📚 Treatment of Uncertainties | 不确定度的处理
Every measurement in physics carries a degree of uncertainty. Whether using a ruler, a stopwatch or a digital sensor, no reading is ever perfectly precise. Understanding how to handle these uncertainties is at the heart of experimental science — it allows you to judge the reliability of your results and to communicate findings honestly. In CIE A-Level Physics, the treatment of uncertainties is examined both in the practical papers and in the theory, forming the backbone of good scientific practice.
物理中的每一次测量都带有一定程度的不确定度。无论使用的是直尺、秒表还是数字传感器，任何读数都不可能完全精确。理解如何处理这些不确定度是实验科学的核心——这能让你判断结果的可靠性，并诚实地呈现发现。在 CIE A-Level 物理中，不确定度的处理在实验卷和理论卷中都会考查，是良好科学实践的根基。
1. What is Uncertainty? | 什么是测量不确定度？
you make is only an estimate of the true value, and the uncertainty gives the size of the 'doubt' has an uncertainty of at least ±0.1 cm because your eye must judge the last decimal place.
不确定度是指被测量的真值预期所在的范围。你进行的每次测量都只是真值的一个估计，而不确定度给出了与该估计相关的“怀疑”大小。例如，如果你用一把毫米刻度直尺测得一根导线的长度为 15.7 cm，由于最后一位小数需要靠肉眼判断，该读数的不确定度至少为 ±0.1 cm。
In physics, uncertainty is not a mistake — it is an inevitable consequence of the limited precision of our instruments and our ability to read them. Systematic errors and random errors must be distinguished from uncertainty, but they all contribute to the overall 'spread' of data. Uncertainty is expressed as an interval and is carried through all calculations so that the final result reflects the quality of the original measurements.
在物理学中，不确定度并不是错误——它是仪器精度和读数能力有限所产生的不可避免的结果。系统误差和随机误差必须与不确定度区分开来，但它们都会影响数据的整体“离散程度”。不确定度被表示为一个区间，并在所有计算中进行传递，以使最终结果能够反映原始测量的质量。
2. Absolute and Percentage Uncertainties | 绝对不确定度与百分比不确定度
of ±0.01 g, the absolute uncertainty is ±0.01 g. This tells you directly how large the margin is in absolute terms.
绝对不确定度是以与被测量相同单位表示的不确定度。如果一台天平读出的质量为 3.24 g，分辨率为 ±0.01 g，则绝对不确定度为 ±0.01 g。它直接告诉你误差范围有多大，用绝对数值表示。
Percentage uncertainty makes comparisons between quantities of different sizes much easier. It is calculated by dividing the absolute uncertainty by the measured value and multiplying by 100%:
百分比不确定度使得比较不同大小的量变得容易得多。它由绝对不确定度除以测量值再乘以 100% 得出：
% uncertainty = (absolute uncertainty / measured value) × 100%
For the mass above, the percentage uncertainty would be (0.01 / 3.24) × 100% ≈ 0.3% quantity is extremely useful when we combine uncertainties in multiplication or division.
对于上述质量，百分比不确定度为 (0.01 / 3.24) × 100% ≈ 0.3%。这种无量纲的量在我们进行乘除合并不确定度时极为有用。
3. Sources of Uncertainty in Measurements | 测量中不确定度的来源
, so the absolute uncertainty in any one reading is ±0.5 mm. In many A-Level experiments, you will take the difference between two readings (e.g. start and end positions), so the uncertainty in the length is often estimated as ±1 mm because two readings are involved.
标尺单次读数的不确定度通常取最小刻度的一半。对于毫米刻度的直尺，最小刻度为 1 mm，所以单次读数的绝对不确定度为 ±0.5 mm。在许多 A-Level 实验中，你需要读取两个读数的差值（如起点和终点位置），因为涉及两次读数，长度的不确定度通常估计为 ±1 mm。
Digital instruments have an uncertainty equal to the smallest digit on the display, unless stated otherwise by the manufacturer. For a stopwatch measuring to 0.01 s, the resolution is 0.01 s, but human reaction time introduces a much larger random uncertainty, often around 0.2 s. Thus, reaction time becomes the dominant source of uncertainty in timing.
数字仪器的不确定度等于显示屏最小数字位，除非制造商另有说明。对于可测量到 0.01 s 的秒表，分辨率为 0.01 s，但人的反应时间会引入大得多的随机不确定度，通常在 0.2 s 左右。因此，反应时间成为计时中的主要不确定度来源。
Environmental factors, parallax error and variations in measured quantities all add to the uncertainty budget. Identifying the largest source of uncertainty helps you improve the design of an experiment. Reducing the dominant uncertainty has the greatest effect on the quality of your final result.
环境因素、视差以及被测量的变动都会增加不确定度。找出最大的不确定度来源有助于改进实验设计。减小主要的不确定度对最终结果质量的改善效果最大。
4. Combining Uncertainties: Addition and Subtraction | 合并不确定度：加法和减法
When two or more measurements are added or subtracted, the absolute uncertainty in the result is found by adding the absolute uncertainties of each measurement. This is known as the 'worst-case' method
当两个或多个测量值进行加减运算时，结果的绝对不确定度是将每个测量值的绝对不确定度相加。这被称为“最坏情况”法。
If R = A + B or R = A – B, then ΔR = ΔA + ΔB
the absolute uncertainty is 0.1 + 0.1 = ±0.2 cm. The result is reported as 20.8 ± 0.2 cm.
例如，一根金属杆分成两段测量：一段为 12.5 ± 0.1 cm，另一段为 8.3 ± 0.1 cm。总长度为 20.8 cm，绝对不确定度为 0.1 + 0.1 = ±0.2 cm。结果应报告为 20.8 ± 0.2 cm。
difference is 19.5 °C and the uncertainty is 0.5 + 0.5 = ±1.0 °C. Always add absolute uncertainties — they never cancel out in the worst-case scenario.
即使在相减时也适用此规则。如果你测得温度从 T₁ = 25.5 ± 0.5 °C 变为 T₂ = 45.0 ± 0.5 °C，温差为 19.5 °C，不确定度为 0.5 + 0.5 = ±1.0 °C。始终相加绝对不确定度——在最坏情况下，它们不会相互抵消。
5. Combining Uncertainties: Multiplication and Division | 合并不确定度：乘法和除法
When quantities are multiplied or divided, you add the percentage (or fractional) uncertainties. The absolute uncertainty in the result is then found by multiplying the final value by the combined percentage uncertainty.
当量进行乘法或除法运算时，需要将百分比（或分数）不确定度相加。结果的绝对不确定度则用最终值乘以合并后的百分比不确定度得出。
If R = A × B or R = A ÷ B, then %ΔR = %ΔA + %ΔB
tangle from length L = 5.0 ± 0.2 cm and width W = 2.0 ± 0.1 cm. The area A = 10.0 cm². The % uncertainty in L is (0.2/5.0)×100% = 4%. The % uncertainty in W is (0.1/2.0)×100% = 5%. The combined % uncertainty is 9%. Thus the absolute uncertainty in area is 9% of 10.0 cm² = 0.9 cm². The area is reported as 10.0 ± 0.9 cm².
考虑从长度 L = 5.0 ± 0.2 cm 和宽度 W = 2.0 ± 0.1 cm 计算矩形面积。面积 A = 10.0 cm²。L 的百分比不确定度为 (0.2/5.0)×100% = 4%。W 的百分比不确定度为 (0.1/2.0)×100% = 5%。合并的百分比不确定度为 9%。因此面积的绝对不确定度为 10.0 cm² 的 9%，即 0.9 cm²。面积应报告为 10.0 ± 0.9 cm²。
ce / time inherits the sum of the percentage uncertainties of distance and time. This approach is fundamental when planning your experiment — u know which measurement needs to be made more precisely to control the final uncertainty.
在除法中，规则完全相同。像速度 v = 路程 / 时间这样简单的量，其不确定度继承了路程与时间百分比不确定度之和。这种方法在制定实验方案时非常基本——你能知道哪个测量需要更精确以控制最终不确定度。
6. Combining Uncertainties: Raising to a Power | 合并不确定度：幂函数
If a quantity is raised to a power n, the percentage uncertainty is multiplied by n. This rule applies when a measurement is part of a formula involving a square, square root, cube, etc.
如果一个量被求 n 次幂，其百分比不确定度要乘以 n。当测量值出现在包含平方、平方根、立方等公式中时，此规则适用。
If R = Aⁿ, then %ΔR = n × %ΔA
ed as constant and v = 10.0 ± 0.5 m s⁻¹. The percentage uncertainty in v is (0.5/10.0)×100% = 5%. Since v is squared, n = 2, the % uncertainty in Ek is 2 × 5% = 10% uncertainty is 5 J. This shows how squaring a quantity magnifies its relative uncertainty.
例如，动能 Ek = ½ mv²，其中 m 视为常数，v = 10.0 ± 0.5 m s⁻¹。v 的百分比不确定度为 (0.5/10.0)×100% = 5%。由于 v 是平方，n = 2，Ek 的百分比不确定度为 2 × 5% = 10%。因此，如果 Ek 的最佳值为 50 J（为简化忽略 m），绝对不确定度为 5 J。这说明平方一个量会放大其相对不确定度。
uch as square roots (n = ½). If the period of a pendulum T = 2.0 ± 0.1 s and you calculate √T, the % uncertainty in √T is half that of T, i.e. (0.1/2.0)×100% ÷ 2 = 2.5% ropagation for any power-law relationship, this multiplying factor must always be applied.
该规则也适用于分数次幂，如平方根（n = ½）。如果单摆的周期 T = 2.0 ± 0.1 s，而你需要计算 √T，那么 √T 的百分比不确定度是 T 的一半，即 (0.1/2.0)×100% ÷ 2 = 2.5%。在进行任何幂律关系的不确定度传播时，必须始终应用这个乘数因子。
7. Repeated Measurements and the Mean | 重复测量与平均值
When you take several readings of the same quantity under the same conditions, the best estimate of the true value is the arithmetic mean. The uncertainty in the mean can be estimated using the range of the readings. A simple method acceptable at A-Level is to quote the uncertainty as half the range of the repeated values.
当你在相同条件下多次测量同一物理量时，真值的最佳估计是算术平均值。均值的测量不确定度可以用读数的范围来估计。在 A-Level 中，一种可接受的简单方法是将不确定度取为重复测量值范围的一半。
= 0.04 mm, so the uncertainty is ±0.02 mm. The result is expressed as 0.51 ± 0.02 mm, with the mean rounded to the same decimal place as the uncertainty.
例如，对导线直径进行五次测量得到：0.50 mm, 0.52 mm, 0.51 mm, 0.49 mm, 0.53 mm。平均值为 0.510 mm。范围是 0.53 - 0.49 = 0.04 mm，因此不确定度为 ±0.02 mm。结果表示为 0.51 ± 0.02 mm，平均值修约到与不确定度相同的小数位。
More advanced treatments use the standard deviation of the mean, but the half-range method gives a quick, conservative estimate of precision. The key point is that repeating measurements reduces the random uncertainty, not the systematic one. If the same zero error affects all readings, the accuracy is still limited by that systematic effect.
更高级的处理会使用平均值的标准偏差，但半范围法能给出一个快速且偏保守的精度估计。关键在于重复测量减少的是随机不确定度，而非系统不确定度。如果同一个零误差影响所有读数，准确度仍会受限于该系统效应。
8. Drawing Error Bars | 绘制误差棒
When plotting experimental data, error bars represent the absolute uncertainty in each measurement. On a graph of, say, voltage against current, each plotted point must have both a vertical error bar (uncertainty in voltage) and a horizontal error bar (uncertainty in current). If one uncertainty is negligible, the corresponding error bar may be omitted, but you should state this.
在绘制实验数据时，误差棒代表每个测量值的绝对不确定度。例如在电流-电压图上，每个数据点必须同时具有垂直误差棒（电压的不确定度）和水平误差棒（电流的不确定度）。如果某一不确定度可以忽略，对应的误差棒可以省略，但应当说明。
the horizontal error bar will extend from 1.95 A to 2.05 A. This visual representation lets you see immediately which points carry more weight and how much scatter there is.
每个误差棒的长度是绝对不确定度的两倍（点上方一个不确定度，点下方一个不确定度）。对于 2.00 ± 0.05 A 的电流读数，水平误差棒将从 1.95 A 延伸到 2.05 A。这种直观的表示方式能让你立刻看出哪些点的权重更大，以及数据离散程度如何。
Error bars are essential for judging whether a straight-line relationship is valid. If the line of best fit passes through all the error bars, the proposed model is supported by the data. If many error bars are missed even by the best line, systematic errors or an incorrect model may be indicated.
误差棒对于判断直线关系是否成立至关重要。如果最佳拟合线穿过所有误差棒，则表明所提出的模型得到数据支持。如果连最佳拟合线都未能触及许多误差棒，则可能暗示着系统误差或模型不正确。
9. Finding Uncertainties from a Graph: Gradient and Intercept | 从图中求斜率与截距的不确定度
Once a graph is plotted with error bars, you must draw a line of best fit and, to estimate the uncertainty in the gradient, additional 'worst-fit' lines. The best fit is the single straight line that passes closest to all the points. To determine the uncertainty in the gradient m, draw the steepest possible straight line that still passes through all the error bars and the shallowest possible line that does the same.
当图上画好误差棒后，必须绘制一条最佳拟合线，并为了估计斜率的不确定度，还要绘制“最差拟合线”。最佳拟合线是一条最接近所有点的直线。为了确定斜率 m 的不确定度，需要画出一条尽可能陡的直线，但仍要穿过所有的误差棒，再画出一条尽可能平的直线，同样穿过所有的误差棒。
Δm = ½ (mmax – mmin)
If the steepest line has gradient mmax = 4.2 V A⁻¹ and the shallowest line has mmin = 3.8 V A⁻¹, the uncertainty in the gradient is (4.2 – the y-axis: draw the highest and lowest plausible intercepts and take half the difference.
如果最陡直线的斜率为 mmax = 4.2 V A⁻¹，最平直线的斜率为 mmin = 3.8 V A⁻¹，则斜率的不确定度为 (4.2 – 3.8)/2 = 0.2 V A⁻¹。斜率报告为 4.0 ± 0.2 V A⁻¹。同样的原理也适用于 y 轴截距：画出可能的最大和最小截距，取差值的一半。
In many practical examinations, you will be required to construct these worst-fit lines and use them to quote a final measured quantity with an uncertainty. Practise drawing them neatly: the worst lines must pivot around the centroid of the data points (or use extreme error bar limits) while remaining straight.
在许多实验考试中，你会被要求绘制这些最差拟合线，并用它们来给出最终测量量及其不确定度。要练习整齐地绘制这些线：最差拟合线必须绕数据点的质心（或利用误差棒的极限端点）转动，同时保持直线。
10. Reporting Results: Significant Figures and Rounding | 报告结果：有效数字与修约
Experimental results should be quoted with an uncertainty that reflects the precision of the measurement. The uncertainty is normally given to one significant figure (occasionally two if the first digit is 1). The measured value is then rounded to the same decimal place as the uncertainty.
实验结果应当附带反映测量精度的测量不确定度。通常，不确定度保留一位有效数字（若首位数字为 1，偶尔保留两位）。测量值则修约至与不确定度相同的小数位。
ificant figure. The speed then becomes 5.8 m s⁻¹, and the final result is (5.8 ± 0.1) m s⁻¹. Carrying excessive digits would falsely suggest higher precision than is justified.
例如，若计算得出速度为 5.763 m s⁻¹，不确定度为 ±0.12 m s⁻¹，先按一位有效数字将不确定度表示为 0.1 m s⁻¹，速度则修约为 5.8 m s⁻¹，最终结果即为 (5.8 ± 0.1) m s⁻¹。保留过多数字会虚假地暗示更高的精度。
| Rule | Example |
|---|---|
| Uncertainty to 1 significant figure | Δd = 0.06 mm → 0.06 mm (already 1 sf) |
| Measured value matches decimal places of uncertainty | d = 2.348 mm → 2.35 mm |
| Result: (2.35 ± 0.06) mm | Both numbers end at hundredths |
Always check that the magnitude of the uncertainty is consistent with your reading precision. If your calculated uncertainty is smaller than the reading resolution of the instrument, adjust it to at least the resolution – you cannot have a result more precise than the device allows.
始终检查不确定度的大小是否与你的读数精度一致。如果计算出的不确定度小于仪器的读数分辨率，至少要将其调整为分辨率——你不可能获得比仪器允许的更精确的结果。
Following these conventions ensures that your work meets the standards expected in A-Level Physics and beyond. Treating uncertainties rigorously turns raw data into reliable evidence, whether you are verifying Ohm’s law or measuring the acceleration of free fall.
遵循这些惯例可确保你的工作达到 A-Level 物理乃至更高层次所期望的标准。无论是验证欧姆定律还是测量自由落体加速度，严谨处理不确定度都能把原始数据转化为可靠的证据。
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