Statistics Definitions > Percent Error

**Contents**:

## What is Percent Error?

Percent errors tells you how big your errors are when you measure something in an experiment. Smaller percent errors mean that you are close to the accepted or real value. For example, a 1% error means that you got very close to the accepted value, while 45% means that you were quite a long way off from the true value. Measurement errors are mostly unavoidable: equipment can be imprecise, hands can shake, or your instruments just might not have the capability to measure accurately. Percent error will let you know how badly these unavoidable errors affected your results.

## Formula

The formula for percent error is:

**PE = (|accepted value – experimental value| \ accepted value) x 100%.**

**Example question:** The accepted distance to the moon is 238,855 miles.* You measure the distance as 249,200 miles. What is the percent error?

**Solution:**

Step 1: Insert your data into the formula:

PE = (|accepted value – experimental value| \ accepted value) x 100% =

((|238,855 miles – 249,200|) \ 238,855 miles) x 100% =

Step 2: Solve:

(10345 \ 238,855 miles) x 100% =

0.0433 * 100% =

**4.33%.**

*That’s the average distance, but let’s assume it’s the distance on the day you’re taking the measurement!

**Note**: in some sciences, the absolute value sign is sometimes (but not always) omitted. You may want to refer to your textbook to see if the author is omitting the absolute value sign. If you aren’t sure, the most common form is **with** the absolute value sign.

## Alternate Wording

Accepted value is sometimes called the “true” value or “theoretical” value, so you might see the formula written in slightly different ways:

- PE = (|true value – experimental value| \ true value) x 100%.
- PE = (|theoretical value – experimental value| \ theoretical value) x 100%.

All three versions of the formula mean the exact same thing — it’s just different wording.

## Alternative Definition using Relative Error

The percentage error is sometimes reported as being 100% times the relative error. Be careful though, because there are actually **two types of relative error**: one for precision and one for accuracy (not sure of the difference between the two? See: Accuracy and Precision). The definition “100% times the relative error” is only true if you are using the “accuracy” version of relative error:

- RE
_{accuracy}= (Absolute error / “True” value) * 100%.

The definition does *not *work if you’re using the RE for precision:

- RE
_{precision}= absolute error / measurement being taken.

## What is Percent Difference?

Percent difference is practically the same as percent error, only instead of one “true” value and one “experimental” value, you compare two experimental values. The formula is:

**Where**:

- E
_{1}is the first experimental measurement. - E
_{2}is the second experimental measurement.

**Example question:** You make two measurements in an experiment of 20 mL and 22 mL. What is the percent difference?

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