more variables. For instance the first-order Taylor Expansion for the function f(x,y,z) about x = a ± δa, y = b ± δb, and z = c ± δc is

f(a±δa,b±δb,c±δc) @ f(a,b,c) ± δa´¶f/¶x½_x=a,y=b,z=c ± δb´¶f/¶y½_x=a,y=b,z=c ± δc´¶f/¶z½_x=a,y=b,z=c.

So the uncertainty for any function is

δ{f(A,B,C,...)} @da´¶f/¶x + δb´¶f/¶y + δc´¶f/¶z + ... ,

where all the derivates are evaluated at the principle values of the measurements A, B, C, etc.

At this point our "sloppy" notation becomes downright atrocious. It is common to write the uncertainty in the function as

δ{f(A,B,C,...)} @dA´¶f/¶A + δB´¶f/¶B + δC´¶f/¶C + ... .

The symbols A, B, and C now stand for the measurements, the principle values, and the variables of differentiation.

Also note that we assume that each term is positive. It would be more correct to write

δ{f(A,B,C,...)} @ |δA´¶f/¶A| + |δB´¶f/¶B| + |δC´¶f/¶C| + ... .

although the absolute value signs are usually omitted.

Examples

(i) Let A and B be measured quantities. Find an expression for δ(F) where F = AB.

According to our rule we need to do two partial derivatives,

¶F/¶A = B , and

¶F/¶B = A .

Our formula for the uncertainty in the product AB is thus

δF = δA B + δB A.

(ii) Let A,B, C, and D be measured quantities. Find an expression for δ(F) where F = (A/B + C/D)^½.

According to our rule we need to do four partial derivatives,

¶F/¶A = ½ (A/B + C/D)^-½ (1/B) = ½ (1 / BF),

¶F/¶B = ½ (A/B + C/D)^-½ (-A/B²) = ½ (-A / B²F) ,

¶F/¶C = ½ (A/B + C/D)^-½ (1/D) = ½ (1 / DF) , and

¶F/¶D = ½ (A/B + C/D)^-½ (-C/D²) = ½ (-C / D²F) .

Our formula for the uncertainty in the product F is thus

δF = ½ (δA/B + |- δBA/B²| + δC/D + |- δDC/D²|) / F .

This is normally written without the absolute signs

δF = ½ (δA/B + δBA/B² + δC/D + δDC/D²) / F .

(iii) Let A, B, and θ be measured quantities. Find an expression for δ(F) where F = A - Bcos(θ).

According to our rule we need to do three partial derivatives,

¶F/¶A = 1 ,

¶F/¶B = -cos(θ) , and

¶F/¶q = -Bsin(θ) .

Our formula for the uncertainty in the product F is thus

δF = (δA 1 + |δB × -cos(θ)| + |Δθ × -Bsin(θ)|) .

Simplifying, we get

δF = δA + δB cos(θ) + Δθ Bsin(θ) .

Note Δθ must be in radians!

(iv) Let I, L, and R be measured quantities and μ₀ be exact. Find an expression for the relative uncertainty in F = (μ₀/2π)(I²L/R).

According to our rule we need to do two derivatives,

¶F/¶I = (μ₀/2π)(2I/R) , and

¶F/¶R = (μ₀/2π) (-I²/R²) .

Our formula for the uncertainty in the product F is thus

δF = (μ₀/2π) {|δI (2I/R)| + |δR (-I²/R²)|}.

Simplifying, we get

δF = (μ₀/2π) {2δI/R + δR(I/R)²}.

(iv) Let M, G, and A be measured quantities. Find an expression for δ(X) where X = M(G-A).

According to our rule we need to do three derivatives,

¶X/¶M = G - A ,

¶X/¶G = M , and

¶X/¶A = -M .

Our formula for the uncertainty in the product X is thus

δX = |δM(G - A)| + |δGM| + |δA × -M| .

Simplifying, we get

δX = δM(G - A) + δGM + δAM .

Exercises

Find expressions for the relative uncertainty in the following. Treat μ₀ and ε₀ as exact.

i)		ii)
iii)		iv)
v)		vi)

Solutions

i)
ii)
iii)
iv)
v)
vi)

Questions?mike.coombes@kwantlen.ca