**Key Difference – Absolute Error vs Relative Error**

Absolute error and relative error are two ways of indicating errors in experimental measurements though there exists a difference between absolute error and relative error based on their calculation. Most of the measurements in scientific experiments comprise of errors, due to instrumental errors and human errors. In some cases, for a particular measuring instrument, there is a pre-defined constant value for absolute error *(The smallest reading. Eg: – ruler = +/- 1 mm.)* It is the difference between true value and the experimental value. However, relative error varies depending on the experimental value and the absolute error. It is determined by taking the ratio of absolute error and the experimental value. Thus, the **key difference **between absolute error and relative error is, **absolute error** **is the** **magnitude of the difference between the exact value and the approximation** whereas **relative error is calculated by dividing the absolute error by the magnitude of the exact value.**

**What is Absolute Error?**

Absolute error is an indication of the uncertainty of a measurement. In other words, it measures to what extent, the true value can vary from its experimental value. Absolute error is expressed in the same units as the measurement.

*Example:* Consider we want to measure the length of a pencil using a ruler with millimeter marks. We can measure its length to the nearest millimeter value. If you get the value as 125 mm, it is expressed as 125 +/- 1 mm. The absolute error is +/- 1 mm.

**What is Relative Error?**

Relative error is dependent on two variables; absolute error and experimental value of the measurement. Therefore, those two parameters should be known, to calculate the relative error. Relative error is calculated by the ratio of absolute error and the experimental value. It is expressed as a percentage or as a fraction; so that it has no units.

## What is the difference between Absolute Error and Relative Error?

### Definition of Absolute Error and Relative Error

**Absolute error:**

Absolute error is a Δx value (+ or – value), where x is a variable; it is the physical error in a measurement. It is also known as the actual error in a measurement.

In other words, it is the difference between true value and the experimental value.

## Absolute Error = Actual Value – Measured Value |

**Relative error:**

Relative error is the ratio of absolute error (Δx) to the measured value (x). It is expressed either as a percentage (percentage error) or as a fraction (fractional uncertainty).

### Units and Calculation of Absolute Error and Relative Error

#### Units

**Absolute error:**

It has the same units as the measured value. For example, if you measure the length of a book in centimeters (cm), the absolute error also has the same units.

**Relative error:**

Relative error can be expressed as a fraction or as a percentage. However, both do not have a unit in the value.

#### Error Calculation

Example 1:The actual length of a land is 500 feet. A measuring instrument shows the length to be 508 feet. |

**Absolute error:**

Absolute error = [Actual value – measured value] = [508-500] feet = 8 feet

**Relative error:**

*As a percentage:*

*As a fraction:*

*Example 2:*

A student wanted to measure the height of a wall in a room. He measured the value using a meter ruler (with millimeter values), it was 3.215m. |

**Absolute error:**

Absolute error = +/- 1 mm = +/- 0.001m *(The smallest reading which can be read using the ruler)*

**Relative error:**

Relative error = Absolute error÷ Experimental value = 0.001 m÷ 3.215 m * 100 = 0.0003%

Image Courtesy: “Absolute error” by DEMcAdams – Own work. (CC BY-SA 4.0) via Wikimedia Commons “Relative error of a Monte Carlo integration to calculate pi” by Jorgecarleitao – python and xmgrace. (CC BY-SA 3.0) via Wikipedia
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