Appendix 2: Propagation of Uncertainty (last edited 11/24/2004 )

**Comparison of Uncertain Quantities**

After you perform an experiment and analyze the data, you need to publish your results. This could range from a journal article to an internal company memo to a lab report read only by the TA. You may be comparing your values with theoretical predictions or the results from another experiment, or you may just be stating your results. In all these cases, the respective error magnitudes are just as important as the values themselves.

An estimate of uncertainty is essential to the proper interpretation of any experiment.

The numerical value one assigns to experimental uncertainty has only a statistical meaning. We assume that the inherent randomness in measuring a quantity can be typified by the normal or Gaussian distribution discussed in Appendix 1. To reiterate some of the major points: If you were to measure something 100 times, about 68 of them will cluster within the experimental error of the mean (+1σ) and close to 95 of them will be within twice the error (+2σ). Therefore, 95% of the time any one measurement will be within 2σ of the average of a large number of measurements. In this class, this is the breakpoint for agreement.

If our result A differs from an exact quantity B by less than two times the
measurement uncertainty of A, u{A}, then we say that they agree *within the
accuracy of the experiment*. If they are further away than that, we say that
they don’t agree and state the per cent difference. Doing this comparison is
performing the *Standard Equivalency Test (SET*) and is expressed
mathematically by

(1)

A typical statement for a false SET is, “At the 95% confidence level, my measurement is inconsistent with the theory. The discrepancy is 5%.”

Note that this is the only place where stating a per cent difference is okay.
You should *always* have an independent estimation of the uncertainty in a
quantity, regardless of knowing what the quantity is supposed to be.

As an example, suppose we perform an experiment and obtain an answer of 53 ± 3 meters for some given distance. We compare that measurement to theoretical predictions as follows:

A. If the theory predicts 57 meters exactly, then the difference (4 meters) is
*less* than twice the experimental uncertainty (6 meters), the SET is true,
and we say something to the effect of, “The measurement is in agreement
with the theoretical prediction to within the accuracy of the experiment.”

B. If the theory predicts 59 meters exactly, then the difference (7 meters) is
*more* than twice the uncertainty, the SET fails, and we report, “At the
95% confidence level, the experiment disagrees with the predictions. The
discrepancy is 12%.”

The comparison of two quantities A and B which are *both* uncertain requires
a different analysis. In the statistical sense when we ask if two quantities are
the same, we are really asking if the difference of those two quantities is
zero. Using rules laid out in the following section, the complete mathematical
translation of the SET is

(2)

If B is exact, this reduces to Eq. 1.

Suppose that A were in the cell A1 in a spreadsheet, B were in B1, u{A} in A2, and u{B} in B2. The SET is a logical formula that you can type in a labeled cell. To compare A and B, you would type in one of the following:

=ABS(A1-B1)<=2*SQRT(A2^2+B2^2)

or

=ABS(A1-B1)<=2*SQRT(SUMSQ(A2,B2))

or

=ABS(A1-B1)<=2*SqrtSumSqs(A2,B2), (If you have the User Defined function SqrtSumSqs.)

Excel will calculate both sides of the inequality and do the comparison, returning TRUE if it passes and FALSE if it doesn’t.

**Rules for the Propagation of Error**

Assume we measure two values A and B, using some apparatus. We know these values are uncertain. By physical reasoning, testing, repeated measurements, or manufacturer’s specifications, we estimate the magnitude of their uncertainties. u{A} is the absolute error in A, and u{B} is the absolute error in B. The relative errors are u{A}/A and u{B}/B.

What are the uncertainties in a calculation that uses A and B? What follows are rules that give the uncertainty expression for basic arithmetic operations using A and B, then what to do for more complicated expressions.

The square of the uncertainty in the sum or difference of two numbers is the sum of the squares of individual absolute errors.

(3)

For example, if

• A = 2.5 grams,

• u{A} = 0.4 grams,

• B = 4.1 grams,

• u{B} = 0.3 grams,

then,

You state the sum of these two masses as, “6.6 ± 0.5 g” (g being the abbreviation for grams). The mass difference is “1.6 ± 0.5 g.”

This procedure gives an error on the sum or difference that is larger than either individual uncertainty, but smaller than u{A} +u{B}. This is appropriate because if our measurement of A strays from the true value in one direction (either greater than or less than), our measurement of B is just as likely to fluctuate in the opposite direction as well as in the same direction. Because B can fluctuate in the same direction, it will make the error on the sum larger than u{A}. But because it can also go in the opposite direction, it will not increase the error as much as the maximum error u{A} + u{B}.

Another example of the application of the rule for the error on a difference is
when you are asked if two measurements are consistent. Are they the same
numbers? You perform the SET. Suppose your first measurement of the
oscillation period of a pendulum is t_{1} = 4.0
± 0.1 seconds and the second is
t_{2} = 3.85
± .05 seconds. To find if the two measurements are consistent, you
calculate the positive difference (t_{1} - t_{2} ) = 0.15 seconds and the error on
that difference:

Since the difference is less than twice the error (0.15 ≤ 2×0.11 = 0.22), the SET is true, and you say that the two measurements are consistent.

Multiplication and Division: The Product and Quotient Rule

The *relative* uncertainty in a product or a quotient will be the square root of the
sum of the squares of the *relative *errors in the individual factors.

Translated to mathematics, this is saying that

(4)

These are expressions for relative errors. To get the absolute uncertainty, multiply both sides by the appropriate denominator:

(5)

For example, suppose

• F = 25.0 newtons,

• u{F} = 1.0 newtons,

• x = 6.4 meters,

• x = 0.4 meters.

Then the relative error in both F×x and F/x is

You declare your results as, “F×x = 160 N·m
± 7%,” or “F/x= 3.9 N/m
**±****
7%**.”
(Remember to round off the error to one significant figure.) You could also
report, “F×x = 160
± 12 Nm,” or “F/x= 3.9
**±****
0.3 N/m**.” It is a good idea to
calculate the absolute uncertainty to assure yourself of reporting the correct
number of digits in the answer.

Multiplication of an uncertain quantity by a constant is a special case of the multiplication rule, but one that comes up frequently enough that it is good to state it explicitly:

When an uncertain quantity is multiplied by a constant, the *absolute* error on the
product is the constant times the *absolute* error of the original quantity. The
*relative *error on the product is the same as the *relative* error on the original
quantity.

For a constant k,

(6)

The previous law for multiplication and division assumed that the error on each of the factors was not correlated with the error on the other; i.e., if the measurement on the first variable was too large, the error on the second had an equal probability of being too large and too small.

While A^{2} is the product of A × A, the error on each A is most certainly
correlated. Therefore, we need a different rule for dealing with factors with
exponents.

The relative error on an uncertain quantity raised to an exponent is the exponent times the relative error.

(7)

Note that n can be any real number, not just an integer.

For example, if

• t = 2.03 seconds,

• **u{t}/t = 1%,**

• A = 16.07 meters^{2}^{},

• u{A} = 0.06 meters^{2}^{},

then

Your stated results would be, “t^{5} = 34.5 s^{5}
± 5%” and “
= 4.009
± .007
m.”

Combined Operations: The Chain Rule

What do we do if we have a more complicated expression, something
beyond A + B or A × B? We use what is called a *chain rule*. The idea is that
you treat a function as if it were a variable. You keep applying the rules
given above for each operation until you have an expression containing the
errors in the functions. Then you apply the chain rule to the uncertainties in
the functions.

More formally, u{_} is an operator. We keep applying the given rules until the expression is reduced to elements that are known numbers.

For example, in the addition of two functions f and g of two or more uncertain quantities A, B, ... ,

(8)

The next step is to find f and g. This, of course, depends on how they are defined.

The product and quotient rule for two functions is

(9)

Again this is just one step. To continue, you need to find expressions for the relative errors of f and g.

You may notice the similarity between Eqs. 3 and 8 and Eqs. 4 and 9. This is the essence of the chain rule:

Any rule for variables holds for functions.

Example 1

As an example of the chain rule, consider the velocity of an accelerating
body, v=v_{0}+at. We have the values and uncertainties of v_{0}, a, and t. To find
u{v}, first let f=v_{0} and g=at and apply the addition rule (Eq. 8). This gives you
an expression with u{at}. Both a and t are variables with known
uncertainties, so you can use the product rule (Eq. 5).

The propagation of uncertainty is a mathematical derivation. As such, you have to justify each line, either by applying one of the propagation rules or using algebra. Using the rules as listed in the Lab References, what you write down for the propagation of the uncertainty of v is as follows:

As another example, a_{c}=mv^{2}/r , the centripetal acceleration of a mass m
moving at a constant speed v in a circle of radius r. Assuming known values
and uncertainties of m, v, and r, finding the expression to calculate u{a_{c}}
proceeds thusly:

Finally, consider the uncertainty of an average quantity, where the error in each measurement is the same:

This is the error of the mean as stated at the end of Appendix 1.

The ± notation and the rules we have given for the propagation of errors have assumed three things:

1. The errors are small compared to the measurements.

2. The errors are statistically independent (i.e., the direction and amount of a fluctuation in A does not depend in any manner on the direction or amount of a fluctuation in B).

3. The underlying distribution which these errors represent is Gaussian like the one dealt with in Appendix 1.

In some experiments, one or more of these assumptions may be incorrect. For the purpose of this course, we will assume that these conditions hold.

Suppose we take measurements of the quantities A and B and that we can define some function f of these variables. We can expand this function in a Taylor series about the most probable values and . If the measurements are close to these values, we can approximate this infinite series to the first order:

The first term is a constant, the next two are partial derivatives evaluated at the most probable values, after differentiation. (To take a partial derivative, assume the other variables are constants and take the normal derivative.)

The error in the function, from this expression and the rules outlined above, is

(10)

Extending this to more than two variables is obvious:

(11)

Note that all of the rules for the propagation of error given previously are specific applications of this general formula.

For example, suppose we have V=V_{0}e^{-t/RC}, where the quantity RC is exact.
Taking the partial derivatives we get

(12)

Plugging these into our generalized formula for the uncertainty gives us

(13)

Dividing both sides of the equation by V leads to an expression for the relative error in V:

(14)