Question Video: Identifying a Formula for the Absolute Error of a Measurement Physics • First Year of Secondary School

Video Transcript

Which of the following is the correct formula for the value of the absolute error of the measurement Δ𝑥, given the accepted value 𝑥 naught and the measured value 𝑥? (a) Δ𝑥 is equal to 𝑥 divided by 𝑥 naught. (b) Δ𝑥 is equal to 𝑥 naught divided by 𝑥. (c) Δ𝑥 is equal to 𝑥 minus 𝑥 naught. (d) Δ𝑥 is equal to 𝑥 naught minus 𝑥. (e) Δ𝑥 is equal to the absolute value of 𝑥 naught minus 𝑥.

This question is asking us to identify the correct formula for a quantity, here, the absolute error of a measurement. As with all such questions, we could immediately find the right answer by simply recalling the correct formula. Here, that answer is choice (e). And we know that (e) is correct because we have recalled the definition of absolute error as the absolute value of the difference between the accepted value and the measured value. Now it is sometimes necessary and almost always quite useful to memorize relevant physics formulas. However, it is neither efficient nor possible to memorize every possible formula. Instead, we want to be able to work out that choice (e) is the correct answer given what we know about absolute error and measurement error in general.

This is also useful because even if we have memorized a formula, sometimes we’ll forget it when we need to use it. Okay, so let’s start by recalling that the absolute error is the total amount by which a measured value differs from the accepted value. Looking at our answer choices, we can immediately eliminate choices (a) and (b). Since the quantities 𝑥 divided by 𝑥 naught and 𝑥 naught divided by 𝑥 do not give an amount by which things differ. They rather compare the relative sizes of two quantities. On the other hand, choices (c) through (e) are all some kind of difference between the measured value and the accepted value. And the absolute error is defined as the total amount by which these two quantities differ.

Now we need to determine the correct order for the difference, or we need to select the absolute value where order doesn’t matter. Our first clue is that the absolute error is defined as the total amount by which the two values differ rather than how much larger one specific value is than another specific value. So we’re not interested in how much larger the measured value is than the accepted value, nor are we interested in how much larger the accepted value is than the measured value. Rather, we are interested in a quantity that does not depend on whether the measured or accepted value is larger, and that is the absolute value of the difference between them.

As an example, let’s consider measuring the length of this pencil with a ruler that is 10 centimeters long and has gradations of one centimeter each. To the nearest gradation, the length of this pencil is seven centimeters. However, because the gradations are one centimeter apart from each other, the measurement error from this ruler is a maximum of half a centimeter, so we can write the length of the pencil measured by this ruler as seven centimeters plus or minus half a centimeter because we at most overestimate or underestimate our measurement by one-half of a centimeter.

One-half centimeter is the absolute error of the measurement. And that is because it makes reference to the error between the measured gradation and the true length of the pencil. It doesn’t depend on the actual number that we measured on the ruler.

The important thing to understand is that we have expressed this value as plus or minus some positive number. The plus or minus essentially means that we are allowing for the possibility of either overestimating or underestimating the true value. We can now understand why we always use a positive number for the absolute error. That is, we express it in terms of an absolute value. The sign of the difference, whether it is positive or negative, only tells us if the measured value is an overestimate or an underestimate of the accepted value. However, whether the measured value is an overestimate or underestimate, the positive number, that is, the absolute value of this difference, still tells us exactly how much the two values differ, which is the absolute error that we have defined.