Catherine has a master's degree in Mathematical Biology and teaches math at the college level.
Dan has a 36-inch-long floorboard in his living room that needs to be replaced. He goes to the store to purchase a new floorboard, which he measures at exactly 36 inches long, but when he tries to install it in his floor, the new floorboard is about half an inch too long. What's going on here?
Dan didn't make a mistake. It turns out that the tools we use to measure properties of objects, like their size or weight, are not completely accurate. They are mostly accurate, but there's always some error in any tool of measurement. In this lesson, we'll talk about the accuracy of measuring tools and how this influences two types of error.
Measuring Tools & Accuracy
Tools of measurement are said to be accurate to a certain degree. Let's say that the measuring tape Dan used to measure his new floorboard was accurate to 1 inch. This means that there is a 1-inch range, or ''window'', around a measurement where the true value could actually be located. From the measured value, that's half an inch on either side. Dan measured the new floorboard at 36 inches long, but the true length could be anywhere between 35.5 inches and 36.5 inches long. In this case, the floorboard was actually on the longer end. We say that the measurement is 36 ± 0.5 inches.
There are two types of error that are affected by the accuracy of measuring tools. The absolute error is defined as the absolute value (or magnitude) of the difference between the measured value and the true value. Thus, let:
- ea = the absolute error
- xm = the measured value
- xt = the true value
The formula for computing absolute error is:
ea = | xm - xt |
There's only one catch - we don't usually know what the true value is. In that case, we just need to think about the accuracy of the measuring tool. Let's use Dan's measuring tape as an example. It is accurate to 1 inch, which means that the true value of any measurement could be up to half an inch smaller or half an inch larger than the actual measurement. The maximum possible difference between the measurement and its true value is half an inch. Therefore, the absolute error is 0.5 inches.
There is another type of error that is affected by the accuracy of measuring tools. The relative error is defined as the absolute error relative to the size of the measurement. All you need to do is divide the absolute error by the measured value. In addition to the variables, let:
- er = the relative error
Then the formula for computing relative error is:
- er = ea / xm
In Dan's situation, the absolute error is 0.5 inches, and he measured the floorboard at 36 inches long, so the relative error is:
er = 0.5 inches/36 inches = 0.014 (rounded to the nearest thousandth)
The relative error doesn't have units because they cancel out during the calculation. The relative error is a proportion, so we can also express it as a percentage by multiplying the relative error by 100%, like this:
er = 0.014 * 100% = 1.4%
We can say that the absolute error is 1.4% of the measured value.
Comparing Relative Errors
The relative error depends on both the absolute error and the measured value. Let's see what happens when you measure two objects of different sizes with the same measuring tool. Say Dan used the same measuring tape to measure the width of the floorboard at 6 inches. The absolute error is still 0.5 inches. What is the relative error? It is:
er = 0.5 inches/6 inches = 0.083 (rounded to the nearest thousandth)
It can also be expressed as a percentage like this:
er = 0.083 * 100% = 8.3%
The relative error is larger when the size being measured is small. This makes sense because half an inch of absolute error makes a big difference when you're measuring something only 6 inches long, compared to 36 inches long.
What happens to the relative error when you take the same measurement using two different measuring tools? Let's compare Dan's measuring tape, which is accurate to 1 inch, to a ruler that is accurate to 0.5 inches (which means the absolute error is 0.25 inches). We'll use the width of the floorboard as an example. Both the measuring tape and ruler say the floorboard is 6 inches long. Let's calculate the relative errors:
- Measuring tape: er = 0.5 inches/6 inches = 0.083 * 100% = 8.3%
- Ruler: er = 0.25 inches/6 inches = 0.042 * 100% = 4.2%
The relative error using the ruler is smaller than the relative error using the measuring tape because the absolute error of the ruler is smaller than the absolute error of the measuring tape.
Let's review. Absolute error is defined as the absolute value of the difference between the measured value and the true value of a measurement and is usually given as the maximum possible error given a measuring tool's degree of accuracy. The absolute error has the same units as the measurement. Relative error is defined as the absolute error relative to the size of the measurement, and it depends on both the absolute error and the measured value. The relative error is large when the measured value is small, or when the absolute error is large. Relative error has no units.
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