Calculating Percent Error

As an example, let's say you look down from a window into a city square and see a large crowd.

You do your best to estimate the number of people in the crowd, and you decide that it looks like about 1,000.

This is our measured value.

In actuality, all of these people brought tickets to enter some event, and there were 912 tickets torn at the gate.

That would be the actual value.

Assuming all these people showed tickets upon entry, how far off the true value was your estimation? One way to report this error is to say that you were off by 88 people, which is the absolute error.

But we need to do more than to simply state the difference between the two numbers, because the magnitude alone doesn't convey how far off you were relative to the true value.

If you guess 89 people when there was only one, you made a horrible estimation, whereas if you guess 1 million when there are 999,912, you are a genius, or just a very lucky guesser.

So these are incredibly different levels of accuracy, even though both of these scenarios involved an absolute error of 88.

We need to describe the error as a percentage of the true value so we can get an actual proportion that provides the context needed to assess the accuracy of the measurement.

To do this, we take our estimated value and subtract the true value.

Whatever we get, we take the absolute value so that we can disregard the sign, and then divide that by the true value, which gives us the relative error.

And lastly, we can multiply the result by 100.

This will express the relative error as a percentage, giving us the percent error associated with our measurement.

For example, with the crowd outside the window, we guessed 1,000.

From that, let's subtract the actual value of 912 to get 88.

This is already positive, so the absolute value won't affect it.

88 divided by 912 times 100 gives us 9.65%, so we were less than 10% off with our estimate, which isn't too bad.

As an example, let's say you look down from a window into a city square and see a large crowd.

You do your best to estimate the number of people in the crowd, and you decide that it looks like about 1,000.

This is our measured value.

In actuality, all of these people brought tickets to enter some event, and there were 912 tickets torn at the gate.

That would be the actual value.

Assuming all these people showed tickets upon entry, how far off the true value was your estimation? One way to report this error is to say that you were off by 88 people, which is the absolute error.

But we need to do more than to simply state the difference between the two numbers, because the magnitude alone doesn't convey how far off you were relative to the true value.

If you guess 89 people when there was only one, you made a horrible estimation, whereas if you guess 1 million when there are 999,912, you are a genius, or just a very lucky guesser.

So these are incredibly different levels of accuracy, even though both of these scenarios involved an absolute error of 88.

We need to describe the error as a percentage of the true value so we can get an actual proportion that provides the context needed to assess the accuracy of the measurement.

To do this, we take our estimated value and subtract the true value.

Whatever we get, we take the absolute value so that we can disregard the sign, and then divide that by the true value, which gives us the relative error.

And lastly, we can multiply the result by 100.

This will express the relative error as a percentage, giving us the percent error associated with our measurement.

For example, with the crowd outside the window, we guessed 1,000.

From that, let's subtract the actual value of 912 to get 88.

This is already positive, so the absolute value won't affect it.

88 divided by 912 times 100 gives us 9.65%, so we were less than 10% off with our estimate, which isn't too bad.