📚 Precision, Accuracy and Measurement Uncertainty | 精确度、准确度与测量不确定度
Measurement lies at the heart of experimental physics. Every reading, whether taken with a metre rule or a digital multimeter, carries an inherent degree of doubt called uncertainty. A rigorous understanding of accuracy, precision and uncertainty is essential not only for Paper 3 practicals but also for analysing data in theory papers.
测量是实验物理学的核心。无论是用米尺还是数字万用表进行的每一次读数，都带有一种固有的怀疑程度，即不确定度。对准确度、精确度和不确定度的严谨理解，不仅对 Paper 3 实验考试至关重要，对理论试卷中的数据分析同样不可或缺。
1. Accuracy vs Precision | 准确度与精确度的区别
Accuracy describes how close a measured value is to the true or accepted value of the physical quantity. A measurement is accurate when the systematic error is small, meaning the average of the readings coincides with the true value.
准确度描述测量值与该物理量真实值或公认值之间的接近程度。当系统误差较小时，测量是准确的，即读数的平均值与真实值一致。
Precision describes how closely repeated measurements agree with one another. A precise measurement has a small random error, meaning the spread of readings around the mean is narrow—even if the mean itself is far from the true value.
精确度描述重复测量结果之间彼此一致的程度。精确的测量具有较小的随机误差，即读数在平均值周围的分散程度很小——即使平均值本身远离真实值。
A useful analogy is that of a dartboard. An accurate thrower hits the bullseye; a precise thrower clusters darts tightly together. The ideal experiment is both accurate and precise, but the two qualities are independent. Four combinations are possible, as summarised in the table below.
一个有用的类比是飞镖靶盘。准确的投掷者命中靶心；精确的投掷者使飞镖紧密聚拢。理想的实验应该既准确又精确，但这两个性质相互独立。共有四种组合，如下表所示。
| Situation | 情形 | Accuracy | 准确度 | Precision | 精确度 | Example | 示例 |
| Accurate, not precise 准确但不精确 |
Mean close to true value 平均值接近真实值 |
Large scatter 分散度大 |
Darts scattered around the bullseye 飞镖散布在靶心周围 |
| Precise, not accurate 精确但不准确 |
Mean far from true value 平均值远离真实值 |
Tight grouping 聚集紧密 |
Darts tightly grouped off-centre 飞镖密集聚集但偏离中心 |
| Both accurate and precise 既准确又精确 |
Mean close to true value 平均值接近真实值 |
Tight grouping 聚集紧密 |
All darts clustered on the bullseye 所有飞镖集中在靶心 |
| Neither accurate nor precise 既不准确也不精确 |
Mean far from true value 平均值远离真实值 |
Large scatter 分散度大 |
Darts scattered randomly off-target 飞镖杂乱散布且未中靶 |
2. Random Errors | 随机误差
Random errors cause repeated readings of the same quantity to fluctuate unpredictably above and below the true value. Their origins include fluctuating environmental conditions such as temperature drift, vibration, air currents, parallax in judging a scale, and variations in human reaction time when using a stopwatch.
随机误差导致同一物理量的重复读数在真实值上下不可预测地波动。其来源包括波动的环境条件，如温度漂移、振动、气流、判断刻度时的视差，以及使用秒表时人体反应时间的变化。
Because random errors are equally likely to push a reading upwards or downwards, they directly affect precision. However, their effect on the mean is reduced by repeating the measurement many times and averaging: the random fluctuations tend to cancel one another as the sample size increases.
由于随机误差使读数偏高或偏低的概率相等，因此它们直接影响精确度。然而，通过多次重复测量并取平均，随机误差对平均值的影响会被削弱：随着样本量增大，随机波动往往会相互抵消。
In the laboratory, random errors are also revealed by the resolution of the instrument. A metre rule marked in millimetres cannot distinguish differences smaller than 1 mm, so even under perfect conditions, readings will scatter within that limit.
在实验室中，随机误差也由仪器的分辨率体现。以毫米为刻度的米尺无法分辨小于 1 mm 的差异，因此即使在理想条件下，读数也会在该极限内分散。
3. Systematic Errors | 系统误差
Systematic errors displace every measurement consistently in one direction—always too high or always too low. Common causes include a poorly calibrated balance, a ruler with a worn or damaged zero end, a voltmeter that has not been zeroed, or a technique such as failing to read the meniscus at eye level.
系统误差使每次测量一致地偏向一个方向——总是偏高或总是偏低。常见原因包括天平校准不佳、直尺零端磨损或损坏、电压表未调零，或类似未从液面弯月面等高处读数的错误技术。
Unlike random errors, systematic errors cannot be reduced by repeating measurements or averaging. They affect accuracy and remain hidden unless the result is compared with a known standard or an independent method. A classic example is the zero error of a micrometer screw gauge: if the anvil and spindle do not close to zero before use, every reading carries the same offset.
与随机误差不同，系统误差无法通过重复测量或取平均来减小。它们影响准确度，并且除非将结果与已知标准或独立方法比较，否则会一直隐藏。一个经典例子是千分尺的零误差：如果在使用前砧座和轴杆未闭合归零，则每次读数都带有一个相同的偏移。
In exam questions, you should always check whether the instrument was zeroed before measurement and whether the experimental method could introduce parallax or heat loss. Identifying one systematic error often earns the full mark for that part of the question.
在考试题目中，你应始终检查仪器在测量前是否已归零，以及实验方法是否可能引入视差或热量损失。识别出一个系统误差往往能为该小问赢得满分。
4. Absolute, Fractional and Percentage Uncertainty | 绝对、分数与百分比不确定度
Absolute uncertainty, denoted Δx, is the actual range within which the true value is expected to lie. For example, a length recorded as 25.0 cm ± 0.1 cm has an absolute uncertainty of 0.1 cm. The measured value must always be quoted together with its absolute uncertainty.
绝对不确定度（记作 Δx）是真实值预期所在的实际范围。例如，记录为 25.0 cm ± 0.1 cm 的长度，其绝对不确定度为 0.1 cm。测量值必须始终与其绝对不确定度一并给出。
Fractional uncertainty is the ratio of the absolute uncertainty to the measured value. It expresses the uncertainty as a dimensionless fraction:
分数不确定度是绝对不确定度与测量值的比值。它将不确定度表示为一个无单位的分数：
Fractional uncertainty = Δx / x
分数不确定度 = Δx / x
Percentage uncertainty is the fractional uncertainty multiplied by 100%:
百分比不确定度是分数不确定度乘以 100%：
Percentage uncertainty = (Δx / x) × 100%
百分比不确定度 = (Δx / x) × 100%
For the length 25.0 cm ± 0.1 cm, the percentage uncertainty is (0.1 / 25.0) × 100% = 0.4%. Note that a larger measured value generally gives a smaller percentage uncertainty for the same absolute uncertainty, which is why choosing a longer length or a larger time interval improves the quality of a measurement.
对于 25.0 cm ± 0.1 cm 的长度，百分比不确定度 = (0.1 / 25.0) × 100% = 0.4%。注意，在绝对不确定度相同的情况下，测量值越大通常得到的百分比不确定度越小，这就是为什么选择更长的长度或更大的时间间隔能够改善测量的质量。
5. Combining Uncertainties: Addition and Subtraction | 不确定度的合成：加法与减法
When two measured quantities are added or subtracted, their absolute uncertainties are always added. This rule applies regardless of whether the operation is addition or subtraction:
当两个测量量相加减时，其绝对不确定度始终相加。无论运算是加还是减，该规则均适用：
If y = a + b or y = a − b, then Δy = Δa + Δb
若 y = a + b 或 y = a − b，则 Δy = Δa + Δb
For example, suppose a = 5.0 ± 0.1 cm and b = 3.0 ± 0.2 cm. Then a + b = 8.0 cm and the uncertainty is Δ = 0.1 + 0.2 = 0.3 cm, so the result is written as 8.0 ± 0.3 cm. Likewise, a − b = 2.0 ± 0.3 cm.
例如，设 a = 5.0 ± 0.1 cm，b = 3.0 ± 0.2 cm。则 a + b = 8.0 cm，不确定度 Δ = 0.1 + 0.2 = 0.3 cm，因此结果写为 8.0 ± 0.3 cm。同理，a − b = 2.0 ± 0.3 cm。
This is a conservative rule: the combined uncertainty is never smaller than the larger of the two individual uncertainties. A particularly common application is the measurement of extension, where the initial reading and the final reading each carry an uncertainty, and subtracting them doubles the absolute uncertainty.
这是一个保守的规则：合成不确定度永远不会小于两个单项不确定度中较大者。一个特别常见的应用是伸长量的测量：初始读数和最终读数各带有不确定度，相减后绝对不确定度变为原来的两倍。
6. Combining Uncertainties: Multiplication and Division | 不确定度的合成：乘法与除法
When measured quantities are multiplied or divided, their percentage uncertainties are added. This rule applies whether the operation is multiplication, division, or a combination of both:
当测量量相乘或相除时，其百分比不确定度相加。该规则无论运算为乘法、除法还是两者的结合均适用：
If y = a × b or y = a / b, then %Δy = %Δa + %Δb
若 y = a × b 或 y = a / b，则 %Δy = %Δa + %Δb
For example, if a = 4.0 ± 0.2 m and b = 2.0 ± 0.1 s, then the area-like product y = a × b = 8.0 m·s. The percentage uncertainties are 0.2/4.0 = 5% and 0.1/2.0 = 5% respectively, so %Δy =
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