Percent Error: Definition, Formula & Examples

Percent error compares an estimate to a correct value and expresses the difference between them as a percentage. This statistic allows analysts to understand the size of the error relative to the true value. It is also known as percentage error and % error.

In this context, the estimate and correct values can be the following:

Estimate: A measurement, approximation, experimentally derived value, or a guess.
Correct Value: A quantity that has been proven or generally accepted as being valid. It can be a standard measurement for an item used in testing measurement systems. Or, a known value that is correct on a theoretical basis, such as the circumference of a circle.

Why Assess Percent Error?

Percent error is a valuable statistic when your estimate targets a known, correct value. In general terms, use it to quantify how close an estimate is to that true value. Smaller errors occur when an approximate value is close to the correct value. As the estimates move further away from the actual value, the percent error increases.

The measurement instrument, estimation process, personnel, or a combination of factors can cause these errors. When the error becomes large enough, it can invalidate your estimates. At that point, you’ll need to take corrective measures. However, there is no standard cutoff point because it varies by subject area.

The purpose for calculating the percentage error depends on the context. In scientific studies and quality management projects, analysts use it to compare measured values to known values to assess the validity of their measurements. Alternatively, researchers use it to compare a value from an experiment to a theoretical or true value to understand the validity of their experimental calculations.

To calculate this type of measurement error, you must know the correct value. When you don’t know it, you’ll need to use another method, such as evaluating measurement variability.

After covering the formula, I’ll go over several examples of using it in different contexts.

Percent Error Formula

Finding the percent error involves three steps:

Calculate the error, which is the Estimate – Correct Value.
Divide by the Correct Value.
Multiply by 100 to produce a percentage.

When calculating this statistic, some fields of study retain the plus or minus values to indicate whether the Estimate is above or below the Correct value. However, other areas use the absolute value of the error, which always produces positive values. In the percent error equations below, the bars (|) indicate using the absolute value.

When you don’t use the absolute value of the error, you’ll obtain positive percentages when the Estimate is greater than the Correct value and negative values when the Estimate is lower. However, the absolute value form always produces positive values. Check to see which version is the norm for your field!

Below are the percent error formulas:

$Percent Error = {\displaystyle \frac {{\text{Estimate - Correct}}}{{\text{Correct}}}} \times100$

$Absolute Percent Error = {\displaystyle \frac {{\vert \text{Estimate - Correct}\vert}}{{\text{Correct}}}} \times100$

Examples of Percent Error

For these percent error examples, I use the percent error formula that retains the positive and negative signs because it provides more information. Remove the negative signs to produce the absolute value form.

Guesses / Rough Estimations

Imagine you’re planning a party and estimate that 15 people will attend. In reality, 18 people attend.

${\displaystyle \left( \frac {15-18}{18}\right)} \times100 = -16.67 \%$

Your guess was in error by -16.67%, meaning that it was too low.

Assessing Measurements

Measurements are inexact. They are approximations of the actual characteristic. Human error and device limitations can contribute to measurement error.

For example, quality control analysts are assessing the measurement system for their inspection process. They need to obtain valid measurements of part lengths. A standard part they use for testing has an agreed upon length of 5.0mm. An inspector measures this part and records 5.2mm.

${\displaystyle \left( \frac {5.2-5.0}{5.0}\right)} \times100 = 4.0 \%$

The inspector’s measurement contains 4% error and it was too high.

When you’re using percent error to compare measurements to a known standard item, smaller errors represent measurements that are close to the correct value. If your measurements have more significant errors, you might need to make adjustments to your measurement system.

Related post: Accuracy vs. Precision

Comparing Experimental Values to Known Values

In the third century BCE, Eratosthenes, a Greek librarian in Egypt, used the Sun’s positions at two locations at the same time to estimate the Earth’s circumference as being 43,075km. Currently, the Earth’s circumference has a known value of 40,096km.

${\displaystyle \left( \frac {43,075-40,096}{40,096}\right)} \times100 = 7.4 \%$