How Do You Calculate Percentage Error?

Calculate percentage error by dividing the absolute difference between an approximate value and the exact value by the magnitude of the exact value, then multiplying by $100\%$. The formula is:

$$\text{Percentage error} = \frac{\left|\text{approximate value} - \text{exact value}\right|}{\left|\text{exact value}\right|} \times 100\%$$

This is part of SL 1.6, Approximating and estimating in IB Mathematics: Applications and Interpretation and applies to both SL and HL students.

The Method

First, calculate the absolute error by finding the magnitude of the difference. The absolute value makes the result non-negative because error describes the size of the discrepancy, not its direction. Then divide by the magnitude of the exact value and convert the result into a percentage.

$$\text{Absolute error} = \left|\text{approximate value} - \text{exact value}\right|$$

Suppose an object's exact length is $48\text{ cm}$, but a student measures it as $46.5\text{ cm}$. The absolute error is:

$$\left|46.5-48\right|=1.5\text{ cm}$$

Therefore,

$$\text{Percentage error} = \frac{1.5}{48}\times 100\% = 3.125\%$$

The percentage error is therefore $3.13\%$, correct to three significant figures.

Percentage error is a relative measure, so it allows errors from differently sized quantities to be compared fairly. For example, an absolute error of $1\text{ cm}$ is substantial for a $5\text{ cm}$ object but small for a $500\text{ cm}$ object. An approximate value below the exact value is an underestimate; one above it is an overestimate. However, percentage error itself remains non-negative because of the absolute value.

A common misconception is dividing by the approximate value. The denominator must be the exact value, unless the question explicitly defines another comparison. Percentage error is undefined when the exact value is zero because the calculation would require division by zero. Unlike absolute error, percentage error has no physical unit.

IB Exam Technique

Show the subtraction, absolute value, division, and multiplication by $100\%$. Keep full GDC precision throughout and round only the final answer to the requested accuracy. Premature rounding can change the result and lose an accuracy mark; also check that the denominator is the exact value before calculating.