Propagation of Error

When doing experiments, we often measure two or more distinct physical quantities and then combine the
measurements mathematically to obtain some final result. For example, to de termine the average speed of an object
we could measure the amount of time it takes to travel a certain distance and then use

The uncertainty in the calculated average speed depends upon the uncertainty in the distance as well as the
uncertainty in the time. Propagation of error refers to the methods used to determine how the uncertainty in a
calculated result is related to the uncertainties in the in dividual measurements .

The methods used for determining how uncertainties are combined can be quite sophisticated. In this course we
will use approximate methods that are easier to use, but still give reasonable agreement with more advanced
methods.

THE MIN-MAX METHOD

The basic method we will use to propagate errors is called the min-max method. To use this method we define
a minimum and maximum value for each of the measurements used to calculate the final result. The minimum and
maximum values are simply (best value - uncertainty) and (best value + uncertainty). Then we use these values to
calculate a minimum and maximum value for the calculated result. From the minimum and maximum values for the
calculated result, we deduce the uncertainty in the result. Here are some examples.

1) Addition of measurements
When two resistors are connected in series, the total resistance is the sum of the individual resistances. Suppose
that two resistors in series have resistances of 78 ± 1 ohms and 135 ± 2 ohms. The best estimate of the total
resistance is 78 + 135 = 213 ohms. Based on the uncertainty, it is quite likely that the resistance of the first resistor
is between 77 and 79 ohms and the resistance of the second resistor is between 133 and 137 ohms. These are our
minimum and maximum values. Using the maximum values for each resistor we get a maximum total resistance of
79 + 137 = 216 ohms. Similarly, the minimum total resistance is 77 + 133 = 210 ohms. The difference between the
maximum resistance and the minimum resistance is 6 ohms so the total resistance with uncertainty is 213 ± 3 ohms.
In this case the total uncertainty is just the sum of the two uncertainties.

Notice that we have used the terms minimum and maximum a little carelessly. For example, 78 ± 1 ohms does
not mean that the resistance is guaranteed to be within 77 and 79 ohms. Instead it means that there is about a 68%
probability that the resistance falls within that range. The same is true for the other resistor and for the total
resistance. For the remainder of this handout we will use the terms maximum and minimum, but we must always
keep in mind that these are not rigid limits on the values.

2) Subtraction of measurements
The mass of an empty thermos bottle is found to be 78.3 g ± 0.2 g. When filled with liquid nitrogen its mass is
167.7± 0.3 g. The best estimate for the mass of liquid nitrogen in the flask is 167.7 - 78.3 = 89.4 g. The minimum
and maximum values for the mass of the thermos bottle are 78.1 g and 78.5 g. The minimum and maximum values
for the mass of the thermos bottle plus nitrogen are 167.4 g and 168.0 g. To find the maximum mass for the liquid
nitrogen we subtract the minimum mass of the thermos bottle, 78.1 g, from the maximum mass of the thermos +
liquid nitrogen, 168.0 g. The result is that the maximum value for the mass of liquid nitrogen is 168.0 g - 78.1 g =
89.9 g. To find the minimum mass of the liquid nitrogen we subtract the maximum mass of the thermos, 78.5 g,
from the minimum mass of the thermos + nitrogen, 167.4g. The result is 167.4 g - 78.5 g = 88.9 g. Thus the mass
of the liquid nitrogen is 89.4 ± 0.5 g. Note that the masses are subtracted, but the uncertainties add.

3) Multiplication by a constant
The time for a pendulum to complete 5 swings is found to be t = 12.6 ± 0.2 s. The time for 1 swing, T, is given
by T = (1/5)t. The best estimate for T is (1/5) x 12.6 = 2.52 s. The maximum time for one swing is (1/5) x 12.8 =
2.56 s and the minimum time is (1/5) x 12.4 s = 2.48 s. Thus the time for one swing is T = 2.52 ± 0.04 s. Note that
the uncertainty is 1/5 of the uncertainty in the original measurement.

4) Multiplication of measurements
A rectangular plot of land is found to be 163 ± 1 ft by 386 ±2 ft. The best estimate for the area of the plot is 163
ft x 386 ft = 62918 ft². The maximum area is 164 ft x 388 ft = 63632 ft². The minimum area is162 ft x 384 ft =
62208 ft². The maximum area is larger than the best estimate of the area by 714 ft². The minimum area is smaller
than the best estimate by 710 ft². Taking the average we have that the area is 62918 ± 712 ft². We recall that
uncertainties are normally quoted to only one or two significant figures , and the precision of the result should match
that of the uncertainty. Thus the uncertainty is 700 ft², and the result is 62900 ± 700 ft². It is worth noting that if we
were using rules for significant figures to decide how to round off the answer, we would give the answer to 3
significant figures, namely 62900 ft². With our analysis we not only find out how to round off the final result, but
we also get information about the actual size of the uncertainty.

5) Division of measurements
A sprinter runs 400 ± 2 m in 65.31 ± 0.05 s. Her average speed is 400m/65.31s = 6.12464 m/s. To find the
maximum value for her speed we take the maximum distance and the minimum time (similar to what we did with
subtraction above) and get a speed of 402m/65.26s = 6.15998 m/s. The minimum value for her speed is
398m/65.36s = 6.08935 m/s. The maximum and minimum are both different from the best value by about 0.035
m/s. Rounding to 1 sig fig we get that the uncertainty is 0.03s. Rounding the best estimate properly we have an
average speed of 6.12± 0.03 m/s.

6) Raising to a power and multiplying by a constant
The radius of a circle is found to be 7.5 ± 0.1 cm. The best estimate for the area of the circle (A = πR²) is A =
3.1416 x 7.5² = 176.715 cm². The maximum area is 3.1416 x 7.6² = 181.459 cm² and the minimum area is 3.1416 x
7.42 = 172.034 cm². The difference between the maximum value and the best estimate is 4.744 cm² and the
difference between the minimum and the best estimate is 4.681 cm². Thus the uncertainty is about 4.7 cm², or to 1
significant figure, 5 cm². The final result for the area is A = 177 ± 5 cm².

RULES FOR PROPAGATING UNCERTAINTIES

The above examples illustrate how the min-max method may be applied in a number of specific cases. We can
also apply the method more generally and develop some simple rules for propagating errors. We will not work
through the derivations now, but will instead just present the results. The derivations may be found in any error
analysis textbook and you will be learning them next semester.

Rule #1
When two measurements are added or subtracted, the absolute uncertainty in the result is the sum of the
absolute uncertainties of the individual measurements.

Rule #2
When a measurement is multiplied by a constant, the absolute uncertainty in the result is equal to the absolute
uncertainty in the measurement times the constant, and the relative uncertainty in the result is the same as the
relative uncertainty in the measurement.

Rule #3
When two measurements are multiplied or divided, the relative uncertainty in the result is the sum of the
relative uncertainties in the individual measurements. This rule could be stated equivalently in terms of percentage
uncertainties, since relative and percentage uncertainties are simply related by a factor of 100.

Rule #4
When a measurement is raised to a power, including fractional powers such as in the case of a square root ,
the relative uncertainty in the result is the relative uncertainty in the measurement times the power.

Comments:
Clearly, Rule #1 agrees with examples 1 and 2 given for the min-max method.

Rule #2 also obviously agrees with the result from example 3, but it contains the extra information that the
relative error remains the same when a quantity is multiplied by a constant. Checking this we see that the relative
error in the original time measurement is 0.2 s/12.6 s = 0.0159, and the relative error in the time for one swing is
0.04 s/2.52 s = 0.0159.

Now we will apply Rule #3 to example 5 - division of two measurements - to check that it gives the same
result. The relative uncertainty in the distance run is 2 m/400 m = 0.005. The relative uncertainty in the time taken
is 0.05 s/65.31 s = .00077. Adding the relative uncertainties we get 0.005 + 0.00077 = 0.00577. According to Rule
#3, this is the relative uncertainty in the speed. However, we want to know the absolute uncertainty in the speed.
Since

we can rearrange to get

absolute uncertainty = relative uncertainty * best estimate.

Thus we find that the absolute uncertainty in the speed = 0.00577 x 6.12464 m/s = 0.035 m/s. This is the same
answer that we got in example 5.

In addition to giving the same answer, application of Rule #3 has another benefit. By calculating the
relative errors in the measurements we obtain information about which measurement contributes most to the
uncertainty in the result. In this example we see that most of the uncertainty in the speed is due to the uncertainty in
the distance measurement. This type of information is useful when trying to decide how to improve an experiment
or when trying to sort out why an actual result does not agree with an expected result. Often, to simplify our
calculations, we will look for the most significant source of error in an experiment and disregard smaller sources of
error.

Now we will apply Rules #2 and #4 to example 6. According to Rule #2, multiplying by a constant does
not change the relative error, and according to Rule #4, the relative error in R² is 2 times the relative error in R.
Thus we have that the relative error in the area of the circle is twice the relative error in the radius. The relative
error in the radius is 0.1 cm/7.5 cm = 0.0133. Multiplying by 2 we get that the relative error in the area of the circle
is 0.0266. The absolute error in the area is 0.0266 x 176.715 cm² = 4.7 cm², which agrees with the earlier result.

One final word about the methods we will use to find uncertainties. As stated earlier, these methods are
approximate. More sophisticated methods for finding uncertainties will typically give results equal to, or smaller
than those obtained by our methods. In other words, the methods we use tend to overestimate the uncertainty. The
reason is that we have assumed a worst-case scenario in which errors from two measurements always add together.
In reality , there is a chance that the errors from two independent measurements will partially cancel. More
sophisticated methods of error analysis take this possibility into account. However, the difference is often not very
important. In many cases we will make only rough estimates of the uncertainty in our measurements and so even if
we were to apply fancier methods of error propagation, we would not have a lot of confidence in our final
uncertainty. In this course, what we want to achieve is a rough idea of the uncertainty, say to within a factor of 2.
Thus the approximate methods described in this handout are quite sufficient.