📚 Combined 1.6.1: Mastering Uncertainty Calculations | 综合1.6.1：掌握不确定度计算
Every measurement in physics carries a degree of doubt – the uncertainty. To compare experimental results with theoretical models, to judge the reliability of a conclusion, you need to know how to quantify and combine these uncertainties. In Edexcel A-level Physics, specification point 1.6.1 requires you to understand absolute, fractional and percentage uncertainties and, crucially, to be able to combine them correctly. This article walks you through every technique you will meet in your practical work and in examination questions.
物理中每个测量值都带有一定程度的疑问——这就是不确定度。为了将实验结果与理论模型进行比较，并判断结论的可靠程度，你需要知道如何量化并合成这些不确定度。在Edexcel A-level物理中，考纲1.6.1节要求你理解绝对不确定度、分数不确定度和百分比不确定度，最关键的是要能正确地将其合成。本文将带你系统掌握所有在实验作业和考试题目中会出现的方法。
1. What is Uncertainty? | 什么是不确定度？
od, and the observer. Expressing a measurement as (5.00 ± 0.05) cm tells you that the length is almost certainly between 4.95 cm and 5.05 cm. This range is not a mistake – it is a fundamental feature of experimental science.
不确定度是预期被测量真值所在的一个范围。它来源于测量仪器、方法和观测者的局限性。将测量结果表示为 (5.00 ± 0.05) cm 表示长度几乎一定在 4.95 cm 至 5.05 cm 之间。这个范围不是错误，而是实验科学的基本特征。
2. Types of Uncertainties | 不确定度的类型
Uncertainties are usually classified as random or systematic. Random uncertainties cause readings to be scattered about the true value; they can be reduced by taking repeat readings and averaging. Systematic uncertainties shift all readings in one direction – for example, a zero error on a meter. Combining uncertainties typically deals with random uncertainties, but identifying systematic errors is essential for evaluating your experiment.
不确定度通常分为随机不确定度和系统不确定度。随机不确定度导致读数围绕真值分布；它们可以通过反复测量取平均值来减小。系统不确定度使所有读数朝一个方向偏移——例如，仪表的零误差。合成不确定度主要处理随机不确定度，但识别系统误差对于评价你的实验至关重要。
3. Absolute, Fractional and Percentage Uncertainty | 绝对、分数和百分比不确定度
If a quantity A is measured as A ± ΔA, then ΔA is the absolute uncertainty. The fractional uncertainty is ΔA/A, and the percentage uncertainty is (ΔA/A) × 100% te uncertainty of 2 g, a fractional uncertainty of 0.01, and a percentage uncertainty of 1%. Converting between these forms is a key skill.
若某物理量 A 的测量值为 A ± ΔA，那么 ΔA 是绝对不确定度。分数不确定度为 ΔA/A，百分比不确定度为 (ΔA/A) × 100%。例如，质量 (200 ± 2) g 的绝对不确定度为 2 g，分数不确定度为 0.01，百分比不确定度为 1%。在这些形式之间进行转换是一项关键技能。
4. Rules for Combining Uncertainties: Addition and Subtraction | 合成不确定度的规则：加减法
When two measurements are added or subtracted, their absolute uncertainties add in quadrature? No – for A-level, the simple rule is: add the absolute uncertainties directly. If C = A + B or C = A – s (25.0 ± 0.1) cm, the total length is 145.0 cm with an uncertainty of 0.3 cm (0.2 + 0.1).
当两个测量值相加或相减时，它们的绝对不确定度直接相加。若 C = A + B 或 C = A – B，则 ΔC = ΔA + ΔB。例如，如果桌面长度测量为 (120.0 ± 0.2) cm，伸出部分为 (25.0 ± 0.1) cm，则总长度为 145.0 cm，不确定度为 0.3 cm (0.2 + 0.1)。
ΔC = ΔA + ΔB
This simple addition rule always gives the maximum possible uncertainty – sometimes called the worst-case estimate – and is what Edexcel expects you to use.
这一简单相加规则始终给出可能的最大不确定度——有时称为最坏情况估计——这正是Edexcel希望你会使用的。
5. Rules for Combining Uncertainties: Multiplication and Division | 合成不确定度的规则：乘除法
For multiplication or division, you work with fractional or percentage uncertainties hen the fractional uncertainty in Q is the sum of the fractional uncertainties in A and B:
对于乘法或除法，应使用分数或百分比不确定度。若 Q = A × B 或 Q = A / B，则 Q 的分数不确定度等于 A 和 B 的分数不确定度之和：
ΔQ/Q = ΔA/A + ΔB/B
Equivalently, percentage uncertainty in Q = %uncertainty in A + % ± 0.2) cm and width (5.0 ± 0.1) cm, first calculate the percentage uncertainties: length 2%, width 2%. The area percentage uncertainty is therefore 4%.
等价的表达是：Q 的百分比不确定度 = A 的百分比不确定度 + B 的百分比不确定度。例如，求一个矩形面积，长度 (10.0 ± 0.2) cm，宽度 (5.0 ± 0.1) cm，先计算百分比不确定度：长度 2%，宽度 2%，因此面积百分比不确定度为 4%。
6. Combining Uncertainties for Powers and Roots | 幂次和方根的不确定度合成
er, the fractional uncertainty is multiplied by the modulus of the power. If Q = Aⁿ, then:
当物理量进行幂次运算时，其分数不确定度要乘以幂的绝对值。若 Q = Aⁿ，则：
ΔQ/Q = |n| × ΔA/A
s tripled. This rule often appears in energy or gravitational field calculations, e.g. v = √2gh, where the uncertainty in v depends on half the combined fractional uncertainty of g and h.
对于平方根 (n = ½)，分数不确定度是原值的一半。对于立方 (n = 3)，则变为三倍。这一规则经常出现在能量或引力场计算中，例如 v = √2gh，其中 v 的不确定度取决于 g 和 h 合成分数不确定度的一半。
7. Worked Example: Measuring Density | 例题：测量密度
Let' rst, calculate volume V = π(d/2)² L. The radius r = 1.00 cm, so r² = 1.00, but we need the uncertainty in r and r². Since r = d/2, percentage uncertainty in r = percentage uncertainty in d = 1%. Then for r², n = 2, so %U in r² = 2 × 1% = 2%. Volume V = π r² L, so %U(V) = %U(r²) + %U(L) = 2% + (0.2/8.0 × 100% = 2.5%) = 4.5%. Density ρ = m/V, so %U(ρ) = %U(m) + %U(V) = (5/250 × 100% = 2%) + 4.5% = 6.5%. Absolute uncertainty in ρ is then 6.5% of the calculated density value.
让我们把规则组合起来。一名学生测定圆柱体的密度：质量 m = (250 ± 5) g，长度 L = (8.0 ± 0.2) cm，直径 d = (2.00 ± 0.02) cm。首先计算体积 V = π(d/2)² L。半径 r = 1.00 cm，因此 r² = 1.00，但我们需要 r 和 r² 的不确定度。r = d/2，所以 r 的百分比不确定度等于 d 的百分比不确定度 = 1%。对于 r²，n = 2，所以 r² 的百分比不确定度 = 2 × 1% = 2%。体积 V = π r² L，因此 V 的百分比不确定度 = r² 的% + L 的% = 2% + (0.2/8.0 × 100% = 2.5%) = 4.5%。密度 ρ = m/V，所以 ρ 的百分比不确定度 = m 的% + V 的% = (5/250 × 100% = 2%) + 4.5% = 6.5%。ρ 的绝对不确定度即为计算密度值的 6.5%。
8. Graphical Analysis and Uncertainty | 图形分析与不确定度
Uncertainty bars on graphs make the spread of data visible. The best-fit line and the worst-fit lines (steepest and shallowest that still pass through most error bars) allow you to determine the uncertainty in the gradient and intercept. The absolute uncertainty in gradient is (steepest gradient – shallowest gradient) / 2. This method is widely used in the core practicals, such as measuring the acceleration of free fall or the Young modulus.
图表上的误差棒使数据的分散程度可视化。最佳拟合直线与最差拟合直线（仍然穿过大多数误差棒的最陡和最平缓的直线）可用于求出斜率和截距的不确定度。斜率的不确定度为（最陡斜率 – 最平缓斜率）/ 2。这一方法广泛用于核心实验中，例如测量自由落体加速度或杨氏模量。
9. Significant Figures and Uncertainty | 有效数字与不确定度
fied. The measured value should be quoted to the same number of decimal places as the absolute uncertainty. For instance, (4.56 ± 0.03) V is correct; (4.561 ± 0.03) V implies unrealistic precision.
不确定度通常应给出一位有效数字，除非其首位数字是 1 或 2，此时可给两位有效数字。测量值的小数位数应与绝对不确定度的小数位数一致。例如，(4.56 ± 0.03) V 是正确的；而 (4.561 ± 0.03) V 则暗示了不切实际的精度。
10. Common Pitfalls | 常见误区
- Forgetting to double the percentage uncertainty for a squared term. Always check the power first.
- 直接相加百分比不确定度时忘记平方项加倍。务必先检查幂次。
- Using absolute uncertainty addition for multiplication. This overestimates the uncertainty excessively and is not the accepted method.
- 在乘法中使用绝对不确定度相加。这会严重高估不确定度，且不是可接受的方法。
- Miscounting the resolution of digital and analogue instruments. e is ± half the last digit; for analogue it is usually ± half the smallest scale division.
- 数错数字仪器和模拟仪器的分辨率。数字设备的分辨率不确定度为 ± 末位数字的一半；模拟仪表通常为 ± 最小分度值的一半。
11. Summary & Exam Tips | 总结与考试技巧
Mastering uncertainty calculations is about applying a small set of rules consistently. In exam questions, always identify whether quantities are added or multiplied to choose the right rule. Show your working neatly, convert to percentage uncertainties for multi-step problems, and check that your final absolute uncertainty is quoted to the appropriate number of significant figures. With practice, the process becomes automatic and will give you confidence in all practical assessments.
掌握不确定度计算在于一贯地应用一小套规则。在考题中，始终先判断物理量是相加还是相乘，以选择正确的规则。清晰展示你的计算过程，在多步骤问题中转换为百分比不确定度，并检查最终的绝对不确定度是否给出了恰当的位数。通过练习，这个过程将变得自动，让你在所有实验考核中充满信心。
Published by TutorHao | Physics Revision Series | aleveler.com
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