Absolute and Relative Error
Why Do Errors Matter?
In the real world, measurements are never perfectly exact. Whether you're measuring the length of a table, the weight of an object, or the speed of a car, there's always some degree of approximation. Understanding error helps us quantify how far off our approximation is from the true value.
Absolute Error
Definition: The absolute error of an approximate value is the absolute value of the difference between the exact value and the approximate value.
Where:
- $ \Delta x $ — absolute error
- $ x_{\text{exact}} $ — the true (exact) value
- $ x_{\text{approx}} $ — the approximate value
Key point: Absolute error is always non-negative (≥ 0) and has the same units as the measured quantity.
Relative Error
Definition: The relative error of an approximate value is the ratio of the absolute error to the absolute value of the approximate value.
Where:
- $ \delta x $ — relative error (dimensionless)
- $ \Delta x $ — absolute error
- $ |x_{\text{approx}}| $ — absolute value of the approximate value
Key point: Relative error is unitless and is often expressed as a percentage: $ \delta x \times 100\% $