📚 Combining Experimental Uncertainties | 实验不确定度的合成
In Edexcel A-Level Physics and other experimental sciences, no measurement is perfectly exact. Every reading from a ruler, thermometer, voltmeter, or balance carries a small range of doubt called an uncertainty. When you use measured values to calculate a final result, such as density, resistance, or acceleration, the individual uncertainties combine to give an overall uncertainty in that result. This article explains how to combine uncertainties correctly using absolute and percentage forms, with clear rules for addition, subtraction, multiplication, division, and powers. It also includes worked examples and exam technique tips to help you answer practical-based questions with confidence.
在 Edexcel A-Level 物理及其他实验科学中，任何测量都不可能是完全精确的。尺子、温度计、电压表或天平的每一个读数都带有一定范围的怀疑，称为不确定度。当你用测量值计算最终结果（如密度、电阻或加速度）时，各个不确定度会合并，最终结果也会产生一个总不确定度。本文介绍如何正确合成绝对和百分比形式的不确定度，并给出加减、乘除和幂运算的清晰规则。文中还包括例题和考试技巧，帮助你自信地解答实验类题目。
1. Why Uncertainties Matter | 为什么不确定度很重要
Uncertainty tells us how reliable a result is. A calculated value such as 9.81 m s⁻² for g means little unless we know whether the uncertainty is ±0.01 m s⁻² or ±0.5 m s⁻². In Edexcel practical assessments and written papers, you may be asked to estimate uncertainties, combine them, or evaluate whether a result agrees with an accepted value. Good uncertainty analysis is also essential for evaluating the quality of experimental design and for suggesting improvements.
不确定度告诉我们结果有多可靠。例如 g 的计算值 9.81 m s⁻²，如果我们不知道其不确定度是 ±0.01 m s⁻² 还是 ±0.5 m s⁻²，这个结果意义不大。在 Edexcel 实验评估和笔试中，你可能需要估计不确定度、合成不确定度，或判断某个结果是否与公认值一致。良好的不确定度分析对于评价实验设计质量以及提出改进建议也至关重要。
Uncertainties are not mistakes. A mistake is reading 25.5 V as 26.5 V or using the wrong unit. An uncertainty is the unavoidable spread in readings caused by instrument precision, human reaction time, or environmental fluctuations. We always quote a result as best estimate ± uncertainty, such as L = 1.250 ± 0.005 m.
不确定度不是错误。错误是把 25.5 V 读成 26.5 V 或使用错误单位。不确定度是由于仪器精度、人的反应时间或环境波动造成的不可避免的读数散布。我们总是把结果写成“最佳估计值 ± 不确定度”，例如 L = 1.250 ± 0.005 m。
2. Absolute and Percentage Uncertainty | 绝对不确定度与百分比不确定度
Absolute uncertainty is the uncertainty expressed in the same unit as the measurement. For example, a length measured as 25.0 ± 0.2 cm has an absolute uncertainty of 0.2 cm. Percentage uncertainty expresses this uncertainty as a fraction of the measured value, multiplied by 100.
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，长度测量为 25.0 ± 0.2 cm，其绝对不确定度为 0.2 cm。百分比不确定度是将该不确定度表示为测量值的一个分数，再乘以 100。
percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%
For the example above, the percentage uncertainty is (0.2 ÷ 25.0) × 100% = 0.8%. Percentage uncertainty is dimensionless, which allows us to compare uncertainties across different quantities. This becomes especially useful when measurements are multiplied or divided.
对于上例，百分比不确定度为 (0.2 ÷ 25.0) × 100% = 0.8%。百分比不确定度没有量纲，因此我们可以比较不同物理量的不确定度。这在测量值相乘或相除时尤其有用。
| Form | Definition | Example |
| Absolute uncertainty | Uncertainty in the same unit as the reading | 0.05 mm for a digital calliper |
| Percentage uncertainty | Absolute uncertainty ÷ reading × 100% | 1.5% for a current of 0.40 ± 0.006 A |
3. Combining Uncertainties for Addition and Subtraction | 加减运算的不确定度合成
When you add or subtract measured quantities, the absolute uncertainties simply add together. This is because the maximum possible difference between the true values occurs when both uncertainties act in the same direction. Suppose a total length is found from L = L₁ + L₂, where L₁ = 20.0 ± 0.2 cm and L₂ = 35.0 ± 0.3 cm. The total length is 55.0 cm and the absolute uncertainty is 0.2 + 0.3 = 0.5 cm.
当你对测量值进行加减运算时，绝对不确定度直接相加。这是因为真实值之间的最大可能差异发生在两个不确定度同方向作用时。假设总长度由 L = L₁ + L₂ 求得，其中 L₁ = 20.0 ± 0.2 cm，L₂ = 35.0 ± 0.3 cm。总长度为 55.0 cm，绝对不确定度为 0.2 + 0.3 = 0.5 cm。
The same rule applies for subtraction. If you find a temperature change Δθ = θ₂ − θ₁, with θ₁ = 21 ± 0.5 °C and θ₂ = 43 ± 0.5 °C, then Δθ = 22 °C and the absolute uncertainty is 0.5 + 0.5 = 1.0 °C. Do not subtract uncertainties for subtraction; they still add.
减法也遵循同样规则。如果你求温度变化 Δθ = θ₂ − θ₁，其中 θ₁ = 21 ± 0.5 °C，θ₂ = 43 ± 0.5 °C，那么 Δθ = 22 °C，绝对不确定度为 0.5 + 0.5 = 1.0 °C。减法时不要把不确定度相减；它们仍然相加。
4. Combining Uncertainties for Multiplication and Division | 乘除运算的不确定度合成
When measured values are multiplied or divided, percentage uncertainties add. This is because relative errors accumulate multiplicatively. For example, density is calculated from ρ = m ÷ V. If mass m = 50.0 ± 0.5 g and volume V = 25.0 ± 0.5 cm³, then the percentage uncertainty in ρ is the percentage uncertainty in m plus the percentage uncertainty in V.
当测量值相乘或相除时，百分比不确定度相加。这是因为相对误差以乘法方式累积。例如，密度由 ρ = m ÷ V 计算。如果质量 m = 50.0 ± 0.5 g，体积 V = 25.0 ± 0.5 cm³，那么 ρ 的百分比不确定度等于 m 的百分比不确定度加上 V 的百分比不确定度。
%U(m) = (0.5 ÷ 50.0) × 100% = 1.0%
%U(V) = (0.5 ÷ 25.0) × 100% = 2.0%
%U(ρ) = 1.0% + 2.0% = 3.0%
The calculated density is 2.00 g cm⁻³. To convert the percentage uncertainty back to an absolute uncertainty, multiply the density by 3.0%: 2.00 × 0.030 = 0.060 g cm⁻³. So the final result is ρ = 2.00 ± 0.06 g cm⁻³. Notice that the final absolute uncertainty is quoted to one or two significant figures, matching the precision of the result.
计算得到的密度为 2.00 g cm⁻³。要把百分比不确定度转换回绝对不确定度，将密度乘以 3.0%：2.00 × 0.030 = 0.060 g cm⁻³。因此最终结果为 ρ = 2.00 ± 0.06 g cm⁻³。注意最终绝对不确定度保留一位或两位有效数字，与结果的精密度相匹配。
5. Raising a Measurement to a Power | 测量值的幂运算
If a measured quantity is raised to a power, multiply the percentage uncertainty by that power. This arises from the binomial expansion of small errors. For example, if a cube has side length a = 10.0 ± 0.2 cm, then volume V = a³. The percentage uncertainty in a is (0.2 ÷ 10.0) × 100% = 2.0%. The percentage uncertainty in V is therefore 3 × 2.0% = 6.0%.
如果测量值被乘方，百分比不确定度要乘以该幂次。这源于小误差的二项式展开。例如，立方体边长 a = 10.0 ± 0.2 cm，体积 V = a³。a 的百分比不确定度为 (0.2 ÷ 10.0) × 100% = 2.0%。因此 V 的百分比不确定度为 3 × 2.0% = 6.0%。
The volume is 10.0³ = 1.00 × 10³ cm³. The absolute uncertainty is 1.00 × 10³ × 0.060 = 60 cm³. So V = 1.00 × 10³ ± 60 cm³. The same rule applies to square roots: a square root corresponds to a power of ½, so the percentage uncertainty is halved. For example, if T² is measured and you calculate T = √(T²), the percentage uncertainty in T is half that in T².
体积为 10.0³ = 1.00 × 10³ cm³。绝对不确定度为 1.00 × 10³ × 0.060 = 60 cm³。因此 V = 1.00 × 10³ ± 60 cm³。平方根也遵循同样规则：平方根对应 ½ 次幂，因此百分比不确定度减半。例如，如果测量了 T² 并计算 T = √(T²)，T 的百分比不确定度是 T² 的一半。
6. Repeated Measurements and Mean Uncertainty | 重复测量与平均值不确定度
When you take several repeat readings for the same quantity, the best estimate is the mean. The uncertainty in the mean can be estimated using the half-range method: take the maximum reading and the minimum reading, subtract them, and divide by two. For example, if five diameter readings are 1.52 mm, 1.50 mm, 1.53 mm, 1.51 mm, and 1.54 mm, the mean is 1.52 mm and the half-range is (1.54 − 1.50) ÷ 2 = 0.02 mm. Thus the diameter is 1.52 ± 0.02 mm.
当你对同一物理量进行多次重复读数时，最佳估计值是平均值。平均值的不确定度可以用半范围法估计：取最大读数与最小读数之差，再除以二。例如，五个直径读数为 1.52 mm、1.50 mm、1.53 mm、1.51 mm 和 1.54 mm，平均值为 1.52 mm，半范围为 (1.54 − 1.50) ÷ 2 = 0.02 mm。因此直径为 1.52 ± 0.02 mm。
Some courses also accept the standard deviation as a measure of spread, but Edexcel A-Level practical questions often expect the half-range method. Always check whether the question asks for absolute uncertainty or percentage uncertainty. A common exam task is to calculate the mean, the half-range, and then express the final value as mean ± half-range.
一些课程也接受标准差作为离散程度的度量，但 Edexcel A-Level 实验题通常要求使用半范围法。答题时务必看清题目要求的是绝对不确定度还是百分比不确定度。常见的考试任务是计算平均值、半范围，并将最终值表示为平均值 ± 半范围。
7. Reading Uncertainties from Instruments | 仪器读数不确定度
The uncertainty in a single reading depends on the type of instrument. For an analogue scale, such as a ruler, a protractor, or an analogue voltmeter, the reading uncertainty is usually half of the smallest scale division. For example, a ruler with millimetre divisions has a reading uncertainty of ±0.5 mm for a single reading. However, if you measure a length between two marks, you take two readings, so the absolute uncertainty becomes ±1 mm.
单次读数的不确定度取决于仪器类型。对于模拟刻度，如尺子、量角器或模拟电压表，读数不确定度通常是最小刻度分度值的一半。例如，具有毫米分度的尺子，单次读数不确定度为 ±0.5 mm。但如果你测量两个标记之间的长度，需要读取两个位置，因此绝对不确定度变为 ±1 mm。
For digital instruments, such as digital multimeters, digital stopwatches, or digital balances, the reading uncertainty is usually taken as the smallest displayed digit or the manufacturer’s quoted accuracy. For instance, a digital stopwatch displaying 12.35 s has an uncertainty of at least ±0.01 s. However, human reaction time for starting and stopping is typically ±0.1 s to ±0.3 s, so the dominant uncertainty is often much larger than the display precision. Always consider the largest realistic source of uncertainty.
对于数字仪器，如数字万用表、数字秒表或数字天平，读数不确定度通常取最小显示位或制造商给出的准确度。例如，数字秒表显示 12.35 s，其不确定度至少为 ±0.01 s。然而，人类启动和停止秒表的反应时间通常为 ±0.1 s 至 ±0.3 s，因此主要不确定度往往远大于显示精度。务必考虑最大的实际不确定度来源。
8. Worked Example: Density of a Metal Wire | 例题：金属丝密度的不确定度合成
A student measures the mass, length, and diameter of a metal wire to determine its density. The mass is m = 4.50 ± 0.05 g. The length is l = 1.200 ± 0.005 m. The diameter is d = 0.52 ± 0.02 mm. The density is given by ρ = m ÷ (π d² l ÷ 4). Calculate the density and its absolute uncertainty.
一名学生测量金属丝的质量、长度和直径以确定其密度。质量 m = 4.50 ± 0.05 g，长度 l = 1.200 ± 0.005 m，直径 d = 0.52 ± 0.02 mm。密度公式为 ρ = m ÷ (π d² l ÷ 4)。计算密度及其绝对不确定度。
First convert to SI base units: d = 0.52 mm = 0.52 × 10⁻³ m, so the radius squared term uses d² = (0.52 × 10⁻³)² = 2.704 × 10⁻⁷ m². The cross-sectional area is A = π d² ÷ 4 = π × 2.704 × 10⁻⁷ ÷ 4 = 2.124 × 10⁻⁷ m². The volume is V = A × l = 2.124 × 10⁻⁷ × 1.200 = 2.549 × 10⁻⁷ m³. The mass in kg is 4.50 × 10⁻³ kg, so the density is ρ = 4.50 × 10⁻³ ÷ 2.549 × 10⁻⁷ = 1.766 × 10⁴ kg m⁻³.
首先转换为 SI 基本单位：d = 0.52 mm = 0.52 × 10⁻³ m，因此 d² = (0.52 × 10⁻³)² = 2.704 × 10⁻⁷ m²。截面积 A = π d² ÷ 4 = π × 2.704 × 10⁻⁷ ÷ 4 = 2.124 × 10⁻⁷ m²。体积 V = A × l = 2.124 × 10⁻⁷ × 1.200 = 2.549 × 10⁻⁷ m³。质量以 kg 表示为 4.50 × 10⁻³ kg，因此密度 ρ = 4.50 × 10⁻³ ÷ 2.549 × 10⁻⁷ = 1.766 × 10⁴ kg m⁻³。
Now calculate percentage uncertainties. For mass: %U(m) = (0.05 ÷ 4.50) × 100% = 1.11%. For length: %U(l) = (0.005 ÷ 1.200) × 100% = 0.417%. For diameter: %U(d) = (0.02 ÷ 0.52) × 100% = 3.85%. Since d is squared in the formula, the percentage uncertainty contribution from d is 2 × 3.85% = 7.70%. The percentage uncertainty in ρ is therefore 1.11% + 0.417% + 7.70% = 9.23%.
现在计算百分比不确定度。质量：%U(m) = (0.05 ÷ 4.50) × 100% = 1.11%。长度：%U(l) = (0.005 ÷ 1.200) × 100% = 0.417%。直径：%U(d) = (0.02 ÷ 0.52) × 100% = 3.85%。由于公式中 d 被平方，直径对百分比不确定度的贡献为 2 × 3.85% = 7.70%。因此 ρ 的百分比不确定度为 1.11% + 0.417% + 7.70% = 9.23%。
Convert back to absolute uncertainty: Δρ = 1.766 × 10⁴ × 0.0923 = 1.63 × 10³ kg m⁻³. The final result is ρ = 1.77 × 10⁴ ± 1.6 × 10³ kg m⁻³. The diameter measurement dominates the uncertainty, so improving the precision of the diameter measurement would have the greatest effect on reducing the overall uncertainty.
转换回绝对不确定度：Δρ = 1.766 × 10⁴ × 0.0923 = 1.63 × 10³ kg m⁻³。最终结果为 ρ = 1.77 × 10⁴ ± 1.6 × 10³ kg m⁻³。直径测量主导了不确定度，因此提高直径测量的精度对降低总不确定度效果最大。
9. Worked Example: Resistance from V and I | 例题：由 V 和 I 求电阻的不确定度合成
A resistor is connected to a variable power supply. The potential difference is measured as V = 6.00 ± 0.05 V and the current as I = 0.40 ± 0.01 A. Resistance is calculated from R = V ÷ I. Calculate the resistance and its percentage uncertainty.
一个电阻连接到可调电源。测得电位差 V = 6.00 ± 0.05 V，电流 I = 0.40 ± 0.01 A。电阻由 R = V ÷ I 计算。计算电阻及其百分比不确定度。
The resistance is R = 6.00 ÷ 0.40 = 15 Ω. The percentage uncertainty in V is (0.05 ÷ 6.00) × 100% = 0.833%. The percentage uncertainty in I is (0.01 ÷ 0.40) × 100% = 2.5%. Since V and I are divided, percentage uncertainties add: %U(R) = 0.833% + 2.5% = 3.33%. The absolute uncertainty in R is 15 × 0.0333 = 0.50 Ω. Therefore R = 15.0 ± 0.5 Ω.
电阻为 R = 6.00 ÷ 0.40 = 15 Ω。V 的百分比不确定度为 (0.05 ÷ 6.00) × 100% = 0.833%。I 的百分比不确定度为 (0.01 ÷ 0.40) × 100% = 2.5%。由于 V 和 I 相除，百分比不确定度相加：%U(R) = 0.833% + 2.5% = 3.33%。R 的绝对不确定度为 15 × 0.0333 = 0.50 Ω。因此 R = 15.0 ± 0.5 Ω。
Notice that the current measurement contributes about three times more to the total percentage uncertainty than the voltage measurement. In a practical question, you might be asked to suggest how to reduce the uncertainty. Using a more precise ammeter or taking repeated current readings would be more effective than improving the voltmeter precision.
注意电流测量对总百分比不确定度的贡献大约是电压测量的三倍。在实验题中，你可能会被要求提出降低不确定度的方法。使用更精确的电流表或对电流进行多次重复读数，会比提高电压表精度更有效。
10. Significant Figures and Rounding | 有效数字与修约
Final uncertainties are usually quoted to one significant figure, or occasionally two if the first digit is 1 or 2. The final result should be quoted to the same decimal place as the absolute uncertainty. For example, if you calculate g = 9.812 m s⁻² and Δg = 0.063 m s⁻², the uncertainty becomes 0.06 m s⁻² and the result becomes 9.81 m s⁻². Do not quote more decimal places than the uncertainty allows.
最终不确定度通常保留一位有效数字，如果首位数字为 1 或 2，偶尔保留两位。最终结果的小数位数应与绝对不确定度的小数位数一致。例如，如果计算出 g = 9.812 m s⁻²，Δg = 0.063 m s⁻²，不确定度变为 0.06 m s⁻²，结果变为 9.81 m s⁻²。不要保留比不确定度允许的更多小数位。
Percentage uncertainties are normally given to two or three significant figures during calculations to avoid rounding errors, but the final absolute uncertainty is rounded at the end. Always show your working clearly in Edexcel questions so that examiners can see how you combined uncertainties, even if a small rounding difference appears in the final answer.
百分比不确定度在计算过程中通常保留两位或三位有效数字，以免产生舍入误差，但最终绝对不确定度在最后才进行修约。在 Edexcel 题目中务必清晰展示计算过程，这样即使最终答案出现小的舍入差异，阅卷人也能看出你是如何合成不确定度的。
11. Summary of Combination Rules | 不确定度合成规则总结
The table below summarises the key rules for combining uncertainties. Use it for quick revision before your Edexcel practical-based questions. Remember that these rules give the maximum possible uncertainty, which is the standard approach at A-Level. More advanced statistical methods are not required in Edexcel papers.
下表总结了不确定度合成的关键规则。可在 Edexcel 实验题前的快速复习中使用。请记住，这些规则给出的是最大可能不确定度，这是 A-Level 的标准方法。Edexcel 考试不要求更高级的统计方法。
| Operation | Rule | Example |
| Addition or subtraction | Add absolute uncertainties | (10.0 ± 0.2) + (5.0 ± 0.3) = 15.0 ± 0.5 cm |
| Multiplication or division | Add percentage uncertainties | R = V ÷ I gives 3.33% for V = 6.00 ± 0.05 V, I = 0.40 ± 0.01 A |
| Power n | Multiply percentage uncertainty by n | V = a³ with 2.0% in a gives 6.0% in V |
| Square root (power ½) | Multiply percentage uncertainty by ½ | T = √(T²) with 4.0% in T² gives 2.0% in T |
| Repeated readings | Use half-range as uncertainty | Readings 1.52, 1.50, 1.54 mm give 1.52 ± 0.02 mm |
12. Final Exam Tips | 考试终极技巧
When answering Edexcel A-Level practical or data analysis questions, always state the formula you are using, show the substitution of absolute or percentage uncertainties, and convert to the correct form at the end. If a question gives uncertain values in different units, convert all measurements to SI units before combining. Pay attention to whether the quantity is squared, square-rooted, or appears multiple times in a formula.
在解答 Edexcel A-Level 实验或数据分析题时，务必写出所用公式，展示绝对或百分比不确定度的代入过程，并在最后转换为正确形式。如果题目给出的不确定值单位不同，合成前应先将所有测量值转换为 SI 单位。
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