Propagation of Measurement Uncertainties

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

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 there are a few rules that we can use to make these calculations simpler. The rules are usually approximations and are based on the assumption that the relative uncertainty in any one number is small, say under 10%, so the precision of all your measurements must be that high.

Notation

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

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.

As we will see later, this is

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

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.

Adding or Subtracting

Derivation of Rule for Addition

This result can be derived quite simply. First, consider the addition of two measurements

- Finding the maximum possible value when adding

The biggest this sum can be is

- Finding the minimum possible value when adding

The smallest this sum can be is

- Result for addition

Combining the result for the maximum and minimum values, the range of values for addition is

It should be clear that no matter how many numbers we add together, we should get the general result

So the maximum uncertainty in the sum A₁ + A₂ + ... + A_n is

In "sloppy" notation, this is

Derivation of Rule for Subtraction

Next, consider the subtraction of two measurements

- Finding the maximum possible value when subtracting

The biggest this difference can be is

- Finding the minimum possible value when subtracting

The smallest this difference can be is

- Result for subtraction

Combining the result for the maximum and minimum values, the range of values for subtraction is

It should be clear that no matter how many numbers we subtract, we should get the general result

So the maximum uncertainty in the difference B₁ - B₂ - ... - B_n is

In "sloppy" notation, this is

Note that whether we are adding or subtracting, we add absolute uncertainties.

Example

Let A = 5.23 <± 0.07, B = 4.67 <± 0.04, and C = 7.11 <± 0.09. Find F = A + B - C.

First

Next, using our rule

Thus the answer is

We can only keep one figure in the uncertainty, thus

Multiplying and Dividing

Derivation of Rule for Multiplication

This result can be derived as follows. First, consider the multiplication of two measurements

- Finding the maximum possible value when multiplying

The biggest the product AB can be is

Now δ(A) and δ(B) are supposed to be small compared to P(A) and P(B). As well we keep the uncertainty only to one figure. Hence δ(A)δ(B) is negligible and can be dropped. Thus we have

We usually rewrite the above expression as

- Finding the minimum possible value when multiplying

The smallest the product AB can be is

Now δ(A) and δ(B) are supposed to be small compared to P(A) and P(B). As well we keep the uncertainty only to one figure. Hence δ(A)δ(B) is negligible and can be dropped. Thus we have

We usually rewrite the above expression as

- Result for multiplication

Combining the result for the maximum and minimum values, the range of values for multiplication is

Remember that this result is an approximation, good only when the relative uncertainties are small.

When we multiply many numbers, the pattern is the same

So the maximum uncertainty in the product A₁ × A₂ × ... × A_n is

In "sloppy" notation, this is

Derivation of Rule for Division

Next consider the division of two measured quantities

First we rewrite this in relative form as

- Finding the maximum possible value for division

The biggest A/B can be is

There is a mathematical formula that we can use here

So the biggest value of A/B can be written as

Multiplying out the terms involving the relative uncertainty, we get

Now δ(A)/P(A) and δ(B)/P(B) are supposed to be small compared to 1. As well we keep the uncertainty only to one figure. Hence {δ(A)/P(A) × δ(B)/P(B)} and smaller terms are negligible and can be dropped. Thus we have

We usually rewrite the above expression as

- Finding the minimum possible value for division

The smallest A/B can be is

Using the mathematical formula

the smallest value can be written as

Multiplying out the terms involving the relative uncertainty, we get

Now δ(A)/P(A) and δ(B)/P(B) are supposed to be small compared to 1. As well we keep the uncertainty only to one figure. Hence {δ(A)/P(A) × δ(B)/P(B)} and smaller terms are negligible and can be dropped. Thus we have

We usually rewrite the above expression as

Combining the results for the maximum and minimum values we find the range of values for the division A/B to be

So the maximum uncertainty in A/B is

In "sloppy" notation, this is

Note that the method of finding the maximum uncertainty is the same for division and multiplication.

Remember that this result is an approximation, good only when the relative uncertainties are small.

Example

Let A = 5.23 <± 0.07, B = -4.67 <± 0.04, and C = 7.11 <± 0.09. Evaluate F = AB/C.

First, the principle value is

Next applying our rule

So we have

Rounding down to one significant figure in the uncertainty yields

Note how we had to handle the negative signs so that the relative and absolute uncertainties were always positive.

Powers and Roots

Derivation of Rule for Powers

Consider a number raised to some value z

- Finding the maximum possible value for powers

The maximum value of will A^z be

This may be rewritten as

There is a mathematical identity called the Binomial Expansion which states that

So

Now δ(A)/P(A) is supposed to be small compared to 1. As well we keep the uncertainty only to one figure. Hence {δ(A)/P(A)}² and smaller terms are negligible and can be dropped. Thus we have

- Finding the minimum possible value for powers

The minimum value of A^z will be

This may be rewritten as

Applying the Binomial Expansion, we get

Now δ(A)/A is supposed to be small compared to 1. As well we keep the uncertainty only to one figure. Hence (δ(A)/A)² and smaller terms are negligible and can be dropped. Thus we have

- Result for powers

Combining the results for the maximum and minimum values we find the range of values for a number to some power

So the maximum uncertainty in A^z is

In "sloppy" noatation, this is

Remember that this result is an approximation, good only when the relative uncertainties are small.

Example

Let A = 5.23 <± 0.07. Evaluate F = A³.

First, the principle value is

Next applying our rule

So we have

Rounding down to one significant figure in the uncertainty yields

Functions and Calculus

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

This can be rearranged to give,

Or if we had considered f(a-δa)

And thus we have the general result that

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

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

WARNING: If you are dealing with trigonometric functions, δx must be in radians!

Examples

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

First set your calculator to radians and determine the principle value

Next, using the appropriate rule from the table above

Thus the answer is

We can only keep one figure in the uncertainty, thus

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

First set your calculator to degrees and determine the principle value

Next, using the appropriate rule from the table above

Thus the answer is

We can only keep one figure in the uncertainty, thus

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

Quick and Dirty

This can be written as

Example

Let A = 0.374 <± 0.009 . Evaluate F = arctan(A).

First set your calculator to radians or degrees and determine the principle value

The arctan function (labelled tan^-1 on most calculators) is not in our table of functions, so we will use the quick approach.

First we evaluate the extremes

and

Thus the uncertainty is

or

Hence we find F to be

Where we have kept only one figure in the uncertainty.

Applying the Rules to Complicated Functions

where A, B, C, and D are measurements and thus have uncertainties. We want to find an expression for the uncertainty in F, δF, which depends only on P(A), P(B), P(C), P(D), δ(A), δ(B), δ(C), and δ(D). Such an expression could be used over and over, in a computer spreadsheet for instance, for many different values of A, B, C, and D.

If we were doing this for one set of values of A, B, C, and D on a calculator, we would do bits at at a time. We would set X = A/B and Y = C/D so that

Next we would set Z = X + Y so that

At this point we have something for which we do have a rule. We use the power rule to find that the uncertainty in F in terms of Z is

We know that the uncertainty in Z = X + Y is given by the addition rule, δ(Z) = [δ(X) + δ(Y)]. So we have

In turn, we use our division rule for calculating the uncertainties in X = A/B and Y = C/D. This yields δ(X) = P(A/B)[δ(A)/P(A) + δ(B)/P(B)] and δ(Y) = P(C/D)[δ(C)/P(C) + δ(D)/P(D)]. Hence

Or more simply, this is

where we have made use of the fact that f = P(Z)^½.

The order of mathematical operations is functions, powers, multiplication or division, and then addition or subtraction. However when we write uncertainty expressions, we work from the outside in. This can be seen if we redo the operations above

In "sloppy" notation, this would be written

Examples

(i) Let A, B, and θ = φ±Δφ be measured quantities. Find an expression for δ(F) where F = A - Bcosθ.

The solution is

In "sloppy" notation, this would be written

Note δφ must be in radians!

(ii) 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).

The absolute uncertainty is

Note the use of the "sloppy" notation.

The relative uncertainty is δ(F)/P(F). Thus

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

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

Notice in examples (iii) and (iv) that, although X = Y, we found that δX does not equal δY. Something funny is going on. Which is correct? To find out we would substitute M = m ± δm, G = g ± δg, and A = a ± δa into X or Y and find the maximum value of the uncertainty.

Clearly, the biggest X can be is

Keeping only first order terms, this becomes

Thus the uncertainty in X is indeed

Why was δY incorrect? Let's look at the expansion

What is the biggest possible value of Y? You might suggest

This is what we did in example (iv). Notice, however, that we have used two different values for M, m + δm and m - δm. This is not allowed. A variable can only have one value at a time.

The more complicated an expression is, the harder it is to determine the "correct" error formula. However, we need not be unduly concerned as we only keep one figure in the uncertainty and the difference between δX and δY should mainly be in the second figure if we are dealing with small uncertainties. As a rule of thumb, expressions involving subtraction should be factored, i.e. common variables taken outside a bracket, before the rules are applied.

Exercises

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

Solutions

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

Questions?mike.coombes@kpu.ca