Math Core

Measurements and estimates are almost never perfect. Guessing a jar holds 200 marbles when it really holds 250 is off by 50, and so is weighing a truck 50 pounds too heavy. But one of those is a much bigger miss. Percent error measures how far off you were compared to the true value.

Error compared to the actual value

You already know how to find a percent change: divide the change by the original. Percent error uses the same idea, but it compares the estimate (or measurement) to the actual (true) value.

Percent error

\text{percent error} = \frac{\left| \text{estimate} - \text{actual} \right|}{\text{actual}}

Then write the decimal as a percent. The absolute value makes the error positive, whether the estimate was too high or too low.

Always divide by the actual value, because that's the true amount you're measuring the miss against.

Worked example: An estimate that was too low

Dev guessed that a jar held 200 marbles. It actually held 250. What is his percent error?

The error is $|200 - 250| = 50$ marbles. Divide by the actual value, 250:

\frac{50}{250} = 0.2 = 20\%

Dev's percent error is 20%.

Worked example: A measurement that was too high

A scale shows that a bag of flour weighs 5.2 pounds. The bag really weighs 5 pounds. What is the percent error of the scale?

The error is $|5.2 - 5| = 0.2$ pound.

\frac{0.2}{5} = 0.04 = 4\%

The scale's percent error is 4%.

Comparing errors

Percent error lets you compare mistakes of different sizes fairly. A small error on a small amount can be a bigger percent error than a large error on a large amount.

Worked example: Which estimate is better?

Mia estimated a hallway is 40 feet long. It is actually 50 feet. Jon estimated the school field is 330 feet long. It is actually 300 feet. Whose estimate was more accurate?

Mia: $\dfrac{|40 - 50|}{50} = \dfrac{10}{50} = 0.2 = 20\%$ .
Jon: $\dfrac{|330 - 300|}{300} = \dfrac{30}{300} = 0.1 = 10\%$ .

Jon was off by more feet, but his percent error is smaller. Jon's estimate was more accurate.

Common mistake

Don't divide by the estimate. In Dev's marble example, $\dfrac{50}{200} = 25\%$ is wrong. The error is always compared to the actual value.

Rounding a percent error

Percent errors often don't come out even. Round to the place the problem asks for.

Worked example: Rounding

A recipe calls for 3 cups of water. Nora measures 2.8 cups. What is her percent error, to the nearest tenth of a percent?

\frac{|2.8 - 3|}{3} = \frac{0.2}{3} = 0.0666\ldots \approx 6.7\%

Her percent error is about 6.7%.

Tip

A percent error of $0\%$ means a perfect estimate. The closer to $0\%$ , the more accurate the estimate or measurement.

Practice

Practice 1

Luis estimated that 90 people would come to a school play. Actually, 100 people came. What is his percent error?