📚 Precision in A-Level Physics: Measurements, Uncertainty and Error Analysis | 精度：测量、不确定度与误差分析
In AQA A-level Physics, precision is a core practical skill. Every measurement you take carries an uncertainty, and exam questions regularly ask you to distinguish precision from accuracy, combine uncertainties and comment on experimental quality. This revision guide explains the key ideas with AQA-style examples.
在 AQA A-level 物理中，精度是一项核心实验技能。你进行的每次测量都带有不确定度，考试题经常要求你区分精度与准确度、组合不确定度并评价实验质量。本复习指南通过 AQA 风格的示例解释关键概念。
1. What is Precision? | 什么是精度？
Precision describes how closely repeated measurements agree with one another. If you measure the length of a wire five times and obtain 1.02 m, 1.03 m, 1.02 m, 1.03 m and 1.02 m, your readings are precise because the spread is small.
精度描述重复测量彼此接近的程度。如果你五次测量一根导线的长度，得到 1.02 m、1.03 m、1.02 m、1.03 m 和 1.02 m，则读数很精密，因为离散范围很小。
Precision does not mean the measurement is close to the true value. It only tells you about the consistency or spread of the results. A precise set of readings can still be wrong if every reading is shifted by the same systematic error.
精度并不意味着测量值接近真值。它只说明结果的一致性或离散程度。如果每个读数都被同一个系统误差偏移，一组精密的读数仍然可能是错误的。
2. Precision vs Accuracy | 精度与准确度
Accuracy describes how close a single measurement is to the true or accepted value. Precision describes the spread of repeated results. The classic dartboard picture summarises this: precise but inaccurate darts land tightly grouped away from the bullseye; accurate but imprecise darts are centred on the bullseye but widely scattered.
准确度描述单次测量接近真值或公认值的程度。精度描述重复结果的分散程度。经典飞镖图总结如下：精密但不准确的飞镖紧密聚集却偏离靶心；准确但不精密的飞镖围绕靶心但分布很散。
In an experiment, you can improve precision by using better technique or a higher-resolution instrument, and improve accuracy by removing systematic errors such as zero error or calibration drift.
在实验中，你可以通过改进操作技术或使用更高分辨率的仪器来提高精度，并通过消除零误差或校准漂移等系统误差来提高准确度。
3. Repeatability and Reproducibility | 重复性与再现性
Repeatability means that the same observer using the same equipment under the same conditions gets close agreement between repeated measurements. Reproducibility means that different observers, different instruments or different laboratories also obtain close agreement.
重复性指同一观察者使用相同设备在相同条件下进行重复测量时，结果之间接近一致。再现性指不同观察者、不同仪器或不同实验室也能获得接近一致的结果。
AQA practical questions may give you two sets of results from different groups. If both sets have small internal spread but the mean values differ, the results are repeatable but not reproducible, suggesting a systematic difference between the groups.
AQA 实验题可能给出不同小组的两组结果。如果两组内部离散都很小但平均值不同，则结果具有重复性但不具有再现性，说明两组之间存在系统差异。
4. Resolution and Reading Uncertainty | 分辨率与读数不确定度
Resolution is the smallest change in the measured quantity that an instrument can detect. For a metre ruler, the resolution is 1 mm; for a digital voltmeter, it is the value of the last displayed digit.
分辨率是仪器能够检测到的被测物理量的最小变化。对于米尺，分辨率为 1 mm；对于数字电压表，分辨率是最后一位显示数字的值。
ually ± the last displayed digit, unless the manufacturer gives a different specification.
对于模拟刻度，读数不确定度通常取为最小分度值的一半。对于数字仪器，读数不确定度通常为最后一位显示数字，除非制造商给出不同规格。
Analogue reading uncertainty = ± ½ × smallest scale division
Digital reading uncertainty = ± 1 × last displayed digit
5. Random and Systematic Errors | 随机误差与系统误差
Random errors cause readings to scatter unpredictably about the true value. They arise from human judgement, environmental changes or fluctuations in the instrument. Repeating a measurement and calculating the mean reduces the effect of random error.
随机误差使读数围绕真值不可预测地波动。它们来自人的判断、环境变化或仪器波动。重复测量并计算平均值可以减小随机误差的影响。
Systematic errors shift all readings in the same direction. Examples include a micrometer zero error, a digital balance that reads high because it is not tared, or a stopwatch that runs slow. Averaging many readings does not remove a systematic error.
系统误差使所有读数朝同一方向偏移。例子包括千分尺零误差、未去皮的数字天平读数偏高，或走时慢的秒表。对多次读数取平均不能消除系统误差。
6. Absolute, Fractional and Percentage Uncertainty | 绝对、分数与百分比不确定度
measurement. For example, a length of 12.0 ± 0.5 cm has an absolute uncertainty of 0.5 cm.
绝对不确定度是用与测量值相同单位表示的不确定度。例如，12.0 ± 0.5 cm 的绝对不确定度为 0.5 cm。
Fractional uncertainty is the absolute uncertainty divided by the measured value. Percentage uncertainty is the fractional uncertainty multiplied by 100%. These are useful when comparing the quality of different measurements or when combining uncertainties.
分数不确定度是绝对不确定度除以测量值。百分比不确定度是分数不确定度乘以 100%。它们在比较不同测量质量或组合不确定度时非常有用。
Fractional uncertainty = absolute uncertainty ÷ measured value
Percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%
7. Combining Uncertainties in Calculations | 计算中不确定度的组合
When two measured values are added or subtracted, the absolute uncertainties add. When values are multiplied or divided, the percentage uncertainties add. When a value is raised to a power, the percentage uncertainty is multiplied by that power.
当两个测量值相加或相减时，绝对不确定度相加。当测量值相乘或相除时，百分比不确定度相加。当测量值被乘方时，百分比不确定度需乘以该指数。
If Q = a + b or Q = a − b: ΔQ = Δa + Δb
If Q = a × b or Q = a ÷ b: %ΔQ = %Δa + %Δb
If Q = aⁿ: %ΔQ = n × %Δa
These rules are important for AQA practical analysis, for example calculating the uncertainty in density from measurements of mass and volume, or the uncertainty in resistance from V/I.
这些规则对 AQA 实验分析很重要，例如从质量和体积的测量值计算密度的不确定度，或从 V/I 计算电阻的不确定度。
Published by TutorHao | A-Level Physics Revision Series | aleveler.com
广告 / Sponsored牛津国际 AQA 考试局Oxford International AS & A Level Physics 物理教材…物理 备考教辅 · 正版现货 · 与本文内容相关查看详情 / View on Taobao参考价 ¥598.0
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图