Relative Error — Definition, Formula & Examples

Relative error is the ratio of the absolute error of a measurement to the true (or accepted) value. It tells you how significant an error is relative to the size of what you are measuring.

Given a measured value $x_m$ and a true value $x_t$ , the relative error is defined as $\dfrac{|x_m - x_t|}{|x_t|}$ , provided $x_t \neq 0$ . It is a dimensionless quantity often expressed as a decimal or percentage.

Key Formula

\text{Relative Error} = \frac{|x_m - x_t|}{|x_t|}

Where:

$x_m$ = The measured or approximate value
$x_t$ = The true or accepted value

How It Works

First, find the absolute error by taking the absolute value of the difference between your measured value and the true value. Then divide that absolute error by the true value. The result is a ratio that indicates the size of the error in proportion to the quantity being measured. Multiply by 100 to convert it to a percentage, which is called percent error. A smaller relative error means your measurement is more accurate.

Worked Example

Problem: A student measures the length of a table as 152 cm. The true length is 150 cm. Find the relative error and express it as a percentage.

Find the absolute error: Subtract the true value from the measured value and take the absolute value.

|152 - 150| = 2 \text{ cm}

Divide by the true value: Divide the absolute error by the true value to get the relative error.

\frac{2}{150} \approx 0.0133

Convert to a percentage: Multiply by 100 to express the relative error as a percent error.

0.0133 \times 100 \approx 1.33\%

Answer: The relative error is approximately 0.0133, or about 1.33%.

Why It Matters

Relative error lets you compare the quality of measurements across different scales. An error of 1 cm matters far more when measuring a coin than when measuring a football field. In science labs, engineering, and statistics, relative error is the standard way to judge whether a measurement is acceptably precise.

Common Mistakes

Mistake: Dividing by the measured value instead of the true value.

Correction: Always divide the absolute error by the true (or accepted) value. Using the measured value in the denominator gives a different quantity and does not correctly represent how far off you are from the actual value.