Absolute and Relative Error

Measurement is a major part of scientific calculations. Completely accurate measurement results are absolutely rare. While measuring different parameters, slight errors are common. There are different types of errors, which cause differences in measurement. All the errors can be expressed in mathematical equations. By knowing the errors, we can calculate correctly and find out ways to correct the errors. There are mainly two types of errors – absolute and relative error. In this article, we are going to define absolute error and relative error. Here, we are giving explanations, formulas, and examples of absolute error and relative error along with the definition. The concept of error calculation is essential in measurement. 

Define Absolute Error

Absolute error is defined as the difference between the actual value and the measured value of a quantity. The importance of absolute error depends on the quantity that we are measuring. If the quantity is large such as road distance, a small error in centimetres is negligible. While measuring the length of a machine part an error in centimetre is considerable. Though the errors in both cases are in centimetres, the error in the second case is more important.

Absolute Error Formula

The absolute error is calculated by the subtraction of the actual value and the measured value of a quantity. If the actual value is x₀ and the measured value is x, the absolute error is expressed as,

∆x = x₀- x

Here, ∆x is the absolute error.

Absolute Error Example

Here, we are giving an example of the absolute error in real life. Suppose, we are measuring the length of an eraser. The actual length is 35 mm and the measured length is 34.13 mm. 

So, The Absolute Error = Actual Length - Measured Length

 = (35 - 34.13) mm 

 = 0.87 mm

Classification Of Absolute Error

1. Absolute accuracy error

Absolute accuracy error is the other name of absolute error. The formula for absolute accuracy error is written as E= E exp – E true, where E is the absolute accuracy error, E exp is the experimental value and E true is the actual value.

2. Mean absolute error

MAE or the mean absolute error is the mean or average of all absolute errors. The formula for Mean Absolute Error is given as,

MAE = \[\frac{1}{n}\] \[\sum_{i=1}^{n}\mid x_{i} -x\mid\]

3. Absolute Precision Error

It is a standard deviation of a group of numbers. Standard deviation helps to know how much data is spread.

Relative Error Definition

The ratio of absolute error of the measurement and the actual value is called relative error. By calculating the relative error, we can have an idea of how good the measurement is compared to the actual size. From the relative error, we can determine the magnitude of absolute error. If the actual value is not available, the relative error can be calculated in terms of the measured value of the quantity. The relative error is dimensionless and it has no unit. It is written in percentage by multiplying it by 100.

Relative Error Formula

The relative error is calculated by the ratio of absolute error and the actual value of the quantity. If the absolute error of the measurement is ∆x, the actual value is x0, the measured value is x, the relative error is expressed as, 

                                                     xᵣ = (x₀ - x)/ x₀ = ∆x/x₀

Here, xr is the relative error.

Relative Error Example

Here, we are giving an example of relative error in real life. Suppose, the actual length of an eraser is 35 mm. Now, the absolute error = (35-34.13) mm = 0.87 mm.

So, the relative error = absolute error/actual length

= 0.87/35

= 0.02485

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Relative Error as a Measure of Accuracy

In many cases, relative error is a measure of precision. At the same time, it can also be used as a measure of accuracy. Accuracy is the extent of knowing how accurate the value is as compared to the actual or true value. Students can find the RE accuracy only if they know the true value or measurement. For simplicity, we have the formula for calculating the RE accuracy which is given as

RE\[_{accuracy}\] = Actual error/ true value * 100%

Absolute Error and Relative Error in Numerical Analysis

Numerical analysis of error calculation is a vital part of the measurement. This analysis finds the actual value and the error quantity. The absolute error determines how good or bad the measurement is. In numerical calculation, the errors are caused by round-off error or truncation error.

Absolute and Relative Error Calculation - Examples

1. Find the absolute and relative errors. The actual value is 125.68 mm and the measured value is 119.66 mm.

Solution:

Absolute Error = |125.68 – 119.66| mm

= 6.02 mm

Relative Error = |125.68 – 119.66| / 125.68

= 0.0478

2. Find out the absolute and relative errors, where the actual and measured values are 252.14 mm and 249.02 mm.

Solution:

Absolute Error = |252.14 – 249.02| mm

                           = 3.12 mm

Relatives Error = 3.12/252.14

= 0.0123

Did You Know?

In different measurements, the quantity is measured more than one time to get an average value of the quantity. Mean absolute error is one of the most important terms in this kind of measurement. The average of all the absolute errors of the collected data is called the MAE ( Mean Absolute Error). It is calculated by dividing the sum of absolute errors by the number of errors. The formula of MAE is –

MAE = \[\frac{\displaystyle\sum\limits_{i=1}^n \mid x_{0}-x \mid }{n}\]

Here, 

n is the number of errors, 

x₀ is the actual value, 

x is the measured value, and 

|x₀-x| is the absolute error.