Which of the following is the correct formula for the relative error of the measurement 𝑟 given the accepted value 𝑥 naught and the absolute error Δ𝑥? 𝑟 equals 𝑥 naught times Δ𝑥. 𝑟 equals 𝑥 naught divided by Δ𝑥. 𝑟 equals Δ𝑥 divided by 𝑥 naught. 𝑟 equals 𝑥 naught plus Δ𝑥. 𝑟 equals the absolute value of 𝑥 naught minus Δ𝑥.
This question asks us to identify the correct formula for the relative error of a measurement given the accepted value for the quantity being measured and the absolute error of the measurement. Although we could simply recall the formula having memorized it beforehand, let’s instead start with the definition of relative error and see which of these formulas corresponds to that definition. The relative error is one way to quantify how much a measured value differs from the accepted value of that quantity. And in particular, the relative error expresses this difference as a fraction of the accepted value. Relative error is quite useful. It allows us to quickly assess the accuracy of a measurement and also compare different measurements of different kinds of quantities.
Let’s consider a brief example. We’ll consider two different measurements. One will be the mass of some fruit, and the other will be the length of a book. Now, let’s say that the true mass of the fruit is 1.000 kilograms and the book being quite enormous has a true length of 100 centimeters. When we actually measure these quantities though, we don’t measure 1.000 kilograms for the fruit; we measure 1.020 kilograms. And instead of measuring 100 centimeters for the length of the book, we measure 98 centimeters. Neither of these measurements exactly matches the true value. The measurement of the fruit is 20 grams too large, and the measurement of the book is two centimeters too small.
Now, if we compare 20 grams to one kilogram, which is 1000 grams, we see that 20 grams is two percent of one kilogram. So, the difference between the measured value and the accepted value for the mass of this fruit is two percent of the accepted value. And this is what we mean when we say relative error. On the other hand, 20 grams is the actual amount by which the measured value and accepted value differ. So, this is the absolute error. Looking at our measurement of the book, we see that we’ve already written down the absolute error; it’s two centimeters. And comparing two centimeters to the true length of the book, 100 centimeters, we see that the relative error is again two percent of the accepted value.
We can now notice a few important things. First, the relative error is the same whether the measurement is larger or smaller than the accepted value. Secondly, the relative error is dimensionless, which allows us to directly compare these two measurements. In fact, for these two measurements, the relative error is exactly the same. So, we would say that these two measurements have the same accuracy. Note that this would not be a meaningful comparison if we try to compare the absolute errors because the absolute errors have different dimensions. One is an absolute error with dimensions of mass, and one is an absolute error with dimensions of length, and there is no way to directly compare mass and length.
The same works for comparing measurements on different scales. The scale of our book measurement is centimeters, which are one one hundredth of a meter. If we instead measured an object that was 100 meters long and we found that its length was 98 meters, our absolute error would be two meters, which is much much larger than two centimeters. But two meters in 100 meters is still a relative error of two percent. So, the measurement of our much larger object would be exactly as accurate as our measurement of this book. Alright, we are now ready to figure out which of our answer choices is correct.
We can immediately narrow down our two answer choices to (B) and (C). These are the only two choices that are a fraction, and more importantly, these are the only two answer choices that are dimensionless. As we can see from our example, the accepted value and the relative error have the same dimensions. For the fruit, both had dimensions of mass, and for the book, both had dimensions of length. Note that for the fruit, we had different units, kilograms and grams, but those are both units for mass. So, the dimensions are the same.
In choice (A), we have the accepted value times the absolute error. But since both of those quantities have the same dimensions, their product will have those dimensions squared. Concretely, the product of the accepted value for the mass of the fruit and the absolute error of our measurement would have dimensions of mass times mass or mass squared, and the length of the book times the absolute error in that measurement would have dimensions of length squared. So, choice (A) is not dimensionless and cannot be the correct answer. Similarly, the product and difference of two quantities with the same dimensions also has those same overall dimensions. So, (D) and (E) are also not the correct answer because they are also not dimensionless.
However, when we divide one quantity by another quantity with the exact same dimensions, the result is dimensionless, which is why (B) and (C) are our only possible answer choices. Now for these choices, we have the accepted value divided by the absolute error, which we can also say is the accepted value as a fraction of the absolute error. And we have the reverse, which is the absolute error divided by the accepted value. And this we can say is the absolute error as a fraction of the accepted value. Looking back at our definition for relative error, we see that the quantity we are looking for is a fraction of the accepted value, not a fraction of the absolute error.
So, the correct answer is (C). 𝑟, the relative error, is equal to Δ𝑥, the absolute error, divided by 𝑥 naught, the accepted value for the quantity.