ACARA v9 CONTENT DESCRIPTION “identify the impact of measurement errors on the accuracy of results in practical contexts”
Builds on: Absolute, Relative and Percentage Error. Working with measured lengths, areas and volumes raises a question those calculations quietly assume away: how exact are the measurements themselves? This unit examines measurement error, precision, and the accuracy of results calculated from measured values.
Every measurement has error
A hidden truth runs under all of measurement: no physical measurement is ever exact. A ruler marked in millimetres, a kitchen scale, a stopwatch, each can only be read to the fineness of its scale, and the true value always lies somewhere within a small range around the reading. Recording a length as 5 centimetres to the nearest centimetre does not claim the length is exactly 5; it claims the true length is between 4.5 and 5.5, within half of the smallest division. This unavoidable uncertainty is not carelessness or a fault in the instrument; it is the nature of measuring the real world. Understanding it, rather than pretending numbers are perfect, is what separates honest measurement from false confidence.
No measurement is exact
A measurement is only as precise as the instrument; the true value lies within a range around the reading.
A ruler reads 5 cm, but no measurement is exact. Reveal what reporting 5 cm really tells you about the true length.
Absolute and relative error
To talk about error precisely, two ideas are needed. The absolute error is simply the size of the possible error in the units of the measurement, such as plus or minus 1 millimetre. However, the absolute error alone does not tell you how good a measurement is, because the same error means very different things at different scales. An error of 1 unit in a measurement of 10 is a relative error of 10 percent, fairly poor, while the same 1 unit in a measurement of 1000 is only 0.1 percent, very precise. The relative error, the absolute error expressed as a fraction or percentage of the measurement, is therefore the better guide to quality. When judging or comparing measurements, it is the relative error that matters most.
Absolute versus relative error
Absolute error is the size of the error; relative error compares it to the measurement, showing true precision.
Measuring 10 with an error of 1 means a relative error of 10 percent, quite poor. The absolute error is 1 unit either way. Switch to a larger measurement to see how the same absolute error becomes far less significant.
Significant figures
The way a number is written can itself communicate how precisely it was measured, through its significant figures, the digits that carry real information. Writing 5 centimetres suggests measurement to the nearest centimetre, while 5.0 claims precision to the millimetre and 5.00 to a tenth of that. The trailing zeros are not idle: they assert how fine the measurement was. This carries a discipline. You should record only as many significant figures as your instrument justifies, and when you calculate with measured values, you should not quote an answer to more figures than the measurements support. Writing an area as 38.4713 square centimetres from rough measurements claims a precision that simply was not there.
Significant figures show precision
The digits recorded, including trailing zeros, signal how precisely a quantity was measured.
The number of significant figures shows how precisely something was measured. Writing 5 cm claims precision to the centimetre; 5.0 cm claims the millimetre; 5.00 cm claims even finer. The trailing zeros are not decoration: they state how good the instrument was, so you should record only the digits your measurement truly justifies.
Error in calculated results
Measurement error does not stay put; it grows when measurements are combined in a calculation. Consider a rectangle whose sides are measured as 10 and 4, each to within half a unit. The nominal area is 40, but the smallest the area could be is 9.5 times 3.5, which is 33.25, and the largest is 10.5 times 4.5, which is 47.25. So an area reported as 40 might truly be anywhere from about 33 to 47, a much wider uncertainty than half a unit. This is why a result calculated from measurements is generally less precise than the measurements themselves, and why it is misleading to write such a result to many decimal places. Tracking how error propagates keeps calculated answers honest.
Error grows when measurements combine
Calculating with measured values accumulates their uncertainties, so results are less precise than inputs.
A rectangle has sides 10 and 4, each measured to within half a unit. The nominal area is 40, but reveal how far that area can actually drift once the errors combine.
Tolerance in the real world
Because exact measurement is impossible, the practical world works with tolerance: a stated band around a target value within which a result is acceptable. An engineering part specified as 50 plus or minus 2 millimetres is fine anywhere from 48 to 52 millimetres, and is rejected only outside that range. Tolerance turns the unavoidable reality of measurement error into a clear, usable rule for manufacturing, construction and quality control, where parts must fit together despite never being made to a perfect size. It also reflects good sense: demanding more precision than a task needs is wasteful, while too loose a tolerance lets in parts that do not work. Setting and checking tolerances is measurement error put to practical use.
Tolerance: an acceptable band
Tolerance is the allowed range around a target value within which a measurement is acceptable.
In the real world a measurement is acceptable if it falls within a tolerance, a stated band around a target. A part specified as 50 plus or minus 2 millimetres passes anywhere from 48 to 52. Move the measured value: inside the green band it passes, outside it is rejected. Tolerance turns the unavoidable fact of measurement error into a practical pass-or-fail rule.
Quick self-check
1. A length is recorded as 5 cm, measured to the nearest centimetre. The true length is:
2. An error of 1 unit is most significant when the measurement is:
3. Writing a measurement as 5.00 cm rather than 5 cm indicates:
4. When two measured lengths are multiplied to find an area, the uncertainty in the result:
5. A part is specified as 50 ± 2 mm. A measured value of 53 mm is: