Relative Error in Measurement

What is relative error?

The relative error measures a value's accuracy by dividing the absolute error by the measured value. $$
\text{Relative Error} = \frac{\text{Absolute Error}}{\text{Measured Value}} $$ The percentage error is a form of relative error expressed as a percentage. $$ \text{Percentage Error} = \left( \frac{\text{Absolute Error}}{\text{Measured Value}} \right) \times 100 $$

In science and measurement, achieving accuracy is crucial.

However, every measurement includes a margin of error, which can be expressed in different ways.

For example, absolute error, relative error, and percentage error are key tools for evaluating a measurement’s accuracy.

The absolute error—the difference between the measured value and the true value—isn't always enough to fully gauge measurement accuracy.

$$ x = \bar{x} \pm e_x $$

Where $ \bar{x} $ represents the measured value or the average of measurements, and $ e_x $ is the absolute error.

To get a more meaningful sense of precision, we calculate the relative error.

$$ \epsilon_x = \frac{ e_x }{ \bar{x} } $$

A very small relative error means the error has little impact, resulting in a more precise measurement.

Why is relative error preferred? Relative error is a dimensionless quantity, as it’s a ratio between two values with the same units. It doesn’t rely on units or the scale of the measured values, making it very useful for comparing the precision of different types of measurements.

An Example 

Imagine measuring the length of a bridge.

After taking two measurements, you find slight variations: 149.5 m and 150.5 m.

The average of these values provides an estimate of the bridge’s length:

$$ \bar{x} = \frac{149.5 + 150.5}{2} = 150 \ m $$

The absolute error, representing the average deviation from this central value, is therefore 0.5 meters:

$$ e_x = \frac{150.5 - 149.5}{2} = 0.5 \ m $$

This means the actual bridge length lies within the range of these measurements:

$$ x = \bar{x} \pm e_x $$

$$ x = 150 \pm 0.5 \ m $$

Now we can calculate the relative error to assess how precise the measurement is relative to the average value:

$$ \epsilon_x = \frac{e_x}{ \bar{x}} $$

$$ \epsilon_x = \frac{0.5}{ 150} = 0.0033 $$

Relative error gives us a clearer indication of measurement accuracy, allowing for meaningful comparisons with other measurements, even across different scales.

To calculate the percentage error, simply multiply the absolute error by 100. In this example, it’s 0.33%: $$ \epsilon_x = \frac{0.5}{150} \cdot 100 = 0.33 \ \% $$

Example 2

To further illustrate relative error, let’s consider a practical example where we measure both the length of a bridge and the diameter of a pipe.

We have the following data:

• Bridge length: \( L = (150.0 \pm 0.5) \, \text{m} \)
• Pipe diameter: \( d = (0.8 \pm 0.02) \, \text{m} \)

The bridge length has an absolute error of 0.5 meters, while the pipe diameter has an absolute error of 0.02 meters.

At first glance, the pipe diameter measurement might seem more accurate because the absolute error is smaller, but that’s not necessarily true!

Let’s calculate the relative error for each measurement:

• The relative error of the bridge length is 0.0033 $$ \varepsilon_L = \frac{0.5}{150.0} \approx 0.0033 $$
• The relative error of the pipe diameter is 0.025 $$ \varepsilon_d = \frac{0.02}{0.8} \approx 0.025 $$

Comparing these results, we see that the relative error for the bridge measurement (0.0033) is much smaller than that for the pipe diameter (0.025), even though the bridge has an absolute error of 0.5 m while the pipe’s error is only 0.02 m.

This comparison shows that absolute error alone doesn’t provide a complete picture of measurement accuracy.

Relative error, however, allows us to look beyond the simple difference between measured and true values.

This dimensionless measure lets you accurately compare widely different values, whether it's the length of a bridge or the thickness of a pipe.