Absolute Error

1. Definition

Absolute error is the absolute value of the difference between the true value $x$ and the approximation $\tilde{x}$.

Absolute error measures how far a measured or computed value deviates from the true value, expressed in actual units. It is the most fundamental measure of error.

The main sources of absolute error are (1) measurement error, (2) roundoff error from representing numbers with finitely many digits (Chapter 8), and (3) truncation error from cutting off an infinite process (Chapter 9). This chapter measures all of them together by "how far off" the result is.

In this article the signed error is defined as $\Delta x = \tilde{x} - x$. This quantity is sometimes called the true error, but note that the sign convention varies between references (some define it as true value $-$ approximation). With our convention, $\Delta x > 0$ indicates overestimation and $\Delta x < 0$ indicates underestimation.

2. Calculation Methods and the Case of an Unknown True Value

Computing absolute error involves simple subtraction and taking the absolute value.

In practice, the true value $x$ is often unknown. Alternative approaches include:

• Error bound: finding an upper bound $|x - \tilde{x}| \le \varepsilon$
• A posteriori estimation: in iterative methods, once convergence has been confirmed, using the difference between successive approximations $|\tilde{x}_{n+1} - \tilde{x}_n|$ as a rough indicator of the error. In general it is not the true error itself: if the convergence ratio $r$ is close to $1$ (slow convergence), the true error of $\tilde{x}_{n+1}$ can be about $r/(1-r)$ times this difference, a severe underestimate
• Residual: when solving $f(x) = 0$, using $|f(\tilde{x})|$ as an indicator of how good the approximation is. The residual is a deviation in the function value and does not even have the same units as the error in the solution $|\tilde{x} - x|$; relating the two requires, e.g., that $|f'(x)|$ is sufficiently large near the root. For ill-conditioned problems with small $|f'|$, a small residual does not imply a small error

Measurement results are often written as $\tilde{x} \pm \varepsilon$ to state the error bound: written as an error bound, "length $= 3.14 \pm 0.01\;\text{m}$" means that the absolute error is at most $0.01\;\text{m}$. This is the meaning of $\pm$ used in this article (in metrology, $\pm$ may instead denote a standard uncertainty or a confidence interval).

3. Comparison with Relative Error

Absolute error and relative error are complementary. Near zero the relative error is unstable or undefined, so one usually evaluates with the absolute error or a problem-specific tolerance (for a modified definition that covers $x = 0$, see Chapter 7, §5). When comparing values at different scales, use the relative error as well.

4. Error Propagation: Bounds and First-Order Approximation

Let $\tilde{x} = x + \Delta x$, $\tilde{y} = y + \Delta y$. We describe error propagation through arithmetic operations as a rigorous upper bound for addition and subtraction, and as a first-order approximation that neglects higher-order error terms for multiplication and division.

Addition and Subtraction

In addition/subtraction, the bounds on the absolute errors add. In subtraction where $x \approx y$, the bound on the absolute error is $|\Delta x| + |\Delta y|$ just as for addition, but because the result $x - y$ is small, the relative error tends to become large (catastrophic cancellation).

Multiplication

Indeed, $\tilde{x}\tilde{y} - xy = y\,\Delta x + x\,\Delta y + \Delta x\,\Delta y$. When $|\Delta x|, |\Delta y|$ are sufficiently small, the second-order term $|\Delta x||\Delta y|$ is neglected and one estimates $|\Delta(xy)| \lesssim |y||\Delta x| + |x||\Delta y|$ (first-order approximation). The right-hand side, with absolute values on every term, is a worst-case upper bound; if the signs of $\Delta x$ and $\Delta y$ cancel, the actual error is much smaller.

Division

This is a first-order approximation valid when $y \neq 0$ and $|\Delta y|$ is sufficiently small compared with $|y|$ (so that $\tilde{y}$ does not cross $0$). When $|y|$ is small, the error is greatly amplified.

General Functions

If $f$ is differentiable near $x$ and $|\Delta x|$ is sufficiently small, the first-order estimate is

(the absolute value of both sides of $\Delta f \approx f'(x)\,\Delta x$; unlike the multiplication and division formulas, this approximates the size of the error itself). If a rigorous bound is needed, use the maximum $M$ of $|f'|$ in the neighborhood: $|\Delta f| \le M |\Delta x|$ (mean value theorem). For a multivariate function $f(x_1, x_2, \ldots, x_n)$:

This is the first-order worst-case bound, in which every error is aligned in the least favorable direction. If the errors are independent random quantities, combine them in quadrature, $\sqrt{\sum_i (\partial f/\partial x_i)^2 \sigma_i^2}$ (this is what the reference "Propagation of uncertainty" treats).

5. Significant Digits and Absolute Error Bounds

An approximation $\tilde{x}$ has $n$ significant digits if the absolute error is at most half a unit in the $n$-th digit, i.e.,

and the number of significant digits is the largest non-negative integer $n$ satisfying this inequality. Here $m$ is the exponent of the leading digit of $\tilde{x}$, i.e., the integer with $10^m \le |\tilde{x}| < 10^{m+1}$. For example, $\tilde{x} = 3.1416$ has $m = 0$; if it has 5 significant digits, then $|x - \tilde{x}| \le 0.00005$ (with 4 digits the bound would be $0.0005$).

6. Worked Examples

7. FAQ

8. References

• Wikipedia "Approximation error"
• Wikipedia "Propagation of uncertainty"
• R. L. Burden, J. D. Faires & A. M. Burden, Numerical Analysis, 10th ed., Cengage, 2016.
• J. R. Taylor, An Introduction to Error Analysis, 2nd ed., University Science Books, 1997.