Arithmetic
Measurement Errors
Is your height 175 cm or 175.000000 cm? The difference is enormous: the first is a measurement with ±0.5 cm precision, the second claims nanometer accuracy (which is absurd). Understanding errors is what separates an engineer from a calculator.
- **Engineering:** part tolerances, assembly of mechanisms
- **Science:** processing experimental data
- **Medicine:** diagnostic precision and dosage accuracy
Absolute Error
**Absolute error** is the difference between the approximate and the true value. It is measured in the same units as the quantity itself.
**Absolute error:** Δx = |x̃ - x| where x̃ is the approximate value, x is the true value. Example: π ≈ 3.14, so Δπ = |3.14 - 3.14159...| ≈ 0.00159
Absolute error alone tells us little. An error of 1 cm when measuring a room is excellent; for a microchip it is catastrophic.
A thermometer shows 36.8°C; the true temperature is 36.6°C. What is the absolute error?
Relative Error
**Relative error** is the ratio of the absolute error to the true value. It shows how "serious" the mistake is.
**When to use each:** • **Absolute** - for physical tolerances (±0.1 mm) • **Relative** - for comparing accuracy of different measurements "5% error" is more meaningful than "0.03 m error" (for what?).
Relative error is the universal measure of approximation quality. 1% is good for engineering, 0.01% for science, 0.0001% for metrology.
Measuring 100 m with an error of 50 cm vs. 10 m with an error of 10 cm. Which is more precise?
Error Accumulation
During calculations, errors accumulate. The accumulation rules depend on the operation: addition uses one formula, multiplication another.
**Why subtraction is dangerous:** When subtracting close numbers, absolute errors add while the result is small: a = 10.5 ± 0.1, b = 10.3 ± 0.1 a - b = 0.2 ± 0.2 Relative error: 0.2/0.2 = 100%!
Understanding error accumulation is critical for scientific calculations. Sometimes a problem must be reformulated to avoid dangerous operations.
a = 4 m ± 2%, b = 2 m ± 3%. What is the relative error of a × b?
Estimating Result Error
For complex formulas, errors are estimated using partial derivatives. But simple rules and common sense are often sufficient.
**Rule for writing results:** Round the error up to 1 - 2 significant figures. Round the result to the same decimal place. Wrong: (78.539816... ± 3.1415...) Right: (78.5 ± 3.2) or (79 ± 3)
Proper error analysis is a mark of scientific literacy. A result without an error estimate is only half an answer.
More decimal places = more precise result
Precision is determined by the error, not by the number of digits
Writing (5.123456 ± 0.5) is nonsense. If the error is 0.5, all digits after 5.1 are noise. Correct: (5.1 ± 0.5). A calculator spits out many digits, but that doesn't mean they're meaningful. Always estimate the error and round accordingly.
How should a measurement result be properly written?
Key Ideas
- Δx - absolute error (in measurement units)
- δ = Δx/x - relative error (dimensionless)
- For +/- absolute errors Δ add
- For ×/÷ relative errors δ add
Related Topics
Errors are connected to measurement and computation:
- Approximations — Source of errors
- Significant Figures — Indicator of precision
- Scientific Notation — Explicitly indicating precision
Вопросы для размышления
- Why is relative error more important than absolute error for comparing measurements?
- How can you minimize error accumulation in complex calculations?
- Why is a result without an error estimate an incomplete answer?