Absolute Error and Relative Error: Formula and Equation
Absolute error measures the size of the difference between the estimated value and the true value, whereas relative error expresses this difference as a percentage of the true value. Absolute and relative error are two different methods to calculate the difference between a measured or calculated value and the true or expected value.
In this article, we will understand the difference between both in detail.
Table of Content
- What is Absolute Error?
- Absolute Error Formula
- What is Relative Error?
- Relative Error Formula
- Mean Absolute Error
- Difference Between Absolute and Relative Error
- Difference Between Applications of Absolute Error and Relative Error
- Conclusion
What is Absolute Error?
Absolute error is the gap between the real value and the value we measure. How significant this gap is depends on what we’re measuring. For example, if we have to measure something as big as the distance of a road, then a small mistake in centimetres will not make a difference. But if we have to measure something as small as the length of a machine part, a mistake in centimetres can be a big deal. Therefore, even if the error is the same size, it is more important in the second case.
Absolute Error Formula
Formula for absolute error is:
Absolute Error = ∣Measured Value – True Value∣
It represents the magnitude of the difference between the measured value and the true value.
What is Relative Error?
Relative error helps compare the error in a measurement to the actual value. It gives the accuracy of measurement in comparison to the real size. For example, if we have to measure the weight of a book and find it to be 5% more than its actual weight, the relative error would be 5%. This helps in understanding the size of the mistake we made in our measurement. And, unlike many measurements, relative error does not have any specific unit. It is expressed in percentage by multiplying with 100.
Relative Error Formula
The formula of relative error is:
Relative Error = (Absolute error / True Value)
here,
- Absolute Error is the magnitude of the difference between the measured value and the true value.
- True Value is the actual or known value of the quantity being measured.
This formula helps to quantify how accurate a measurement is relative to the true value, expressed as a percentage.
Mean Absolute Error
Mean Absolute Error abbreviated as MAE, is the average of all absolute errors of the data collected. It is calculated by the formula:
MAE = 1/n.Σ|xi – x|
where:
- |xi – x| is Absolute Errors
- n is Number of Erros
Difference Between Absolute and Relative Error
The difference between absolute error and relative error can be understood from the table given below:
|
Basis |
Absolute Error |
Relative Error |
|---|---|---|
|
Definition |
Magnitude of the difference between the measured value and the true value |
Absolute error expressed as a percentage of the true value |
|
Calculation |
Eabs = (xmeasured – xtrue) |
Erel = (xmeasured – xtrue)/x |
|
Scale |
Constant regardless of the magnitude of the true value |
Varies with the magnitude of the true value |
|
Units |
Same as the units of the measured and true values |
Typically expressed as a percentage or fraction |
|
Interpretation |
Direct measure of the deviation from the true value |
Provides a proportional measure of the error relative to the magnitude of the true value |
Difference Between Applications of Absolute Error and Relative Error
In this section, we have covered the differences between absolute and relative error based on the applications in same sector.
|
Apllications |
Absolute |
Relative |
|---|---|---|
|
Engineering |
It is used when the absolute deviation from the true value is critical. |
It is seful when assessing the significance of the error relative to the scale of the measurement. |
|
Science |
It is helpful in situations where the magnitude of the difference is important, regardless of the scale of the values involved. |
It provides a proportional measure of the error, aiding in comparing errors across different scales or contexts. |
|
Statistics |
It is used in error analysis to quantify the accuracy of measurements or predictions. |
It is used in assessing the precision of estimates or in evaluating the validity of statistical models. |
|
Experimental |
It is useful in experimental setups to evaluate the accuracy of instruments or measurement techniques. |
It helps in interpreting the reliability of experimental results by considering the error relative to the magnitude of the observed values. |
|
Quality Control |
It is important for determining the deviation of manufactured products from the desired specifications. |
It is valuable for comparing the quality of products or processes across different scales or contexts. |
Conclusion
Absolute error measures the gap between the measured and true values, while relative error compares this gap to the actual value, providing a proportionate measure of the error. Both are important in understanding the accuracy of measurements, with absolute error offering a direct measure of deviation and relative error offering a scale-independent comparison. Understanding and calculating both types of errors help ensure the reliability and accuracy of measurements.
FAQs on Absolute Error and Relative Error
What is the difference between mean absolute error and absolute error?
- Absolute error is the magnitude of the difference between measured value and original value.
- Mean absolute error is the mean of magnitudes of the absolute errors in all the measurements throughout the experiment.
What is relative error with an example?
Relative error is the measure of uncertainty of measurement compared to the size of the measurement. For examples, if four weights are measured are 5.04 g, 5.02 g, and 4.98 g. 5.00 g. The absolute error is ± 0.02 g.
How much relative error is acceptable?
Acceptable level of relative error depends on the specific context and the requirements of the measurement or analysis. In many cases, a relative error of less than 5% is considered acceptable, but this threshold may vary depending on factors such as the precision required for the application and industry standards.
What is the best value for relative absolute error?
Relative Absolute Error (RAE) is a measure used to assess how well a prediction model performs. It is expressed as a ratio and falls between 0 and 1. A lower RAE value indicates a better model, with values close to zero being the most desirable. Essentially, the closer the RAE value is to zero, the more accurate the model is considered to be.
Why is relative error more important than absolute error?
Both relative error and absolute error show how big the mistake is compared to the correct value.