📚 Percentage Uncertainty | 百分比不确定性
In A-Level Physics, no measurement is exact. All measurements have an associated uncertainty, and expressing this uncertainty as a percentage of the measured value provides a clear way to compare the precision of different measurements and to combine uncertainties in calculations. This guide covers everything you need to know about percentage uncertainty for the CIE AS and A2 Physics examinations.
在A-Level物理中，没有任何测量是绝对的。所有测量值都带有不确定性，而将这种不确定性表示为测量值的百分比，可以清晰地比较不同测量的精度，并在计算中组合不确定性。本指南涵盖了CIE AS和A2物理考试中关于百分比不确定性所需掌握的全部内容。
1. What is Uncertainty? | 什么是不确定性？
any measurement due to the limitations of instruments or random variations. Absolute uncertainty is expressed in the same units as the measurement, e.g., a length of (5.0 ± 0.1) cm. The '±' symbol indicates the interval within which the true value probably falls.
在实验物理中，不确定性是指真实值预期存在的范围。这不是错误，而是由于仪器限制或随机变化造成的任何测量中固有的部分。绝对不确定性用与测量值相同的单位表示，例如，长度为 (5.0 ± 0.1) cm。'±' 符号表示真实值可能落入的区间。
2. Absolute vs. Percentage Uncertainty | 绝对不确定性与百分比不确定性
Absolute uncertainty (Δx) is the magnitude of the uncertainty in the same units as the quantity. Percentage uncertainty is the absolute uncertainty expressed as a percentage of the measured value:
percentage uncertainty = (Δx / x) × 100%
0 ± 2) cm has an absolute uncertainty of 2 cm and a percentage uncertainty of (2/20) × 100% = 10% nty is dimensionless and allows direct comparison between measurements of different kinds.
绝对不确定性 (Δx) 是与被测量量同一单位的不确定性大小。百分比不确定性是将绝对不确定性表示为测量值的百分比：
百分比不确定性 = (Δx / x) × 100%
例如，测量值 (20 ± 2) cm 的绝对不确定性为 2 cm，百分比不确定性为 (2/20) × 100% = 10%。百分比不确定性是无量纲的，可以对不同种类的测量进行直接比较。
3. Calculating Percentage Uncertainty for a Single Measurement | 单次测量的百分比不确定性计算
lute uncertainty is ±0.5 mm. For a digital instrument, the absolute uncertainty is the smallest unit of the last digit (e.g., ±0.01 g for a balance reading 2.56 g, unless the manufacturer states otherwise). Once you have the absolute uncertainty and the reading, compute the percentage uncertainty using (Δx / x) × 100% A measurement of 15.0 °C with a thermometer of 0.5 °C absolute uncertainty gives about 3.3%.
使用模拟仪器时，绝对不确定性通常取最小刻度的一半。对于以毫米为刻度的直尺，绝对不确定性为 ±0.5 mm。对于数字仪器，绝对不确定性是最后一位数字的最小单位（例如，对读数为 2.56 g 的天平，绝对不确定性为 ±0.01 g，除非厂家另有说明）。得到了绝对不确定性和读数后，使用 (Δx / x) × 100% 计算百分比不确定性。用绝对不确定性为 0.5 °C 的温度计测得 15.0 °C，百分比不确定性约 3.3%。
4. Combining Uncertainties: Addition and Subtraction | 组合不确定性：加法和减法
cm and the absolute uncertainty is Δ(a+b) = 0.5 + 0.2 = 0.7 cm. The percentage uncertainty of the result is then (0.7 / 15.0) × 100% ≈ 4.7% addition or subtraction; you must work with absolute uncertainties and convert at the end.
当两个量相加或相减时，它们的绝对不确定性相加。例如，a = (10.0 ± 0.5) cm，b = (5.0 ± 0.2) cm，那么 a + b = 15.0 cm，绝对不确定性为 Δ(a+b) = 0.5 + 0.2 = 0.7 cm。结果的百分比不确定性则为 (0.7 / 15.0) × 100% ≈ 4.7%。注意，不能简单地将各自的百分比不确定性相加用于加减运算；必须使用绝对不确定性，最后再转换。
5. Combining Uncertainties: Multiplication and Division | 组合不确定性：乘法和除法
the percentage uncertainties (not absolute) add together. If p = a × b or p = a ÷ b, then:
% uncertainty in p = % uncertainty in a + % uncertainty in b
g products and ratios very straightforward. For instance, if a force F = (5.0 ± 0.5) N (10%) acts over a distance d = (2.0 ± 0.2) m (10%), the work done W = F × d = 10 J will have a percentage uncertainty of 10% + 10% = 20%, giving W = (10 ± 2) J.
对于乘法或除法，百分比不确定性（而非绝对）相加。如果 p = a × b 或 p = a ÷ b，那么：
p 的百分比不确定性 = a 的百分比不确定性 + b 的百分比不确定性
这个规则使得处理乘积和比值变得非常简单。例如，力 F = (5.0 ± 0.5) N (10%) 作用在距离 d = (2.0 ± 0.2) m (10%) 上，功 W = F × d = 10 J，其百分比不确定性为 10% + 10% = 20%，得出 W = (10 ± 2) J。
6. Combining Uncertainties: Powers and Roots | 组合不确定性：幂和根
ty is raised to a power, the percentage uncertainty is multiplied by that power. For y = an,
% uncertainty in y = n × (% uncertainty in a)
m v², the constant ½ is exact (zero uncertainty). The percentage uncertainty in K is then %K = %m + 2 × (% iplication also applies to the v² term, which is already accounted for by the factor of 2.
如果量被求幂，百分比不确定性乘以该幂次。对于 y = an，
y 的百分比不确定性 = n × (a 的百分比不确定性)
这包括平方根（n = 0.5）。在诸如动能 K = ½ m v² 的复杂表达式中，常数 ½ 是准确的（不确定性为零）。K 的百分比不确定性则为 %K = %m + 2 × (%v)。注意：乘法的百分比不确定性相加规则也适用于 v² 项，因子 2 已涵盖这一点。
7. Uncertainty from Repeated Measurements | 重复测量的不确定性
ximum value − minimum value). The percentage uncertainty is then (half range / mean) × 100% f range = 0.035 s. The mean is 1.516 s. The percentage uncertainty ≈ (0.035 / 1.516) × 100% ≈ 2.3% s method is standard; standard deviation may be used in some questions but is less common.
当进行多次读数时，绝对不确定性可估计为范围的一半（最大值 − 最小值）。然后百分比不确定性为 (半范围 / 平均值) × 100%。例如，单摆摆动时间的五次测量：1.52 s, 1.48 s, 1.55 s, 1.50 s, 1.53 s。范围是 0.07 s，半范围为 0.035 s。平均值为 1.516 s。百分比不确定性 ≈ (0.035 / 1.516) × 100% ≈ 2.3%。在CIE实验考试中，此方法是标准做法；某些题目可能使用标准差，但较少见。
8. Using Percentage Uncertainty to Compare Precision | 使用百分比不确定性比较精度
Percentage uncertainty is an excellent tool for comparing the precision of different measurements, even if they have different units or magnitudes. A measurement with a smaller percentage uncertainty is more precise. For instance, a 1 mm uncertainty in a 2 cm length represents 5%, while the same 1 mm uncertainty in a 20 cm length represents only 0.5% – the latter is ten times more precise. In experimental design, you should strive to reduce percentage uncertainties, e.g., by measuring larger quantities where possible.
百分比不确定性是用于比较不同测量精度的极佳工具，即便这些测量具有不同的单位或数量级。百分比不确定性较小的测量更精确。例如，1 mm 的不确定性在 2 cm 的长度中代表 5%，而同样的 1 mm 不确定性在 20 cm 的长度中仅代表 0.5%——后者的精度是前者的十倍。在实验设计中，你应致力于减小百分比不确定性，例如尽可能测量较大的量。
9. Worked Example: Complex Equation | 复杂方程实例
.0 ± 0.3) V, t = (120 ± 2) s, E = (4.0 ± 0.1) J. First calculate percentage uncertainties: %V = (0.3/6.0)×100% = 5.0%, %t = (2/120)×100% ≈ 1.67%, %E = (0.1/4.0)×100% = 2.5%. The power rule gives 2 × %V for V². Using the rule for multiplication and division: %R = 2 × (%V) + %t + %E = 2 × 5.0% + 1.67% + 2.5% = 14.17%. The nominal R = (6.0² × 120) / 4.0 = (36 × 120)/4 = 1080 Ω. Absolute uncertainty = 14.17% of 1080 ≈ 153 Ω. Hence, R = (1.08 ± 0.15) × 10³ Ω.
根据 R = (V² t) / E 求电阻 R，已知 V = (6.0 ± 0.3) V，t = (120 ± 2) s，E = (4.0 ± 0.1) J。首先计算百分比不确定性：%V = (0.3/6.0)×100% = 5.0%，%t = (2/120)×100% ≈ 1.67%，%E = (0.1/4.0)×100% = 2.5%。幂次法则给出 V² 的百分比不确定性为 2 × %V。利用乘法和除法的规则：%R = 2 × %V + %t + %E = 2 × 5.0% + 1.67% + 2.5% = 14.17%。名义值 R = (6.0² × 120) / 4.0 = 1080 Ω。绝对不确定性 = 1080 的 14.17% ≈ 153 Ω。因此，R = (1.08 ± 0.15) × 10³ Ω。
10. Error Bars and Percentage Uncertainty | 误差条与百分比不确定性
f the bar indicates the range in which the true value lies. If an exam question asks you to estimate the uncertainty in the gradient, you draw the 'worst fit' responding percentage uncertainty can then be calculated from (Δgradient / gradient) × 100%.
在绘制图表时，你绘制的误差条反映的是每个数据点的绝对不确定性，而不是百分比不确定性。误差条的长度表示真实值所处的范围。如果考题要求你估计斜率的不确定性，你可以画出“最坏拟合”线（可接受的最陡和最平缓的斜率），然后用差值的一半求得斜率的绝对不确定性。相应的百分比不确定性可通过 (Δ斜率 / 斜率) × 100% 计算得出。
11. Common Mistakes and Exam Tips | 常见错误与应试技巧
Always pause and identify the operation: use absolute for ±, percentage for × ÷. For mixed equations, work stepwise, computing percentage uncertainties where powers are involved, then combine appropriately. Also, treat pure constants like 2, π, or ½ as having zero uncertainty. In practical questions, always express final uncertainties to one significant figure (or two if the first digit is 1), and round the measurement to match the same decimal place.
一个常见错误是，在加减法时加了百分比不确定性，或在乘除法时用了绝对不确定性。务必暂停并识别运算类型：加减用绝对，乘除用百分比。对于混合方程式，应分步进行，在涉及幂的地方计算百分比不确定性，然后适当组合。此外，将纯数字如 2、π 或 ½ 视为不确定性为零。在实验题中，最终的不确定性总是表达为一位有效数字（若首位是1则可用两位），并将测量值修约至相同的小数位。
12. Summary and Key Formulae | 总结与核心公式
Percentage uncertainty = (Δx / x) × 100% olute uncertainties Δx + Δy. For multiplication/division: combine percentage uncertainties %x + % te uncertainty. With these rules, you can tackle any CIE uncertainty question systematically and accurately.
百分比不确定性 = (Δx / x) × 100%。对于加/减法：组合绝对不确定性 Δx + Δy。对于乘/除法：组合百分比不确定性 %x + %y。对于幂：将百分比不确定性乘以指数 n。重复测量：近似取 Δx 为范围的一半。误差条显示的是绝对不确定性。掌握这些规则，你就能系统、准确地解答任何CIE不确定度问题。
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