In this explainer, we will learn how to find the uncertainties in the values of quantities that can be calculated from two or more measured values.
Let us first recall that the uncertainty in the value of a physical quantity or measurement is a number describing the precision of our measurement. That is, the uncertainty describes a range of values consistent with a particular measurement and quantifies the fact that no measurement is exact. When the uncertainty is divided by the actual measurement, we find the fractional or percent uncertainty.
If our quantity or measurement is represented by the letter , then the corresponding uncertainty is represented by the symbol and the percent uncertainty is given by .
Every measurement has an uncertainty. But what about when we perform calculations with the results of multiple measurements? For example, if we measure the length and width of a board, we may find the length is m and the width is m. Using this information, we can quickly calculate the area to be . However, both of our measurements have uncertainty, which means that our calculation of the area is also not exact. We need a set of rules for finding uncertainties in situations like this one.
Let us start with a simpler case: adding two measurements together. This sort of calculation might occur when we calculate the total length of two pencils. We can see two pencils and a ruler in the diagram below.
We could measure the length of each pencil using the ruler in the diagram, but since the tips do not align exactly with the calibration marks, our measurement would not be exact. In particular, our uncertainty would be approximately the length between calibration marks. If we add more marks, we can reduce this uncertainty, but there is no ruler with enough calibration marks to perfectly measure every object. Moreover, the marks themselves have a physical width that contribute to the uncertainty. The total length of the two pencils is the sum of each of the individual lengths, each with their own uncertainty. The rule for finding the combined uncertainty in this instance is quite straightforward, and we will call it the sum rule.
Definition: The Sum Rule
When summing individual measurements, the combined uncertainty is the sum of the individual uncertainties from each measurement. This is the sum rule for combining uncertainties.
Symbolically, if , then the sum rule states that
The sum rule turns out to be quite intuitive if we treat the uncertainty in the following way: the minimum value consistent with our measurement is the value we measure minus the uncertainty. Likewise, the maximum value consistent with our measurement is the value we measure plus the uncertainty. Concretely, say we measure a quantity to be s. All values consistent with this measurement are at least 9 s and at most 11 s. Now, say we measure another quantity to be s so that all values consistent with this measurement are at least 22.5 s and at most 23.5 s. To find the minimum and maximum values consistent with the sum of these measurements, we simply add the minimum and maximum values consistent with each individual measurement. Since these minimum and maximum values were found by subtracting or adding the individual uncertainties, it directly follows that the uncertainty of the sum is the sum of the uncertainties.
The sum rule is used any time we have a summation or difference of measurements. The rule applies to both addition and subtraction because subtraction can always be expressed in terms of addition with negative numbers.
Let us see an example of applying the sum rule.
Example 1: Combining Uncertainties by Addition
Two resistors have resistances of Ω and Ω. If the two resistors were placed in series, what would the uncertainty of the two resistors together be?
Recall that the total resistance of two resistors connected in series is given by the sum of the individual resistances. Since we are looking for the uncertainty in the sum of two values, we must use the sum rule for combining uncertainties. Recall that the sum rule tells us that the overall uncertainty is just the sum of each of the individual uncertainties. From the statement, we see these uncertainties are 0.1 Ω and 0.2 Ω. Applying the sum rule, we have as the total uncertainty.
The example we just worked through involved applying the sum rule to a combination of two uncertainties. In fact, the sum rule applies to any number of uncertainties combined by addition or subtraction. This next example shows how to use the sum rule for more than two uncertainties.
Example 2: Combining Multiple Uncertainties by Addition
Three objects have masses of kg, kg, and kg. What is the uncertainty in the total mass of the three objects?
We are looking for the uncertainty in the total, that is the sum, of three physical quantities. Recall that the sum rule for combining uncertainties tells us that the overall uncertainty is just the sum of each of the individual uncertainties. From the statement, we see these uncertainties are 0.1 kg, 0.1 kg, and 0.05 kg. Applying the sum rule, we have as the total uncertainty.
It is worth noting that when adding uncertainties with the sum rule, the combined uncertainty depends only on the uncertainties, not on the particular measurement values. This is not the case for our next rule, the product rule for combining uncertainties, which does depend on the particular measurement values.
At the beginning of this explainer, we wondered how to find the uncertainty of an area given the uncertainties of its corresponding length and width. Since area is length times width, we need the product rule for combining uncertainties.
Definition: The Product Rule
If a quantity , where and are quantities with uncertainties and , respectively, then the uncertainty is given by the product rule
Dividing both sides by and recalling , we see the product rule is equivalent to which is a sum of the fractional uncertainties of each factor.
We use the product rule any time we multiply quantities together. Similarly, because division can always be expressed as multiplication of reciprocals, we also use the product rule when dividing quantities.
Before we examine the mathematical basis for the product rule formula, we will practice applying it with an example.
Example 3: Finding the Combined Uncertainty of a Product of Measurements
An object moves along a straight line at a speed of m/s for s. Work out the distance that the object moves as well as the uncertainty in this value.
Given a speed, , and an interval of time, , the total distance, , an object travels at that speed in that interval is given by
To find , we plug in and , which gives
To find the uncertainty in , we note that is defined as a product, so we use the product rule for combining the uncertainties of and . From the statement, we see these are and . Therefore, the total percent uncertainty is
We have already calculated as 40 m. This means the absolute uncertainty is of 40 m, which is
Combining our results for and , we find the total distance traveled is m.
At this point, it is worth examining the mathematical basis of the product rule. Although the picture is not quite as simple as the sum rule, we can still build up the result in an intuitive way.
Recall that the uncertainty of a measurement multiplied by a number is that number times the uncertainty of the measurement. Symbolically, if is a number, is a measurement, and , then
Now, we observe that, for the product rule, we considered a calculation of the form . The only difference between this and is that has an uncertainty , while does not (equivalently, ).
We can now understand the origin of the product rule for combining uncertainties. Because has an uncertainty, we must treat in the quantity the same way we treated in the quantity . Specifically, as part of finding the total uncertainty, we must multiply by . However, using exactly the same logic, because has an uncertainty, we must treat in the quantity the same way we treated in the quantity . Therefore, in finding the total uncertainty, we must also multiply by . We have now found two terms that must play a role in the final uncertainty, namely and . The first term is the contribution from the uncertainty in and the second is the contribution from the uncertainty in . These two terms are the only contributions to the total uncertainty and, when combined together, exactly form the product rule.
To motivate our last rule, the power rule for combining uncertainties, consider the following example involving a special sort of product.
Example 4: Calculating the Uncertainty of Quantity Raised to a Power
In an experiment, a quantity is found to have a value of . What is the uncertainty in ?
First, observe that , which is a product of two quantities (the same one each time) that both have uncertainty. Using the product rule and the fact that , we calculate the uncertainty in as
In this example, we used the product rule to find the uncertainty. However, let us see what would happen if we used the product rule to write symbolically:
The product rule for this special case of the uncertainty of the power of a quantity has reduced to an especially simple form that requires no addition of terms, only multiplication of appropriate quantities. If we let and divide both sides by like we did for in the usual product rule, we find another useful form: which relates the fractional or percent uncertainties in and .
Now, in light of our success above, we are tempted to find the uncertainty for other powers of , say . Applying the product rule to , we have
But we already know how to simplify using the product rule; it is just . So, we have
At this point, we can see a pattern starting to emerge. By repeatedly applying the product rule, it looks like we will always find that the fractional uncertainty of a quantity raised to a power will be the power times fractional uncertainty of the quantity itself. In fact, this is not only true; it is precisely our power rule for combining uncertainties.
Definition: The Power Rule
The power rule for combining uncertainties is an extension of the product rule to the special case where one quantity is another quantity raised to a power, say , where is a number. In this case, the fractional uncertainty in is related to the fractional uncertainty in by
Now that we have defined our last rule, we can apply it to an example.
Example 5: Finding the Combined Uncertainty of the Power of a Quantity
In an experiment, a quantity is found to have a value of . What is the percent uncertainty in ?
Since we are looking for the percent uncertainty in , which is the power of a quantity, we can apply the power rule for combining uncertainties. Since the exponent on is 2, we have, by the power rule,
Substituting 15 for and 0.3 for , we find
This is the fractional uncertainty in . To find the percent uncertainty, we multiply by to find that is the percent uncertainty in .
Finally, we will compute one last uncertainty with a combination of the product rule and the power rule.
Example 6: Calculating an Uncertainty Using Multiple Steps
An object has a mass of kg and is moving at a speed of m/s. What is the uncertainty in the kinetic energy of the object? Start by calculating the uncertainty in , and then calculate the uncertainty in .
We are looking for the total uncertainty in the kinetic energy defined by the equation
We first notice that the kinetic energy has a constant, , multiplying the quantities with uncertainty, , so we can find the uncertainty in and then divide that result in half to find the uncertainty in the kinetic energy.
Having simplified, our remaining task to finding the uncertainty in , we now notice that this quantity is the product of one factor involving and one factor involving , where the second factor has the specific form . Using the product rule, we can write down
We are given both and in the statement, but we need to calculate . Fortunately, is a power of , and we are given values for both and , so we can use the power rule to compute a value for . By the power rule, we have or, multiplying both sides by ,
Substituting the given values for speed and the corresponding uncertainty, we have
Using this value, as well as and the values for and from the statement, we substitute into our expression for the uncertainty of to find
Finally, multiplying 2.1 by and noting kg⋅m2/s2 is the unit of energy J, we have
In this example, we built up a complicated uncertainty by breaking it down into simpler components, each of which we could calculate using the rules we already knew. This is a general approach to combining uncertainties. Any time we are faced with a quantity that is some combination of sums, products, and powers of other quantities, we can calculate the total uncertainty from the individual uncertainties by repeatedly applying the rules we have learned.
Let us finish by recalling the three rules we have learned for combining uncertainties.
- There are three rules for combining uncertainties:
- the sum rule (),
- the product rule (), or, equivalently,
- the power rule ,
- The three rules can be used together to find the uncertainty of sums of products, products of powers, or any other composition of these three basic combinations.