Source: Toy Car, public domain http://pixabay.com/en/car-porsche-fast-cabrio-29144/ Ruler, public domain http://pixabay.com/en/white-ruler-plastic-transparent-41323/ Plants created by Author
This tutorial covers absolute error and relative error. With absolute error and relative error, it's also good to look at absolute change and relative change, because they're very similar topics. This slide here shows all the four things that we'll be looking at throughout this tutorial. While you don't need to memorize all the nuances right now, it is important to see some of the commonalities and differences between the four terms.
First off, with absolute change and absolute error, the word absolute here is our key that we're looking at differences. So absolute change looks at one kind of difference, absolute error looks at a similar kind of difference. On the other hand, relative change and relative error look at ratios. So relative change looks at one kind of ratio, relative error looks at another kind of ratio.
Now similarly, across the top, the two types of change, they're looking at how things change over time. So it's comparing new values with old values. Across the bottom with the errors, we're looking at errors, and we're looking at mistakes with measurement. So we're looking at differences or ratios between observed values and actual values. Let's go through examples of each.
This example looks at absolute error. Absolute error is the difference between the observed value and the actual value. So the formula way of writing this would be absolute error equals observed value minus actual value. In this case here, let's say we've got our little toy car and are trying to measure how long it is. If you look at from the end of the ruler, it's 3.9 centimeters long. So that is our observed value.
Now for the actual value, sometimes that's pieces information you're going to be given, like if the website told us how long this car was. Otherwise, it's pieces of information that we know to be true. If we look here, I've made a mistake. I didn't line up the end of the car with the 0. Instead, it's with the end of the ruler. So to get the actual value, I need to slide the ruler back so that we're starting at 0. Now the car itself is actually 4.2 centimeters long. So our actual value is 4.2.
To find absolute error, we're going to subtract 3.9 minus 4.2, and we get negative 0.3. Now it's OK to get a negative number. What this is telling us is that our observed value was smaller than our absolute value. Negative numbers are OK here. We'll go through now an example of relative error.
For this example with relative error, we're going to stick with the same problem as last time. We have our toy racecar and we've measured it. We measured it to be 3.9 centimeters long, but in actuality, it was 4.2 centimeters long. We're going to use those same pieces of information to calculate the relative error.
The way we calculate relative error is by the ratio of the absolute error to the actual value. Now this word ratio here is just telling us we're doing a fraction, and the words that come after tell us what the fraction involves. So relative error equals the absolute error over the actual value, and when we say over, that's where our fraction mark goes, that's where we're doing our division. So the absolute error is what we just calculated. It's that negative 0.3 centimeters. And the actual value was the measurement that the car truly is, that 4.2 centimeters, so we're going to use those two numbers to set up a fraction.
So this fraction right here is our relative error. It's more useful to us to divide this out and see what number we get, so I'm going to do that now. Negative 0.0714 and then and so on. I'm going round this, we're not going use all of those places, we're just going to say negative 0.07. So be careful not to forget that negative sign, that's a really common mistake.
So we know that this is our relative error, negative 0.07. More often than not, this is reported as a percent. So if we take that number and multiply it by 100, we can see what percent we were off by. So negative 0.07 times 100, we get 7%. And it's negative 7%. See, I almost forgot the negative there, be sure not to do that.
What this negative is signifying is that we're shrinking, we're to 7% too small with our measurement. So if it was a positive number, we would be over by that much. So this is telling us that our relative error, we're off by 7% and we're too small by 7%-- that's what that negative is telling us.
Absolute change tells us the difference between a new value and its previous value. So if we were measuring the change in the height of a plant over time, we'd want to use absolute change. Let's say it ended up being 3 feet tall, and a month before, it was 2.1 feet tall. So the new value is right here, this is our new, and this is our old or our previous value.
So in order to do absolute change, we're going to do the new value minus the previous value. The new value minus the older value. So in this case, it's 3 feet minus 2.1 feet. When we calculate that out, we get 0.9. That means we've changed by 0.9 feet. Now because it's a positive number, it means we're growing, it's getting bigger. If it was a negative number, it'd mean we were shrinking by that amount.
We'll now calculate relative change. We'll use the same example to calculate relative change. Relative change is the ratio of the absolute change to the previous value. So we'll use the same numbers as before. The new plant was 3 feet, the old plant was 2.1 feet, and the absolute change was 0.9 feet.
So in order to calculate our relative change, we're going to take the absolute change and divide it by the previous value. So 0.9 feet divided by the previous value, the older one, the 2.1 feet. Just like before, it's more easy and useful to divide this out, so we're going to find that now.
So I take my calculator, 0.9 divided by 2.1, and I get 0.4285 and so on. We're going to round this again, so this is going to round to 0.43. So if I wanted to turn this into a percent, I would multiply 100. 0.43 times 100 gets me 43%. So this means that our plant grew by 43% when it grew from here to here.
If we had a negative absolute change, that would mean that our relative change and our percent relative change would also be negative. This is not a problem, it's just telling us that we're shrinking instead of growing.
This has been your tutorial on absolute and relative error and absolute and relative change.