The language of measurement
Introduced| English | Meaning |
|---|---|
| accuracy/ˈækjʊrəsi/ | How close a measurement is to the true value. |
| precision/prɪˈsɪʒn/ | How closely repeated measurements agree. |
| units/ˈjuːnɪts/ | Agreed reference quantities used to express measurements or counts. |
| uncertainty/ʌnˈsɜːtənti/ | An estimate of the possible range of error in a measurement. |
| significant figures/sɪɡˈnɪfɪkənt ˈfɪɡəz/ | The meaningful digits in a measured or rounded number, beginning with its first non-zero digit. |
Close readings can share a fault
- A balance gives 2.001, 2.002 and 2.001 g for a supplied reliable 1.500 g reference mass. The readings are close to one another, but all are far above the reference.
- Repeating the unknown object alone cannot settle whether the balance has an offset. Check a suitable reference and the zero reading before trusting the result.
Distinguish accuracy from precision
- Accuracy 准确度 means closeness to a suitable reference or true value; precision 精密度 describes agreement between repeated readings under stated conditions. Precise results need not be accurate.
- These readings span only 0.001 g but exceed the reference by about 0.501 g. The pattern suggests investigating a systematic fault; it does not identify the cause by itself.
A balance reads 2.001, 2.002, 2.001 g for a mass known to be 1.500 g. What is it?
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The repeats are close but differ from the supplied reference. A systematic offset is one concern to investigate; three readings alone do not establish its cause or identical errors.
Convert units without changing the quantity
- Units 单位 state the scale of a physical quantity. A side of 10.0 cm is 0.100 m. For a square, $A=L^2=(0.100\ \text{m})^2=0.0100\ \text{m}^2$.
- Square the length conversion factor for area: $1\ \text{cm}^2=(0.01\ \text{m})^2=0.0001\ \text{m}^2$. Counts and ratios may be dimensionless but still need a clear definition.
A square side is 10.0 cm. What is its area in square metres?
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L = 0.100 m; A = L² = 0.0100 m². Square the length conversion factor.
State the supplied uncertainty honestly
- Uncertainty 不确定度 can include resolution, procedure and repeat variation. Significant figures 有效数字 communicate supported numerical precision. Extra calculator digits do not establish finer measurements, and half a scale division is not every measurement's full uncertainty.
- For a length supplied as 12.4 cm with uncertainty 0.05 cm, report $12.40\pm0.05\ \text{cm}$. Under the task's interval interpretation, the limits are 12.35 and 12.45 cm; no confidence level is supplied.
Match the value's decimal place to the stated uncertainty when reporting it. Do not invent an uncertainty, a confidence level or a justified exclusion to make data agree with a prediction.
You measure 12.4 cm with a millimetre ruler. Why not write 12.437 cm?
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Extra calculator digits do not establish extra measurement precision. The reporting convention depends on the measured values, procedure and stated uncertainty.
Report the supplied length 12.4 cm with uncertainty 0.05 cm, matching the value's decimal place to the uncertainty.
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Report 12.40 ± 0.05 cm: value, supplied uncertainty and unit. The task gives the uncertainty and no confidence level.
Using the task's interval interpretation, what is the lower limit of 12.40 ± 0.05 cm, in cm?
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Lower limit = 12.40 - 0.05 = 12.35 cm. No confidence level is supplied.
Choose and interpret a derived quantity
- Average speed over an interval is distance divided by elapsed time: $\bar v=d/t$. With supplied distance 3.60 m and time 48.0 s, $\bar v=3.60/48.0=0.0750\ \dfrac{\text{m}}{\text{s}}$.
- The three-significant-figure result reflects the supplied inputs. It does not tell us the speed at every moment; a car can stop and move faster during parts of the interval.
The supplied distance and elapsed time show that the object moved at exactly 0.0750 m/s at every moment.
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False: distance/time gives the average over the interval, not every instantaneous speed.
Check faults instead of averaging them away
- A consistent instrument offset can remain in the mean after many repeats. Compare with a suitable reference, document a suspected fault, and apply a correction only when the evidence justifies it.
- Keep the original readings and distinguish their spread from their difference from a reference. Record the units, procedure and uncertainty so the result can be interpreted.
Precision, accuracy and uncertainty answer different questions. Small repeat spread does not prove small total uncertainty or agreement with the reference.
Repeating a measurement many times removes a systematic error.
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Repeating can help examine random variation, but a consistent offset can remain in the mean. Check the instrument against a suitable reference.
Explain in one sentence how you would detect a systematic error in a balance.
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Example: "Weigh a standard 100 g calibration mass and see whether the balance reads 100 g."