TL;DR: Saying an answer is “close” needs a metric. Absolute error is |approx − true|; relative error scales that gap by the true value. Use both when checking homework or lab numbers in an online mathematical calculator, and state which tolerance your rubric allows.
Two distances, two stories
If the true value is 1000 and your approximation is 1001, absolute error is 1. Relative error is 1/1000 = 0.001 (0.1%). If the true value is 2 and your approximation is 3, absolute error is also 1—but relative error is 50%. Same absolute gap, wildly different quality.
Instructors who mark “within 0.01” usually mean absolute error unless they say percent. Engineers quoting percent tolerance mean relative (sometimes relative to full scale—read the spec).
Computing them without mixing percent steps
- Absolute error E_abs = |x_approx − x_true|
- Relative error E_rel = E_abs / |x_true| when x_true ≠ 0
- Percent relative error = E_rel × 100%
Do not divide by x_approx unless a method explicitly uses that convention. Do not multiply by 100 in the middle of a multi-step symbolic derivation and forget you already switched to percent.
Zero and near-zero truths
Relative error blows up as x_true → 0. If the true value is 0, relative error is undefined; report absolute error only. For tiny targets (10^−8), a “small” absolute error might still be a huge relative miss—decide from the problem context which gate matters.
Calculator display versus true error
A display showing 0.3333333 for 1/3 already carries truncation. Comparing that display to a ten-digit truth without acknowledging display width confuses tool limits with student mistakes. When checking, either increase precision mode or compare symbolic fractions before decimal conversion.
Rounding twice—once in the tool, once when copying to paper—inflates absolute error. Copy full intermediates, round once at the end per instructions.
Propagation: why early slips look late
A 1% relative error in an input can become much larger after division by a small denominator or after exponentiation. When your final check fails relative tolerance, test intermediate absolute/relative errors to find the step that amplified noise. Blindly recomputing only the last line wastes time.
Choosing tolerances for self-checks
- Money to cents — absolute error ≤ 0.005 before standard rounding rules
- Percent problems — match the rubric’s decimal places; state E_abs
- Science labs — often percent relative error against accepted values
- Identity checks (sin²+cos²) — absolute error against 1.0 with a small epsilon
Write the tolerance beside the problem number so “close enough” is auditable.
On group submissions, that written tolerance also settles arguments between collaborators who each assumed a different metric. One line at the top of the shared document prevents two people from checking the same numbers against incompatible standards.
Worked contrast you can reuse
True density 19.3, measured 18.9. E_abs = 0.4. E_rel ≈ 0.0207 ≈ 2.07%. A rubric demanding ≤1% relative fails; one demanding absolute ≤0.5 passes. Same measurement, different verdicts—label the metric on your lab sheet.
Sign handling and the reference value trap
Absolute error is non-negative by construction, so a negative result from your subtraction step is a signal that something is off, not a signed error term. Either you swapped the operands or you are computing a signed difference, which is a different quantity and should be labeled as such.
Relative error needs the magnitude of the true value in the denominator. Writing the denominator as x_true without the absolute value produces a negative relative error whenever the true value is negative, which then flips the comparison against a tolerance. Take the magnitude explicitly and the metric behaves consistently for temperatures, coordinates, and balances below zero.
Watch for the related mistake of dividing by the measurement instead of the reference. Those two denominators differ by exactly the relative error, so a check that passes at 2% one way can fail at 2.04% the other. Small, but enough to flip a borderline lab report.
Reading a tolerance table correctly
A specification written as “0.5% of reading” is relative to the measured value. “0.5% of full scale” is relative to the instrument’s range, which for a small reading can be far more permissive. “±0.5 units” is absolute. Three different gates that all look like a half percent at a glance.
When an instructor or spec sheet gives only one number, quote the metric you used in your answer. A grader cannot fairly mark a number whose definition was never stated, and future you will not remember which convention applied to which problem.
Reporting both numbers in one sentence
A compact format saves readers work: state the absolute difference, then the relative figure in brackets, then the tolerance. One line communicates the measurement quality without a paragraph of explanation.
Reuse that format across a problem set and the comparison between questions becomes immediate, which is exactly the kind of clarity that turns a correct answer into a convincing one.
Running error metrics on www.mathematicalcalculator.com
Compute the subtraction for absolute error first, divide for relative error second, then multiply by 100 only if you need percent. Keep those as three labeled lines. A quick pass on www.mathematicalcalculator.com prevents the classic mistake of treating a 0.02 absolute gap as “2%” when the true value was never 1.
