Errors

1Summary 重點摘要

Measurement error, absolute error, relative error, percentage error.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Understand measurement errorNo measurement can be perfectly accurate: the scale of the instrument, the judgement of the reader and environmental factors all introduce error, so every measured value really stands for an interval. Error is not a mistake but an inherent limit of measuring. The DSE focuses on two kinds: absolute error (the size of the error) and relative or percentage error (the error compared with the true value); the latter compares measurements of very different sizes fairly, such as a pen and a bridge.
2. Step 2: Maximum absolute errorThe maximum absolute error equals half of the smallest scale interval of the instrument. For example, with a ruler marked every 1 cm, a reading of 10 cm really represents any value between 9.5 cm and 10.5 cm, and each end differs from 10 by 0.5 cm, which is the maximum absolute error. From this come the upper and lower limits of the measurement: upper limit = measured value + maximum error, lower limit = measured value − maximum error. A frequent mistake is forgetting to halve the interval and using the full interval as the error.
3. Step 3: Computing absolute errorWhen the true (or accepted) value is known, the absolute error is defined as the absolute difference between the measured value and the true value, that is |measured − true|, always non-negative. The absolute value ensures the size of the error is expressed the same way whether the reading is too large or too small. Find the difference first and take its size if it is negative; never report a raw difference without the absolute value, or an undershoot would produce a 'negative error', which contradicts the definition and loses marks.
4. Step 4: Relative errorThe relative error is the ratio of the absolute error to the true value (or to the measured value, as the question specifies): relative error = absolute error ÷ true value, usually kept as a positive fraction or decimal. Its meaning is to standardise the error: the same 0.5 cm gives a relative error of 0.05 on a 10 cm object but only 0.005 on a 100 cm object, so the larger measurement is clearly the more accurate. The units cancel in the division, so the relative error has no unit.
5. Step 5: Percentage errorThe percentage error is simply the relative error multiplied by 100%: percentage error = relative error × 100%, the form most often required in the DSE. For example, a relative error of 0.05 becomes a percentage error of 5%. Always write the % sign; omitting it counts as an incomplete answer. If rounding is required, follow the instruction exactly (for example, correct to one decimal place), and keeping fraction form in the middle steps is safer than decimals because it stops rounding errors from accumulating into the final answer.
6. Step 6: Comparing accuracyTo judge which of two measurements is more accurate, the standard is the smaller percentage (or relative) error, not the smaller absolute error. For instance, an error of 0.5 cm on 10 cm is 5%, while an error of 1 cm on 200 cm is only 0.5%; the second has the larger absolute error yet is far more accurate. Compute both percentage errors first, then write the comparison in the form 'since this measurement has the smaller percentage error, it is more accurate'; a well-supported statement is needed for full marks.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Any measurement has error
• Max error = half scale interval
• Absolute error = |measured - true|
• Relative error = absolute error / true
• Percentage error = relative error × 100%