Short Answer
Percent error is one of the first quantitative judgments a chemistry student makes about a measurement. It answers: how far is an experimental result from an accepted or theoretical value, expressed as a percentage? The formula is simple, but the grading conventions are not. AP, IB, and high-school chemistry courses differ in how they treat signs, significant figures, and rounding. This reference explains the rule, shows worked examples, compares conventions, and grounds the practice in ASTM, ISO, NIST, and GUM standards. It is part of our precision-and-rounding reference; the significant figures calculator is available for checking results, but the emphasis here is on the reasoning.
Rule Statement
The standard percent error formula in chemistry is:
Percent error = |experimental value − accepted value| ÷ |accepted value| × 100%
Some textbooks write theoretical value or true value in place of accepted value. The denominator must be the reference value, not the experimental value. The absolute-value bars are conventional because percent error is usually reported as a magnitude. If a signed deviation is requested, use (experimental − accepted) ÷ accepted × 100% and label it as signed percent error or percent deviation.
Three precision rules follow immediately:
- The factor 100 is exact and does not limit significant figures.
- Subtraction can reduce significant figures through cancellation; inspect the difference before dividing.
- Round only the final percentage. Carry extra digits through intermediate steps to avoid double-rounding error.
In AP, IB, and high-school grading, the final value is often reported to 2 or 3 significant figures, or to one decimal place when the percentage is large. No single official AP or IB rule mandates a universal decimal place for percent error; the dominant requirement is consistency with measured data and uncertainty. For related rules, see Significant Figures, Rounding Rules, and Error Propagation.
Worked Examples
Example 1: Density with three significant figures
An experiment gives a density of 2.68 g/mL. The accepted density is 2.70 g/mL.
- Difference: |2.68 − 2.70| = 0.02 g/mL.
- Divide by accepted: 0.02 ÷ 2.70 = 0.007407407…
- Multiply by 100: 0.7407407…%.
- Round once: 0.741% if both inputs justify three significant figures; 0.74% if the course convention is two significant figures.
The subtraction 2.68 − 2.70 leaves 0.02, which has one significant figure in strict cancellation logic, but because the values are close and the absolute difference is meaningful in context, many chemistry rubrics retain two or three figures. This is a classic AP/IB grading gray zone: method and consistent sig figs matter more than one extra digit.
Example 2: Titration with cancellation
Experimental amount = 0.00482 mol; accepted amount = 0.00500 mol.
Difference = 0.00500 − 0.00482 = 0.00018 mol. The difference has two significant figures. Divide by 0.00500: 0.00018 ÷ 0.00500 = 0.036. Multiply by 100: 3.6%. Reporting 3.60% or 3.600% overstates precision.
Example 3: Accepted value of zero
If the accepted value is 0, percent error is undefined because division by zero is invalid. Use absolute error (experimental − accepted) or report deviation relative to a detection limit or tolerance range. See Absolute vs Relative Error.
Convention Comparison Table
| Context | Denominator | Sign | Typical sig-fig/rounding practice | Grading note |
|---|---|---|---|---|
| AP Chemistry (lab and FRQ) | Accepted or theoretical value | Absolute value unless signed deviation requested | Correct significant figures; often 2–3 sf or one decimal for percent error | AP expects appropriate sig figs on exam; minor rounding differences usually tolerated if setup is correct. |
| IB Chemistry (IA and exams) | Accepted, theoretical, or literature value | Absolute value; uncertainty may be propagated separately | Final answer consistent with least precise measurement; percent error often 2 sf or one decimal | IA data-analysis criteria reward consistent precision and uncertainty treatment, not a single mandated rounding rule. |
| High school chemistry | Accepted value (sometimes theoretical) | Mostly absolute; some teachers require sign to show direction | Varies: 2 decimal places, 1 decimal place, or 2–3 sf | Follow the teacher’s rubric; the formula is standard, rounding conventions are local. |
| Metrology/industry | Reference value, certified value, or specification | Signed error or bias; magnitude for percent error | Round once; use GUM for uncertainty; ASTM/ISO rounding for conformance | Separate error from uncertainty; do not imply more confidence than the measurement supports. |
Standards Citation
Percent error is a classroom calculation, but its rounding and reporting should follow broader metrological standards when precision matters:
- ASTM E29-13(2019), §6.3 — Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications. It specifies tie-breaking rules, including rounding to the nearest even digit when the discarded portion is exactly one half. This is the basis for many lab-report rounding policies.
- ISO 80000-1:2009, §7.3.3 — Quantities and units — Part 1: General. It recommends one-step rounding to the intended number of significant digits, avoiding cumulative rounding error.
- JCGM 100:2008 (GUM), §7.2.6 — Guide to the Expression of Uncertainty in Measurement. It recommends reporting uncertainty to two significant digits in most cases; percent error should not be confused with uncertainty.
- NIST SP 811, §7.4 — Guide for the Use of the International System of Units (SI). It gives rounding guidance for SI values and unit presentation.
For grading, AP and IB documents govern course expectations; ASTM, ISO, NIST, and GUM govern measurement reporting. When they conflict, the rubric wins for the assignment, but the standards explain the defensible precision.
Common Mistakes
- Using the experimental value in the denominator. Percent error compares to the accepted value. Percent difference compares two measured values and uses their average.
- Forgetting absolute value. A negative result can be useful as signed deviation, but unsigned percent error is normally positive.
- Rounding intermediate differences. If 0.00018 becomes 0.0002 too early, the final percentage shifts.
- Treating 100 as a measured quantity. The 100 in × 100% is exact and never limits significant figures.
- Over-reporting precision. Reporting 0.7407407% from 2.68 and 2.70 g/mL implies precision the data do not support.
- Ignoring subtraction cancellation. The difference between two close values may have fewer significant figures than either input.
- Using percent error when accepted value is zero. The result is undefined; use absolute error instead.
- Confusing percent error with percent yield or percent difference. Percent yield uses theoretical yield in the denominator; percent difference uses the average of two measured values.
Practice Problems
- Measured acceleration = 9.81 m/s²; accepted = 9.80 m/s². Find percent error.
Answer: |9.81 − 9.80| ÷ 9.80 × 100 = 0.10204…% → 0.102% (3 sf) or 0.10% (2 sf). - Measured mass = 12.4 g; accepted = 12.0 g. Find percent error under strict sig-fig rules and under common lab rounding.
Answer: Difference = 0.4 g (1 sf). 0.4 ÷ 12.0 × 100 = 3.333…%. Strict sig figs: 3% (1 sf). Common lab convention: 3.3% (2 sf). This discrepancy is why rubrics often allow a range. - Accepted concentration = 0.00 M; measured = 0.02 M. What is percent error?
Answer: Undefined. Report absolute error = 0.02 M or state that percent error is not applicable.
Quick Reference Table
| Item | Rule or convention |
|---|---|
| Formula | |experimental − accepted| ÷ |accepted| × 100% |
| Denominator | Accepted, theoretical, or literature value; never experimental |
| Sign | Absolute value by default; signed deviation only if requested |
| Exact factor | 100 is exact; does not affect significant figures |
| Rounding | Round once at the final step; ASTM E29 §6.3 for tie-breaking if required |
| AP Chemistry | Correct sig figs; percent error often 2–3 sf or one decimal |
| IB Chemistry | Least precise measurement governs; IA values consistency and uncertainty |
| High school | Follow local rubric; 1–2 decimal places or 2–3 sf are common |
| Accepted value = 0 | Percent error undefined; use absolute error |
Related Rules
- Significant Figures — core counting and reporting rules.
- Rounding Rules — half-up, half-even, and one-step rounding.
- Addition and Subtraction — decimal-place logic behind cancellation in percent error.
- Multiplication and Division — significant-figure limits for the division step.
- Accuracy vs Precision — why percent error measures accuracy, not precision.
- Error Propagation — uncertainty treatment beyond simple percent error.
- Percent Error — broader examples and calculator.
Sources & Further Reading
- ASTM E29-13(2019), Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications, ASTM International, §6.3.
- ISO 80000-1:2009, Quantities and units — Part 1: General, Clause 7.3.3.
- JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM), §7.2.6.
- NIST Special Publication 811, Guide for the Use of the International System of Units (SI), §7.4.
- College Board, AP Chemistry Course and Exam Description; International Baccalaureate Organization, Chemistry Guide (current assessment cycle).
FAQ
Does percent error have a sign?
In most chemistry grading, no. The absolute-value formula reports magnitude. A signed percent error can show whether the experimental value is high or low, but it should be labeled clearly.
How many significant figures should percent error have?
Use the least precise input after accounting for subtraction. In practice, AP and IB lab work often reports 2–3 significant figures or one decimal place. The factor 100 is exact.
What if the accepted value is zero?
Percent error is undefined because division by zero is impossible. Use absolute error or relative error against a non-zero tolerance or detection limit.
Is percent error the same as percent difference?
No. Percent error compares an experimental value to an accepted value. Percent difference compares two measured values, often using their average in the denominator.
Do AP and IB require a specific rounding rule?
They require appropriate significant figures and consistent precision. Neither publishes a single universal decimal-place rule for percent error. ASTM E29 and ISO 80000 provide defensible rounding methods when a precise standard is needed.
Leave a Reply