Percentage error measures how far an estimate or measurement is from the true value, expressed as a fraction of that true value converted to a percentage. A small percentage error means the approximation was close; a large one signals a significant inaccuracy regardless of the units involved.
What is percentage error?
Percentage error compares the size of an error to the size of the quantity being measured. Expressing the error as a percentage — rather than as a raw difference — allows fair comparisons between quantities of very different magnitudes.
The formula:
Percentage error = (|approximate value − exact value| / exact value) × 100%
The vertical bars |…| mean you take the positive (absolute) value of the difference, so percentage error is always ≥ 0%.
How do you calculate percentage error step by step?
- Find the absolute difference: |approximate − exact|.
- Divide by the exact (true) value.
- Multiply by 100 to convert to a percentage.
Worked example 1: A student estimates a table is 130 cm long. The exact length is 125 cm.
- Absolute difference: |130 − 125| = 5 cm
- Divide by exact value: 5 ÷ 125 = 0.04
- Multiply by 100: 0.04 × 100 = 4%
The estimate was 4% above the true value.
Worked example 2: A student uses 3.14 as an approximation for π (exact: 3.14159…).
- Absolute difference: |3.14 − 3.14159| = 0.00159
- Divide by exact value: 0.00159 ÷ 3.14159 ≈ 0.000506
- Multiply by 100: ≈ 0.051%
Using 3.14 introduces only a 0.051% error — very small for most calculations.
Why use percentage error rather than just the raw difference?
The raw difference alone can be misleading:
| Situation | Raw difference | Percentage error |
|---|---|---|
| Estimate 2 kg parcel as 2.4 kg | 0.4 kg | 20% |
| Estimate 200 kg car as 200.4 kg | 0.4 kg | 0.2% |
Both situations involve a 0.4 kg error, but a 0.4 kg mistake on a 2 kg parcel is far more significant than the same error on a 200 kg car. Percentage error captures this distinction clearly.
How does percentage error appear in KS3 maths problems?
The three most common KS3 contexts are:
- Estimating measurements — comparing an estimated length, mass or capacity with the actual value found by measuring accurately.
- Approximating constants — e.g., using 22/7 as an approximation for π.
- Rounding before calculating — if you round 47 to 50 before calculating, the percentage error in the input is |50 − 47| ÷ 47 × 100 ≈ 6.4%.
Worked example — using 22/7 for π:
22/7 ≈ 3.14286 and π ≈ 3.14159.
Percentage error = |3.14286 − 3.14159| ÷ 3.14159 × 100 ≈ 0.00127 ÷ 3.14159 × 100 ≈ 0.040%
22/7 is an excellent approximation.
What are common mistakes when calculating percentage error?
Mistake 1 — Dividing by the approximate value instead of the exact value. The denominator in the formula is always the exact (true) value. Dividing by the approximate value gives a slightly different answer and loses marks.
Mistake 2 — Ignoring the absolute value signs. If the approximation is smaller than the exact value, subtraction gives a negative result. Take the positive (absolute) value before dividing — percentage error is never negative.
Mistake 3 — Forgetting to multiply by 100. The formula produces a decimal (e.g., 0.04); you must multiply by 100 to express it as a percentage (4%). Leaving the answer as 0.04 is incomplete.
Mistake 4 — Confusing percentage error with percentage change. Percentage change (used in finance) shows how a value has changed and can be positive or negative. Percentage error measures the magnitude of an approximation's inaccuracy and is always expressed as a positive percentage.
Frequently asked questions
Can percentage error be greater than 100%?
Yes — if the estimate is more than double the true value. For example, estimating 50 cm when the true length is 20 cm gives |50 − 20| ÷ 20 × 100 = 150%. A percentage error over 100% means the absolute error exceeds the true value itself.
Is percentage error the same as relative error?
Relative error is |approximate − exact| ÷ exact (a decimal). Percentage error is relative error multiplied by 100. They measure the same thing on different scales: a relative error of 0.04 equals a percentage error of 4%.
What is considered an acceptable percentage error in school experiments?
There is no universal rule, but in school science and maths investigations a percentage error below 5% is generally considered acceptable. High-precision engineering may require less than 0.1%. The acceptable level depends entirely on the context and the consequences of inaccuracy.
Why do we divide by the exact value and not the approximate value?
The exact value is the benchmark — it represents reality. Dividing by it answers the question "what fraction of the true quantity was the error?". Using the approximate value in the denominator would answer a different question (the error as a fraction of your estimate) and makes results harder to compare consistently.
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