3.2. Absolute and Relative Errors

You are already familiar with absolute error. Absolute error is the actual value of the error in physical units. For example, let's say you managed to measure the length of your dog L to be 85 cm with a precision 3 cm.

You already know the convention for reporting your result with an absolute error

Suppose you also regularly monitor the mass of your dog. Your last reading for the dog's mass M, with absolute error included, is

Which measurement is more precise? Or in other words, which one has a smaller error? Clearly, we cannot directly compare errors with different units, like 3 cm and 1 kg, just as we cannot directly compare apples and oranges. However, there should be a way to compare the precision of different measurements. Enter the relative or percentage error.

Let's start with the definition of relative error

Let's try it on our dog example. For the length we should divide 3 cm by 85 cm. We get 0.04 after rounding to one significant digit. For the mass we should divide 1 kg by 20 kg and get 0.05. Note that in both cases the physical units cancel in the ratio. Thus, relative error is just a number; it does not have physical units associated with it. Moreover, it's not just some number; if you multiply it by 100, it tells you your error as a percent. Our measurement of the dog's length has a 4% error; whereas our measurement of the dog's mass has a 5% error. Well, now we can make a direct comparison. We conclude that the length measurement is more precise.

Finally, let us see what the convention is for reporting relative error. For our dog example, we can write down the results as follows

The first way of writing is the familiar result with absolute error, and the second and third ways are equally acceptable ways of writing the result with relative error. (Writing the result in the parentheses form might seem a little bit awkward, but it will turn out to be useful later.) Note that no matter how you write your result, the information in both cases is the same. Moreover, you should be able to convert one way of writing into another. You know already how to convert absolute error to relative error. To convert relative error to absolute error, simply multiply the relative error by the measured value. For example, we recover 1 kg by multiplying 0.05 by 20 kg.

Thus, relative error is useful for comparing the precision of different measurements. It also makes error propagation calculations much simpler, as you will see in the next chapter.