# Understanding Absolute and Relative Error

Absolute error is the difference between measured or inferred value and the actual value of a quantity. Because it provides no information about the defect’s significance, the absolute error is insufficient. When calculating the distance in kilometers between cities, a few centimeters of inaccuracy is significant. Let’s look at another situation where a centimeter inaccuracy might be quite significant when measuring small machine parts. Although the two mistakes are both in the range of centimeters, the second one is more serious than the first. Contents

## What is Absolute and Relative Error?

The difference between a measured value and an actual value is known as absolute error. The percentage of the absolute error in relation to the measured value is called the relative error.

## Formula For Absolute and Relative Error

If x is the actual value of a quantity and x0 is the measured value of the quantity, then the absolute error value can be calculated using the formula

Δx = x0-x.

Here, Δx is called an absolute error.

If we consider multiple measurements, then the arithmetic mean of absolute errors of individual measurements should be the final absolute error.

## Mean of Absolute Error

The average of all the absolute mistakes found in the data set is known as the mean absolute error. The acronym for it is MAE, or Mean Absolute Error. It is calculated by dividing the total number of errors by the sum of all absolute errors. The MAE formula is: Here,

• |xi – x| = absolute errors
• n = number of errors

## Example of Absolute Error and Relative Error

The techniques for determining the approximate values and the absolute errors in these computations fall under the purview of numerical analysis. The relative error indicates the size of the error in relation to the correct value, whereas the absolute error indicates the size of the error overall. Errors in numerical calculations can arise for the following reasons.

• Round off error
• Truncation error

### Example 1:

Find the absolute and relative errors of the approximation 125.67 to the value 119.66.

Solution:

• Absolute error = |125.67-119.66|=6.01
• Relative error = |125.67-119.66|/119.66 = 0.05022

### Example 2

For example, 24.13 is the actual value of a quantity and 25.09 is the measure or inferred value, then the absolute error will be:

• Absolute Error = 25.09 – 24.13
• = 0.86

Most of the time it is sufficient to record only two decimal digits of the absolute error. Thus, it is sufficient to state that the absolute error of the approximation 4.55 to the correct value 4.538395 is 0.012.

## Conclusion

To sum up, absolute error and relative error are fundamental ideas in science, engineering, and mathematics. The amount of variation between a measured or computed value and the true value is quantified by absolute error. This variation is expressed as a proportion in relation to the real value using relative error, which is frequently expressed as a percentage mistake. across order to ensure the dependability and caliber of outcomes across a broad range of scientific and practical applications, these ideas are essential for assessing the accuracy and precision of measurements, computations, and data analysis.