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The difference between a â€˜true valueâ€™ and a â€˜measured valueâ€™ or an â€˜estimateâ€™ of the same value is known as an â€˜errorâ€™. This error can be expressed in the form of a percentage. In this article we will discuss the formula to calculate, and hence, express percent error....Read MoreRead Less

The difference between the estimated or measured value and the true or actual value is known as error. In other words, the **deviation** of the estimated or measured value from the actual value is known as error. When the error is compared with the actual value and expressed as a percentage then it is called **percent error** or **percentage error**.

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Calculating the percent error allows us to determine how closely the estimation or a given measurement corresponds to the actual value.Â

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[Note: Measured value is also known as the observed value.]

The formula to calculate percent error is:

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**Percent Error = \( \left ( \frac{Estimated~or~Measured~Value~-~Exact~Value}{Exact~Value} \right )~\times~100\)**

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In the formula we can observe,Â

- The estimated value that is also referred to as the approximated valueÂ
- The measured value that is also referred to as the observed or experimental value
- The exact value that is also referred to as the true or actual value

**Example 1: An event manager is planning a fun fest for children in the month of December. Due to the holiday season he expected around 1000 visitors. The actual number of people who turned up for the fest was 860. What is the percent error of his estimate?**

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**Solution:**

The question states that,Â

Estimated number of visitors = 1000

Actual number of visitors = 860

**Percent Error = \( \left ( \frac{Estimated~Value~-~Exact~Value}{Exact~Value} \right )~\times~100\)Â Â **Â [Formula for percent error]

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Substitute the given values:

Percent Error = \( \left ( \frac{1000~-~860}{860} \right )~\times~100\)

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Â Â Â Â Â Â Â Â Â Â Â Â = \( \left ( \frac{140}{860} \right )~\times~100\)Â Â [Subtract]

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**Â Â Â Â Â Â Â Â Â Â Â Â = **16.28Â Â Â Â Â Â Â Â [Simplify]

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Therefore the percent error in the estimation of the number of visitors is 16.28%.

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**Example 2: Duke weighs himself on a weighing scale. The scale shows 100 pounds. Dukeâ€™s actual weight is 98 pounds. Find the percent error in the observed weight.**

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**Solution:**

As stated in the question,

Actual weight = 98 pounds

Observed or measured weight = 100 pounds

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**Percent Error = \( \left ( \frac{Observed~Value~-~Exact~Value}{Exact~Value} \right )~\times~100\)**Â Â [Formula for percent error]

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Substitute the given values:

Percent Error = \( \left ( \frac{100~-~98}{98} \right )~\times~100\)

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Â Â Â Â Â Â Â Â Â Â Â = \( \left ( \frac{2}{98} \right )~\times~100\)Â Â [Subtract]

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**Â Â Â Â Â Â Â Â Â Â Â Â =** 2.04Â Â Â Â Â Â Â Â [Simplify]

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Therefore, the percent error in Dukeâ€™s weight is 2.04%.

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**Example 3: Rex estimated that he would spend $150 on his weekend trip with his friends. However, due to a few overpriced items in a restaurant he spent a total of $180 on the trip. What is the percent error in his estimate?Â **

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**Solution:**

As given in the question,Â

Estimated spend = 150

Actual spend = 180

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**Percent Error = \( \left ( \frac{Estimated~Value~-~Exact~Value}{Exact~Value} \right )~\times~100\)**Â Â [Formula for percent error]

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Substitute the given values:

Percent Error = \( \left ( \frac{150~-~180}{180} \right )~\times~100\)

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Â Â Â Â Â Â Â Â Â Â Â = \( \left ( \frac{-~30}{180} \right )~\times~100\)Â Â Â [Subtract]

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**Â Â Â Â Â Â Â Â Â Â Â Â = **– 16.67Â Â Â Â Â Â Â Â [Simplify]

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Therefore the percent error in the estimated spend for the weekend trip that Rex took is – 16.67%.

Frequently Asked Questions

The percentage of error is a value expressed as a percentage. This indicates the deviation of the estimated values or experimental results deviate from the actual values.Â

The percent error is negative if the experimental or estimated value is lower than the actual or true value. In order to minimize confusion caused by a negative error, the error can be typically expressed as the absolute difference, a value that is neither negative nor positive.