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Absolute versus relative error

Short answer

Absolute error is the magnitude of difference between true and approximate value in the same unit. Relative error is that divided by true value, a dimensionless ratio. Example: 1000 m bridge measured 999 m, absolute 1 m, relative 0.001; 10 cm pin measured 9 cm, absolute 1 cm, relative 0.1, showing relative error reveals poor accuracy.

Discuss the difference between absolute error and relative error with appropriate examples.

If X is the true value and X′ is the approximate value, the difference between them is the error. It can be stated in absolute or relative terms.

  • Absolute error: Ea = |X − X′|. It gives the size of the error in the same unit as the quantity.
  • Relative error: Er = Ea / |X|. It compares the error with the size of the true value and is unit-free. Percentage error = Er × 100 %.
BasisAbsolute errorRelative error
Formula|X − X′||X − X′| / |X|
UnitSame unit as the quantityNo unit (ratio or %)
ShowsActual size of the mistakeHow serious the mistake is compared with the value
Use for comparisonPoor, depends on scaleGood, quantities of different size can be compared
ValueCan be large even for good accuracySmall when accuracy is good

Example

Bridge length: true 1000 m, measured 999 m. Pin length: true 10 cm, measured 9 cm.

  • Bridge: Ea = 1 m, Er = 1/1000 = 0.001 (0.1 %).
  • Pin: Ea = 1 cm, Er = 1/10 = 0.1 (10 %).

Both have the same absolute error (1 unit) but the pin measurement is far worse. The relative error shows this; the absolute error does not. Hence relative error is the better measure of accuracy.

✓ Answer checked by Claude Opus 5.5 (high effort)

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