📚 Combining Uncertainties (Edexcel A-Level Physics 1.15) | 不确定度的合成（爱德思 A-Level 物理 1.15）
In Edexcel A-Level Physics, every measurement carries an uncertainty, and no experimental result is complete without one. Combining uncertainties lets you estimate the total uncertainty in a final calculated quantity - speed, density, resistance or specific heat capacity - starting from the uncertainties in the raw readings.
在爱德思 A-Level 物理中，每一次测量都存在不确定度，任何实验结果只有给出不确定度才算完整。学会如何合成不确定度，你就能从原始读数的不确定度出发，估算最终计算量（速度、密度、电阻或比热容等）的总不确定度。
This article follows Topic 1.15 of the Edexcel specification. It covers absolute and percentage uncertainty, the rules for adding, subtracting, multiplying, dividing, raising to a power and taking a root, worked examples drawn from typical practical work, and the exam conventions for rounding and significant figures.
本文对应爱德思考纲 1.15 节，涵盖绝对不确定度与百分比不确定度、加减乘除与乘方开方的合成规则、来自典型实验的例题，以及考试中关于舍入和有效数字的书写规范。
1. Why Uncertainties Matter | 为什么不确定度很重要
.5 cm with an absolute uncertainty of ±0.1 cm, the true length is likely to be somewhere between 12.4 cm and 12.6 cm.
不确定度不是错误，而是对真值预期所在范围的说明。例如，用直尺测得长度为 12.5 cm，绝对不确定度为 ±0.1 cm，那么真实长度很可能在 12.4 cm 到 12.6 cm 之间。
When two or more measurements are used in a formula, the uncertainties in each measurement combine to produce an uncertainty in the final result. This process is called propagation of uncertainties, and the rules that govern it depend on which mathematical operation links the quantities.
当两个或多个测量值被用于同一个公式时，各测量值的不确定度会合成，从而产生最终结果的不确定度，这一过程称为不确定度传递。传递所遵循的规则取决于各量之间由哪种数学运算相联系。
g cm⁻³ tells the examiner, and any reader, exactly how precise the experiment was and how much confidence can be placed in the value.
只写 3.6 g cm⁻³ 的结果是不完整的；写成 (3.62 ± 0.15) g cm⁻³，才能让考官和读者准确了解实验的精密程度，以及该数值可信到什么程度。
2. Absolute and Percentage Uncertainty | 绝对不确定度与百分比不确定度
s the measurement itself. A current of (2.0 ± 0.1) A has an absolute uncertainty of 0.1 A.
绝对不确定度是测量值可能波动的范围大小，写作 ±Δx，单位与测量值本身相同。电流 (2.0 ± 0.1) A 的绝对不确定度就是 0.1 A。
Percentage uncertainty compares the uncertainty with the measured value. It is calculated using the expression below and is dimensionless.
百分比不确定度把不确定度与测量值本身作比较。它用下面的表达式计算，无量纲。
percentage uncertainty = (Δx ÷ x) × 100%
A with an absolute uncertainty of ±0.1 A, the percentage uncertainty is (0.1 ÷ 2.0) × 100% = 5%. The same absolute uncertainty of 0.1 A would be only 1% so percentage uncertainty is the fairest way to compare the precision of two measurements.
对于电流 2.0 A、绝对不确定度 ±0.1 A，百分比不确定度为 (0.1 ÷ 2.0) × 100% = 5%。若读数改为 10 A，同样 0.1 A 的绝对不确定度只占 1%，因此百分比不确定度是比较两次测量精密程度最公平的方式。
by the percentage and divide by 100. If a time of 12.0 s has a percentage uncertainty of 4%, the absolute uncertainty is 12.0 × 4 ÷ 100 = 0.48 s.
两个方向都可以转换：要把百分比不确定度换回绝对不确定度，只需用测量值乘以百分比再除以 100。若时间 12.0 s 的百分比不确定度为 4%，则绝对不确定度为 12.0 × 4 ÷ 100 = 0.48 s。
3. Adding and Subtracting Measurements | 测量值的加减
solute uncertainties add directly. If q = a + b or q = a - b, then the rule is as follows.
当测量值相加或相减时，其绝对不确定度直接相加。若 q = a + b 或 q = a - b，则规则如下。
Δq = Δa + Δb
an uncertainty of Δq = 0.2 + 0.1 = 0.3 cm, so the result is written as q = (2.0 ± 0.3) cm.
假设 a = (5.0 ± 0.2) cm，b = (3.0 ± 0.1) cm。差值 a - b = 2.0 cm 的不确定度为 Δq = 0.2 + 0.1 = 0.3 cm，因此结果写作 q = (2.0 ± 0.3) cm。
It often surprises students that the uncertainties still add even though the measured values are subtracted. One value may be at its upper limit while the other is at its lower limit, so the maximum possible difference is larger than the central difference; the worst case is what the rule accounts for.
很多学生感到意外：即使两个测量值相减，不确定度仍然相加。因为一个值可能处于上限，另一个处于下限，使最大可能差值比中心差值更大，而该规则考虑的正是这种最坏情况。
5.0 ± 0.2) cm and b = (4.8 ± 0.2) cm, the difference is only 0.2 cm but the uncertainty is 0.4 cm - the percentage uncertainty of the difference exceeds 100%, and the experiment has failed to measure anything meaningful.
要注意由此带来的实际后果。两个几乎相等的量相减，会得到一个很小却带有很大不确定度的差值。若 a = (5.0 ± 0.2) cm、b = (4.8 ± 0.2) cm，差值只有 0.2 cm，而不确定度却达 0.4 cm，差值的百分比不确定度超过 100%，这样的实验实际上没有测出任何有意义的结果。
4. Multiplying and Dividing Measurements | 测量值的乘除
hen the fractional uncertainty in q is the sum of the fractional uncertainties in a and b.
当测量值相乘或相除时，相加的是百分比不确定度。若 q = a × b 或 q = a ÷ b，那么 q 的相对不确定度等于 a 与 b 相对不确定度之和。
%q = %a + %b
as length (12.0 ± 0.1) cm and width (4.0 ± 0.1) cm. The percentage uncertainties are 0.83% and 2.5%, so the area A = 48.0 cm² has a percentage uncertainty of 0.83 + 2.5 = 3.33%.
考虑一个矩形，测得长 (12.0 ± 0.1) cm、宽 (4.0 ± 0.1) cm。两者的百分比不确定度分别为 0.83% 与 2.5%，因此面积 A = 48.0 cm² 的百分比不确定度为 0.83 + 2.5 = 3.33%。
the absolute uncertainties do not add here: 0.1 + 0.1 = 0.2 cm would be the wrong answer.
再换回绝对不确定度：ΔA = 48.0 × 3.33 ÷ 100 = 1.6 cm²，因此面积写作 A = (48 ± 2) cm²。注意此时绝对不确定度并不相加：若写成 0.1 + 0.1 = 0.2 cm 就是错误答案。
ule. With V = (6.0 ± 0.1) V and I = (0.50 ± 0.01) A, the percentage uncertainties are 1.67% and 2.0%, so R = 12.0 Ω carries an uncertainty of 3.67%, that is 12.0 × 3.67 ÷ 100 = 0.44 Ω, and R = (12.0 ± 0.4) Ω.
V = IR 是检验该规则的好例子。若 V = (6.0 ± 0.1) V、I = (0.50 ± 0.01) A，百分比不确定度分别为 1.67% 与 2.0%，则 R = 12.0 Ω 的不确定度为 3.67%，即 12.0 × 3.67 ÷ 100 = 0.44 Ω，于是 R = (12.0 ± 0.4) Ω。
5. Powers, Roots and Exact Constants | 乘方、开方与精确常数
ercentage uncertainty is multiplied by that power. If q = aⁿ, then the rule is as follows.
当测量值被乘方时，其百分比不确定度乘以该指数。若 q = aⁿ，则规则如下。
%q = n × %a
Take a sphere of radius r = (2.5 ± 0.1) cm, so the percentage uncertainty in r is 4%. The volume is V = (4 ÷ 3)πr³, so the percentage uncertainty in V is 3 × 4 = 12% volume is 65.5 cm³, giving ΔV = 65.5 × 12 ÷ 100 = 7.9 cm³ and therefore V = (65 ± 8) cm³.
取一个半径 r = (2.5 ± 0.1) cm 的球体，r 的百分比不确定度为 4%。体积 V = (4 ÷ 3)πr³，故 V 的百分比不确定度为 3 × 4 = 12%。计算得体积为 65.5 cm³，则 ΔV = 65.5 × 12 ÷ 100 = 7.9 cm³，因此 V = (65 ± 8) cm³。
Roots work the same way, with the fraction as the multiplier. If q = √a, then %q = ½ × %a, so a 9% uncertainty in the area of a square becomes a 4.5% uncertainty in its side length.
开方遵循同样的逻辑，只是乘数为分数。若 q = √a，则 %q = ½ × %a，因此正方形面积的 9% 不确定度会变成其边长的 4.5% 不确定度。
ercentage conversion are exact mathematical numbers, so they never contribute an uncertainty of their own.
精确常数不带任何不确定度。球体公式中的 4 ÷ 3、π，v = 2s ÷ t 中的 2，以及百分比换算中的 100，都是精确的数学数字，它们本身不贡献任何不确定度。
6. A Summary Table of the Rules | 规则汇总表
The four rules above cover almost every calculation you will meet in the Edexcel practical papers. The table below gathers them together for quick revision.
以上四条规则几乎覆盖了爱德思实验卷中会遇到的所有计算。下表把它们汇总起来，便于快速复习。
| Operation | 运算 | Rule | 规则 | Example | 示例 |
| q = a + b or q = a - b | Δq = Δa + Δb | (5.0 ± 0.2) + (3.0 ± 0.1) = (8.0 ± 0.3) |
| q = a × b or q = a ÷ b | %q = %a + %b | 12.0 cm × 4.0 cm = (48 ± 2) cm² |
| q = aⁿ | %q = n × %a | r with 4% gives V with 12% |
| q = √a | %q = ½ × %a | 9% in area gives 4.5% in side |
| q = k × a, k exact | %q = %a | multiplying by 2 or by π changes nothing |
The golden rule is this: for addition and subtraction use absolute uncertainties, and for multiplication, division, powers and roots use percentage uncertainties. Convert between the two whenever the calculation moves from one category to the other.
黄金法则是：加减法用绝对不确定度，乘除、乘方和开方用百分比不确定度。每当运算从一类转到另一类时，就相应地在两者之间转换。
7. Worked Example: Density of a Metal Block | 实例：金属块的密度
ensity is ρ = m ÷ V = 45.2 ÷ 12.5 = 3.616 g cm⁻³, and the task is to find the uncertainty.
某学生用托盘天平测得金属块质量 m = (45.2 ± 0.1) g，用排水法测得体积 V = (12.5 ± 0.5) cm³。密度 ρ = m ÷ V = 45.2 ÷ 12.5 = 3.616 g cm⁻³，接下来要求出其不确定度。
vision, convert both uncertainties to percentages. The mass contributes (0.1 ÷ 45.2) × 100% = 0.22% and the volume contributes (0.5 ÷ 12.5) × 100% = 4.0%. The percentage uncertainty in the density is therefore 4.22%.
由于运算为除法，需把两个不确定度都换算成百分比。质量贡献 (0.1 ÷ 45.2) × 100% = 0.22%，体积贡献 (0.5 ÷ 12.5) × 100% = 4.0%，因此密度的百分比不确定度为 4.22%。
the density to the matching decimal place gives a final answer of ρ = (3.6 ± 0.2) g cm⁻³.
绝对不确定度为 Δρ = 3.616 × 4.22 ÷ 100 = 0.153 g cm⁻³。将不确定度保留一位有效数字，密度保留到与之相同的小数位，最终结果为 ρ = (3.6 ± 0.2) g cm⁻³。
Notice which measurement dominates the final uncertainty. The mass is known to better than 0.3%, while the volume is known to only 4%, so improving the volume measurement - for example by using a narrower measuring cylinder - would reduce the overall uncertainty far more than buying a better balance.
要注意哪个测量主导了最终不确定度。质量的不确定度不到 0.3%，而体积高达 4%，所以改善体积测量（例如换用更细的量筒）比更换更好的天平更能降低总体不确定度。
8. Worked Example: Speed Using a Ruler and a Stopwatch | 实例：用直尺和秒表测速度
at records a time of (0.52 ± 0.01) s. The speed is v = s ÷ t = 1.000 ÷ 0.52 = 1.923 m s⁻¹.
一辆小车通过 1.000 m 的距离，用米尺测量，不确定度为 ±0.001 m；光电门记录的时间为 (0.52 ± 0.01) s。速度 v = s ÷ t = 1.000 ÷ 0.52 = 1.923 m s⁻¹。
The distance contributes (0.001 ÷ 1.000) × 100% = 0.10% and the time contributes (0.01 ÷ 0.52) × 100% = 1.92%. Adding the percentages gives a total of 2.02%, so Δv = 1.923 × 2.02 ÷ 100 = 0.039 m s⁻¹.
距离贡献 (0.001 ÷ 1.000) × 100% = 0.10%，时间贡献 (0.01 ÷ 0.52) × 100% = 1.92%。两者相加得 2.02%，于是 Δv = 1.923 × 2.02 ÷ 100 = 0.039 m s⁻¹。
⁻¹. As always with division, the percentage uncertainties add and the absolute uncertainties do not.
最终结果为 v = (1.92 ± 0.04) m s⁻¹。与所有除法一样，相加的是百分比不确定度，而不是绝对不确定度。
Timing is a recurring weakness in school
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