Formula
Percent error = (|Experimental − Theoretical| ÷ |Theoretical|) × 100. The theoretical value is always the denominator. The absolute value ensures the result is non-negative.
How to use this calculator
Enter the value you measured or calculated (experimental) and the known accepted value (theoretical). The calculator returns the percent error, signed error, absolute difference, and a three-step breakdown of the calculation.
The experimental value is what you observed or measured. The theoretical value (also called the accepted or true value) is the known standard — from a textbook, datasheet, or established scientific constant.
Percent error formula
The standard percent error formula uses absolute values to ensure a non-negative result:
$$\% \text{ Error} = \frac{|\text{Experimental} - \text{Theoretical}|}{|\text{Theoretical}|} \times 100$$
The absolute value of the theoretical value in the denominator accounts for cases where the accepted value itself is negative (e.g. temperature in Celsius below zero). The result is always between 0% and ∞%.
Signed vs absolute percent error
The standard formula discards the sign of the error. A signed version retains it, indicating the direction of deviation:
$$\% \text{ Error (signed)} = \frac{\text{Experimental} - \text{Theoretical}}{|\text{Theoretical}|} \times 100$$
| Version | Formula | Result sign | Use when |
|---|---|---|---|
| Absolute (standard) | |exp − theo| ÷ |theo| × 100 | Always ≥ 0 | Reporting magnitude of error only |
| Signed | (exp − theo) ÷ |theo| × 100 | Positive or negative | Direction matters (over- or under-estimate) |
A positive signed error means the experimental value was higher than theoretical (overestimate). A negative signed error means it was lower (underestimate). This distinction matters in calibration, dosing, and engineering tolerance analysis.
Worked examples
Example 1: chemistry lab — density measurement
A student measures the density of water as 9.6 g/mL. The accepted value is 10 g/mL.
$$\frac{|9.6 - 10|}{|10|} \times 100 = \frac{0.4}{10} \times 100 = 4\%$$
The measurement has a 4% error. The signed error is −4% (underestimate — the measured value was below the accepted value).
Example 2: physics — gravitational acceleration
An experiment measures g = 9.81 m/s². The accepted value is 9.8 m/s².
$$\frac{|9.81 - 9.8|}{|9.8|} \times 100 = \frac{0.01}{9.8} \times 100 \approx 0.102\%$$
The measurement has a ≈0.102% error — a very precise result. The signed error is +0.102% (slight overestimate).
Example 3: temperature measurement
A thermometer reads 23.5°C. The calibrated reference is 25°C.
$$\frac{|23.5 - 25|}{|25|} \times 100 = \frac{1.5}{25} \times 100 = 6\%$$
The thermometer has a 6% error. The signed error is −6% (reads low).
| Experimental | Theoretical | Absolute difference | Percent error |
|---|---|---|---|
| 9.6 | 10 | 0.4 | 4% |
| 9.81 | 9.8 | 0.01 | 0.102% |
| 48 | 50 | 2 | 4% |
| 103 | 100 | 3 | 3% |
| 4.85 | 5 | 0.15 | 3% |
| 11.7 | 12 | 0.3 | 2.5% |
What is an acceptable percent error?
There is no universal threshold — acceptability is defined by the field, instrument precision, and purpose of the measurement:
| Context | Typical acceptable range |
|---|---|
| General chemistry lab (undergraduate) | < 5% |
| Physics experiments (standard instruments) | < 1–2% |
| Analytical chemistry | < 0.5% |
| Industrial quality control | < 0.1–1% (depends on tolerance spec) |
| Medical diagnostic instruments | < 5–10% (device-class dependent) |
| Financial modelling | Varies widely; often < 2–5% |
A large percent error does not automatically mean the experiment failed — it may indicate a systematic error that can be corrected, instrument limitations, or an opportunity to refine the procedure.
Percent error vs percent difference
These two metrics are frequently confused. The distinction lies in whether one value is a known reference:
- Percent error — one value is the accepted truth (theoretical). The denominator is the theoretical value. Use when comparing a measurement to a standard.
- Percent difference — both values are measurements of equal standing. The denominator is their average. Use when neither value is a known standard.
Example: measuring the boiling point of water twice in the same experiment gives two measurements of equal status → percent difference. Comparing your measurement to the textbook value of 100°C → percent error, because 100°C is the accepted standard.
Common mistakes
Using the experimental value as the denominator
The denominator is always the theoretical (accepted) value, not the experimental. Using the experimental value inverts the reference frame and produces a different, incorrect result. Example: |9.6 − 10| ÷ 9.6 = 4.17%, not 4%.
Forgetting the absolute value
Without the absolute value, a result where the experimental is below the theoretical produces a negative percent error. The standard formula uses absolute values so the error magnitude is always non-negative. Only report a negative value when explicitly using the signed version.
Confusing percent error with percent change
Percent change measures how a value has changed over time — from an old value to a new value. Percent error measures how close a measured value is to a known standard. The formulas share the same structure but serve completely different purposes.
Frequently asked questions
What is the percent error formula?
Percent error = (|Experimental − Theoretical| ÷ |Theoretical|) × 100. The theoretical value is always the denominator.
What is the difference between percent error and percent difference?
Percent error compares a measurement to a known standard (theoretical value as denominator). Percent difference compares two independent measurements of equal standing (average as denominator).
What is an acceptable percent error?
It depends on the context. General chemistry labs accept under 5%; physics experiments typically require under 1–2%; analytical chemistry under 0.5%. There is no universal standard.
Can percent error be negative?
The standard absolute version is always non-negative. The signed version can be negative: negative means the experimental value was below the theoretical; positive means it was above.
What does a percent error of 0% mean?
The experimental value exactly equals the theoretical — a perfect measurement. In practice this is rare due to instrument limits and measurement uncertainty.
Why is the theoretical value in the denominator?
The theoretical value is the accepted reference. Dividing by it normalises the error relative to the expected magnitude, making errors comparable across experiments at different scales.