Question Video: Identifying a Formula for the Relative 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 relative error of the measurement 𝑟 given the accepted value 𝑥 naught and the measured value 𝑥? (a) 𝑟 equals 𝑥 divided by the absolute value of 𝑥 naught minus 𝑥. (b) 𝑟 equals 𝑥 divided by 𝑥 naught minus 𝑥. (c) 𝑟 equals the absolute value of 𝑥 naught minus 𝑥 divided by 𝑥. (d) 𝑟 equals the absolute value of 𝑥 naught minus 𝑥 divided by 𝑥 naught. (e) 𝑟 equals 𝑥 naught minus 𝑥 divided by 𝑥.

This question is asking us to recall a formula specifically for the relative error of a measurement. We are asked for this formula in terms of the accepted value 𝑥 naught and the measured value 𝑥.

Now the easiest way to answer this question is to simply recall that the correct answer is choice (d). However, although it is important to memorize some physics formulas, it is far more important to understand how to derive those formulas from definitions. So let’s recall the definition of relative error and then work out this formula.

Whenever we think about measurement error, we are trying to quantify how much a measured value differs from the accepted or actual value. In particular, relative error quantifies the difference between the measured value and accepted value as a fraction of the accepted value.

For example, say we have a beam of wood that is exactly one meter from end to end. However, when we measure the beam of wood, we measure its length to be 1.25 meters, not one meter. Our measurement is one-quarter of a meter longer than the actual value, and this number is 25 percent larger. This number 25 percent is the relative error between the measured value and the accepted value because our measured value is 25 percent larger than the accepted value. In other words, 25 percent expresses how different our measured value is from the accepted value as a fraction of the size of the accepted value.

Looking at our answer choices then, we can immediately see that (a), (b), and also (c) are not correct. (a) and (b) are not correct because they both have the difference between the measured value and accepted value in the denominator. But having the difference in the denominator means that these two fractions measure how large the numerator is relative to the denominator. But we want the reverse, we want to know how large the difference is relative to the accepted value.

Choice (c) is almost correct. It has the difference between the measured and accepted values in the numerator, which is what we want, but the denominator is the measured value. So this gives the difference between the two values as a fraction of the measured value. But we want the difference as a fraction of the accepted value.

This leaves us with choices (d) and (e). Both of them have a difference between the two values in the numerator and the accepted value in the denominator. The only difference between choice (d) and (e) is that in choice (d) the difference between accepted and measured values is inside of absolute value bars, which means that whether the measured value is larger or smaller than the accepted value, this fraction will always be positive. On the other hand, in choice (e), if 𝑥 is greater than 𝑥 naught, the fraction will be negative. If 𝑥 is less than 𝑥 naught, the fraction will be positive. So we need to know if relative error can be negative or if it is always positive.

Let’s look back at our piece of wood. When the measured value was larger than the accepted value, we said that the relative error was 25 percent and that the measured value was 25 percent larger. But if, instead of measuring 1.25 meters we had instead measured 0.75 of a meter, this would be one-quarter of a meter too small. And we would describe the measurement as 25 percent smaller than the accepted value. Note that we are again using positive 25 percent to represent the relative error. The sign of the difference 𝑥 naught minus 𝑥 tells us whether the measured value is larger or smaller than the accepted value. However, the relative error itself is always a positive quantity and depends only on the size of the difference between 𝑥 naught and 𝑥.

Therefore, the correct answer is (d) 𝑟 is equal to the absolute value of 𝑥 naught minus 𝑥 divided by 𝑥 naught.