Define error and its types

Error is the difference between the true (exact) value of a quantity and its approximate value obtained by calculation or measurement: Error = X − X′. In numerical computation, errors cannot be avoided, so we study their type and size.

Types of errors

1. Inherent (input) error: error already present in the data given to the problem, due to faulty measurement or an approximate model of the physical situation. Example: taking π = 3.14 or a length read as 5.2 cm.
2. Round-off error: caused by the computer or calculator keeping only a limited number of digits. A number such as 1/3 = 0.333... is cut or rounded to, say, 4 digits. Example: rounding 37.46235 to 37.46.
3. Truncation error: caused by replacing an infinite process by a finite one, such as cutting a series after a few terms. Example: sin x ≈ x − x³/3! drops the terms x⁵/5! and higher. It also occurs when a derivative is replaced by a finite difference.
4. Modelling error: from simplifying assumptions in the mathematical model (e.g. neglecting air resistance).
5. Blunder (human) error: mistakes such as wrong reading, wrong formula or program bug (e.g. 4.317 written as 4.371). It is avoidable.

Ways of expressing error

• Absolute error = |X − X′|
• Relative error = |X − X′| / |X|
• Percentage error = relative error × 100 %

Other related terms

• Gross error: another name for a blunder.
• Propagated error: error in the inputs carried into the result of later calculations.