Understanding Relative Error, Absolute Error, and Mean Error

When performing measurements or calculations in physics and other sciences, it is crucial to understand the accuracy and precision of the results. Errors in measurement can arise from various sources, such as instrument limitations, observer mistakes, or environmental conditions. To quantify these errors, we use the concepts of relative error, absolute error, and mean error. Let's explore each of these concepts in detail.

Absolute Error

Absolute error is the difference between the measured value and the true value of the quantity. It represents the magnitude of the error without considering the size of the measurement. The absolute error is expressed with the same units as the measurement.

Formula for Absolute Error

If $x_{\text{measured}}$ is the measured value and $x_{\text{true}}$ is the true value, the absolute error $E_{\text{abs}}$ is given by:

$$ E_{\text{abs}} = |x_{\text{measured}} - x_{\text{true}}| $$

Example of Absolute Error

Suppose you measure the length of a table to be 2.00 meters, but the actual length is 1.98 meters. The absolute error in your measurement is:

$$ E_{\text{abs}} = |2.00 \text{ m} - 1.98 \text{ m}| = 0.02 \text{ m} $$

Relative Error

Relative error is the absolute error expressed as a fraction of the true value. It provides a sense of the error's magnitude in relation to the size of the measurement and is often expressed as a percentage.

Formula for Relative Error

The relative error $E_{\text{rel}}$ is calculated as:

$$ E_{\text{rel}} = \frac{E_{\text{abs}}}{|x_{\text{true}}|} $$

When expressed as a percentage, it is:

$$ E_{\text{rel}}\% = \left(\frac{E_{\text{abs}}}{|x_{\text{true}}|}\right) \times 100\% $$

Example of Relative Error

Using the previous example with the table measurement:

$$ E_{\text{rel}} = \frac{0.02 \text{ m}}{1.98 \text{ m}} \approx 0.0101 $$

As a percentage, the relative error is:

$$ E_{\text{rel}}\% \approx 1.01\% $$

Mean Error

Mean error, often referred to as mean absolute error (MAE), is used when multiple measurements are taken. It is the average of the absolute errors of a set of measurements, providing an overall indication of the accuracy of the measurements.

Formula for Mean Error

Given a set of $n$ measurements $x_{\text{measured},i}$ and their corresponding true values $x_{\text{true},i}$, the mean error $E_{\text{mean}}$ is:

$$ E_{\text{mean}} = \frac{1}{n} \sum_{i=1}^{n} |x_{\text{measured},i} - x_{\text{true},i}| $$

Example of Mean Error

Imagine you measure the length of the same table five times, obtaining the following measurements: 2.01 m, 1.99 m, 2.00 m, 2.02 m, and 1.98 m. The true length is 1.98 m. The mean error is:

$$ E_{\text{mean}} = \frac{1}{5} (|2.01 - 1.98| + |1.99 - 1.98| + |2.00 - 1.98| + |2.02 - 1.98| + |1.98 - 1.98|) = \frac{1}{5} (0.03 + 0.01 + 0.02 + 0.04 + 0) = 0.02 \text{ m} $$

Comparison Table

Here is a table summarizing the differences and important points of absolute error, relative error, and mean error:

Concept	Definition	Formula	Example Value	Units
Absolute Error	Difference between measured and true value	$E_{\text{abs}} =	x_{\text{measured}} - x_{\text{true}}	$
Relative Error	Absolute error as a fraction of the true value	$E_{\text{rel}} = \frac{E_{\text{abs}}}{	x_{\text{true}}	}$
Mean Error	Average of absolute errors from multiple measurements	$E_{\text{mean}} = \frac{1}{n} \sum	x_{\text{measured},i} - x_{\text{true},i}	$

Understanding these errors is essential for interpreting the results of experiments and calculations accurately. It allows scientists and engineers to assess the reliability of their measurements and to make informed decisions based on the precision of their data.