Repeated measurements: mean, spread and estimated uncertainty
| English | 中文 | Pinyin |
|---|---|---|
| uncertainty/ʌnˈsɜːtənti/ | 不确定度 | bù què dìng dù |
| range/reɪndʒ/ | 极差 | jí chà |
What would explain this observation?
- Repeated mass measurements can differ slightly even when the same method is used. Reporting only the mean hides evidence about that spread.
- Start with a prediction. State the quantities or features you would compare, then decide what evidence could distinguish two explanations.
Build the model
- Every measurement has some uncertainty 不确定度. Repeat readings help show the distribution of results. Calculate the mean by adding the readings and dividing by their number; calculate the range 极差 as highest minus lowest. A common school estimate is half the range, reported as mean plus or minus that estimate when the distribution supports it. State that convention rather than claiming it is a universal statistical confidence interval.
- uncertainty: An estimate of the doubt associated with a measurement result; range: The largest result minus the smallest result in a set.
Which calculation gives the full range?
For 2.40, 2.42 and 2.44 g, the mean is 2.42 g and the full range is 0.04 g. Half-range is 0.02 g, so the stated estimate is 2.42±0.02 g. Here the mean lies midway between the extremes. For an asymmetric distribution the extreme deviations from the mean differ; display the actual results as well as the chosen uncertainty estimate.
Match each technical term to its precise meaning.
Use the definitions to distinguish related quantities and processes.
Choose evidence that can test it
- For 2.40, 2.42 and 2.44 g, the mean is 2.42 g and the full range is 0.04 g. Half-range is 0.02 g, so the stated estimate is 2.42±0.02 g. Here the mean lies midway between the extremes. For an asymmetric distribution the extreme deviations from the mean differ; display the actual results as well as the chosen uncertainty estimate.
- Plot a dot for each repeat on a labelled mass axis; coincident results can stack vertically. Keep units and sensible decimal places. Investigate an unusual value before excluding it, record any exclusion reason and preserve the original readings. More repeats reveal random spread but cannot by themselves remove a systematic bias, such as an uncorrected balance zero.
Which two habits make the investigation or model in this case more defensible?
Plot a dot for each repeat on a labelled mass axis; coincident results can stack vertically. Keep units and sensible decimal places. Investigate an unusual value before excluding it, record any exclusion reason and preserve the original readings. More repeats reveal random spread but cannot by themselves remove a systematic bias, such as an uncorrected balance zero.
Work from known quantities
- State the known values and their units. Choose the relation because its assumptions fit this case, then rearrange before substitution.
- Known: 5.1, 5.2, 5.2 and 5.3 cm give mean 5.2 cm and range 0.2 cm. Under the stated half-range convention, estimated uncertainty is ±0.1 cm. This says something about the observed repeat spread. It does not prove the true length is inside that interval or that the ruler’s resolution equals 0.2 cm.
Readings are 8.2, 8.4 and 8.6 g. Using half-range, estimate the uncertainty magnitude. Use the same sequence: known quantities → model → relation → substitution → unit and interpretation.
Readings are 8.2, 8.4 and 8.6 g. Using half-range, estimate the uncertainty magnitude.
The result is 0.2 g. Known: 5.1, 5.2, 5.2 and 5.3 cm give mean 5.2 cm and range 0.2 cm. Under the stated half-range convention, estimated uncertainty is ±0.1 cm. This says something about the observed repeat spread. It does not prove the true length is inside that interval or that the ruler’s resolution equals 0.2 cm.
Check the conclusion and its limits
- Uncertainty is not the same as a mistake. Zero range from a coarse instrument does not establish perfect measurement. Do not halve an instrument resolution and label it repeat range. Compare like quantities and express uncertainty in the same unit as the measured value.
- Return to the original observation. Explain what the result supports, which conditions it assumes, and one way to test a competing explanation.
Repeating a measurement always removes a systematic zero error. This claim is false: Uncertainty is not the same as a mistake. Zero range from a coarse instrument does not establish perfect measurement. Do not halve an instrument resolution and label it repeat range. Compare like quantities and express uncertainty in the same unit as the measured value.
Repeated measurements: mean, spread and estimated uncertainty: For 2.40, 2.42 and 2.44 g, the mean is 2.42 g and the full range is 0.04 g. Half-range is 0.02 g, so the stated estimate is 2.42±0.02 g. Here the mean lies midway between the extremes. For an asymmetric distribution the extreme deviations from the mean differ; display the actual results as well as the chosen uncertainty estimate.
Repeating a measurement always removes a systematic zero error.
Uncertainty is not the same as a mistake. Zero range from a coarse instrument does not establish perfect measurement. Do not halve an instrument resolution and label it repeat range. Compare like quantities and express uncertainty in the same unit as the measured value.
An estimate of the doubt associated with a measurement result: write the technical term.
uncertainty means An estimate of the doubt associated with a measurement result.