when we measure things our values aren't always exactly accurate and so it can be important for us to tell how far off we were this is a value called error and we'll talk about how to calculate this in this lesson in the previous lesson we looked at uh groups that were trying to measure a distance that was exactly 500.0 ft long and one group did a number of Trials where they came up with these three values for the length of this distance all of which were off by a little bit so let's see how that we how we'll calculate error of each one of these measurements error here is defined as the measured value minus the actual value it's not a particularly tricky equation so let's go right ahead and look at this uh this first number that we have here so 5035 ft minus that's our measured value minus the actual value which was 5 500.0 ft is going to give us 3.5 ft as our error so for each one of these trials we can indicate an error 3.5 this was 3.5 higher than the actual value so we often indicate whether we were a little bit above or a little bit below by using a plus or minus sign so this error is going to be plus 3.5 ft for this measurement okay for the second measurement once again 502.040 ft above the actual so plus 2.8 now finally for our last trial here 4 97.4 ft minus 500.0 ft is going to give us in this case negative 2.6 feet and so again we'll say minus 2.6 feet to indicate that we low balled it a little bit there this number here 497.org some measurements We compare them to the actual value um and then we uh we calculate the error okay let's say the error comes out to be a th000 feet we might think wow you know we were really off that was a really big error but hey what if the distance that we were measuring was a million feet in that case an error of a th000 feet being a th000 feet off isn't that big of a deal now on the other hand let's say that we had an error of 10t that might not really sound like a big deal at all but what if what we're trying to measure is 100 ft or 200t then those 10 ft become pretty significant and we were very very off from the actual value so instead of just calculating error we also often want to calculate something called percent error and what percent error does is percent error gives us an idea of how big our error is compared to the size of what we were actually measuring so basically is it a big deal should we worry about it so percent error can be described as the the absolute value of the error divided by the actual value times 100% let's look at the percent errors for a couple of these trials here so right here we had an error of 3.5t and we'll divide that by the actual value which was 500.0 ft remember that this is the actual value value sometimes people make the mistake of taking 3.5 and dividing it by 5035 don't do that it's 3.5 divid the actual and then we'll multiply that by 100% the final answer that we're going to get is 7% That's the percent error we are off by 7% so that's not really a huge number okay let's look at this next one the error that we get for the Third situation here now the error here was minus 2.6 we're a little bit below the actual value but remember for percent error we want to calculate the absolute value of the error so with absolute value we always make it positive so it's going to be 2.6 feet don't worry that it's negative because it's absolute value divided Again by the actual value which is 500.0 feet we're going to multiply that by 100% And the answer that that's going to give us is going to be uh 52% there if you're a little bit worried about the significant figures here keep in mind that 100% is part of the definition it's part of the equation so we don't have to worry about significant figures here now percent error we've looked at for these different trials for the measurement of something that was uh 5500 ft let's take the same error 3.5 ft and imagine that we were trying to measure something that was only 17 ft and we ended up with an error of 3.5 in that case we do 3.5 ft divided by 17 ft and multiply that by 100% in that case our error is much bigger it's going to be 21% that's huge now the reason why is because 3.5 is a much bigger part of seven than 3.5 was of 500 but I did this example to show you what I was talking about earlier that 3.5 ft might be a large percentage error or small percentage error depending on the size of what we're measuring in the first place that's why percent error is useful it tells us should we worry about it 7% not really 21% yeah we should that's a really bad error that's way too much now there are a few ways to write the equation for percent error we looked earlier at the equation for error measured value uh minus actual value and then percent error we can also put these two equations together in one big equation for percent error where we take the absolute value of the measured value minus the actual value and then divide it by the actual value and multiply that by 100 as you'll see this is just the combination of these two error is this so we're just taking this guy here and putting it on top of the actual value and that's how we get the third equation but I just tell you this because I don't want you to be confused if you see the definition of percent error being this very large equation that you haven't seen before