How this instrument works
Absolute error — the raw difference between what you measured and what's actually true — tells you almost nothing on its own. Being off by 2 grams is a disaster if you're weighing a 5-gram gemstone and irrelevant if you're weighing a 50-kilogram person. Relative error fixes that by dividing the absolute error by the true value's own size, turning every measurement's accuracy into a percentage that means the same thing regardless of scale or units.
The formula is deliberately simple: take the absolute value of measured minus true, divide by the absolute value of true, and multiply by 100. The absolute values matter because relative error is usually reported as a magnitude of how far off you were, not a signed direction — a measurement that's 2% too high and one that's 2% too low are both '2% relative error,' even though a signed version of the same calculation (sometimes called percent error) would report -2% and +2% respectively.
This calculator is a general-purpose measurement-accuracy tool, not tied to any one field — it applies identically to a chemistry student's titration result, an engineer's tolerance check, a weather forecast's temperature miss, or a manufacturing line's part dimension. Whenever you have both a measured (or approximate) value and a known, accepted, or theoretical value to compare it against, relative error is the standard way to express how close the two are.
- Enter the value you actually got into Measured (approximate) value.
- Enter the known or accepted correct value into True (accepted) value — this cannot be zero, since it's the denominator.
- Read Relative error (%) — this is |measured − true| ÷ |true| × 100, always reported as a non-negative percentage.
- Compare the result against whatever tolerance matters for your context — a 0.5% relative error might be excellent for a hand measurement and unacceptable for a precision instrument.
Worked example — a measurement of 9.8 against a true value of 10
Enter 9.8 into Measured (approximate) value and 10 into True (accepted) value. The instrument computes |9.8 − 10| = 0.2, then divides by |10| to get 0.02, then multiplies by 100.
Relative error (%) reads 2.0000 exactly. So the measurement of 9.8 misses the true value of 10 by exactly 2% — a way of saying 'off by 0.2' that stays meaningful no matter what units or scale the 10 represents, whether it's 10 centimeters, 10 volts, or 10 million dollars.
Questions
What's the difference between relative error and absolute error?
Absolute error is just |measured − true| — a raw number in the same units as your data, with no sense of scale. Relative error divides that absolute error by the true value and expresses it as a percentage, so a 0.2-unit miss on a true value of 10 (2% relative error) and a 0.2-unit miss on a true value of 10,000 (0.002% relative error) are correctly distinguished as very different levels of accuracy, even though the absolute error is identical in both cases.
Why is relative error always positive here?
Because this calculator takes the absolute value of the difference before dividing, so the result reports the magnitude of the miss regardless of whether the measurement was too high or too low. If you need to know the direction as well as the size — for example, to see whether an instrument consistently reads high — you'd want signed percent error, which drops the absolute-value bars and can come out negative.
Can relative error be calculated when the true value is zero?
No — dividing by zero is undefined, so relative error has no meaning when the true (accepted) value is exactly zero. In that specific case you'd fall back to reporting absolute error alone, since there's no nonzero reference size to express the error as a fraction of.
What counts as a 'good' relative error?
It depends entirely on the field and the instrument. A cheap kitchen scale might be considered accurate at 1-2% relative error, while precision laboratory instruments, engineering tolerances, or scientific measurements often demand relative errors under 0.1% or even 0.01%. There's no universal cutoff — always compare your result against the tolerance appropriate to your specific measurement context.
How is relative error different from a percentage change?
The arithmetic looks similar, but the intent differs: percentage change compares two values over time or between conditions (like a price rising from $10 to $12), while relative error specifically compares a measured or approximate value against a known-correct reference, to judge measurement accuracy rather than describe a change.