📚 Combining Uncertainties: Experiment Analysis and Error Propagation | 不确定度合成：实验分析与误差传递
In Edexcel A-Level Physics, experimental work is not judged only by the final result but also by how confidently that result is known. Almost every measurement made with a ruler, stopwatch, ammeter, voltmeter or sensor carries a limit of precision. When several measured quantities are used together in a formula, those individual limits combine to produce an overall uncertainty in the calculated value. This article explains the core rules for combining absolute, fractional and percentage uncertainties, with worked examples that reflect common exam questions.
在 Edexcel A-Level 物理中，实验工作不仅看最终结果，还看我们对该结果的置信程度。使用直尺、秒表、电流表、电压表或传感器所做的几乎每一次测量都存在精度限制。当多个被测量被代入同一个公式时，这些单独的限制会合成为计算值的总体不确定度。本文讲解合成绝对、分数和百分不确定度的核心规则，并配有反映常见考试题的工作示例。
1. Why Combine Uncertainties? | 为什么要合成不确定度？
Suppose you measure a length L and a time T, and then calculate speed using v = L / T. Even if L and T are each measured with a small uncertainty, the calculated speed cannot be more precise than the original measurements allow. The uncertainty in L and the uncertainty in T propagate through the calculation into the uncertainty in v. Combining uncertainties allows you to state a final result as a range, such as v = 1.24 ± 0.03 m s⁻¹, rather than as a misleading single value.
假设你测量了一个长度 L 和一个时间 T，然后用 v = L / T 计算速度。即使 L 和 T 各自的测量不确定度很小，计算出的速度也不可能比原始测量所允许的精度更高。L 的不确定度和 T 的不确定度会通过计算传递到 v 的不确定度中。合成不确定度可以让你把最终结果表示为一个范围，例如 v = 1.24 ± 0.03 m s⁻¹，而不是一个具有误导性的单一数值。
Exam questions often ask you to combine uncertainties when calculating acceleration, density, resistivity, spring constant or refractive index. The rules are straightforward, but only if you first identify which quantities are being added, subtracted, multiplied, divided or raised to a power.
考试题经常要求你在计算加速度、密度、电阻率、弹簧劲度系数或折射率时合成不确定度。规则本身很直接，但前提是你先判断哪些量在做加法、减法、乘法、除法或被乘方。
2. Absolute, Fractional and Percentage Uncertainty | 绝对、分数与百分不确定度
Absolute uncertainty Δx is the uncertainty expressed in the same units as the measurement. For example, a length may be written as x = 12.0 ± 0.1 cm, where 0.1 cm is the absolute uncertainty. Fractional uncertainty is Δx / x, and percentage uncertainty is (Δx / x) × 100%. For the same measurement, fractional uncertainty = 0.1 / 12.0 ≈ 0.0083, and percentage uncertainty ≈ 0.83%.
绝对不确定度 Δx 是用与测量值相同单位表示的不确定度。例如，一个长度可以写为 x = 12.0 ± 0.1 cm，其中 0.1 cm 是绝对不确定度。分数不确定度是 Δx / x，百分不确定度是 (Δx / x) × 100%。对于同一个测量值，分数不确定度 = 0.1 / 12.0 ≈ 0.0083，百分不确定度 ≈ 0.83%。
When a quantity is measured once, the absolute uncertainty is usually taken as half the smallest scale division or the resolution of the digital instrument. When repeated readings are taken, a better estimate can be made using the spread of results, as discussed in Section 7.
当某个量只测量一次时，绝对不确定度通常取最小刻度分度的一半，或数字仪器的分辨率。当进行重复读数时，可以利用结果的离散程度做出更好的估计，如第 7 节所述。
| Type 类型 | Formula 公式 | Example 示例 |
|---|---|---|
| Absolute 绝对 | Δx | 0.1 cm |
| Fractional 分数 | Δx / x | 0.0083 |
| Percentage 百分 | (Δx / x) × 100% | 0.83% |
3. Rule 1: Adding and Subtracting Quantities | 规则一：加减法
When two quantities are added or subtracted, their absolute uncertainties add. If z = a + b or z = a − b, then Δz = Δa + Δb. The rule applies even when one quantity is subtracted, because the uncertainty never cancels; instead, the possible deviations from the true values add up in the worst case.
当两个量相加或相减时，它们的绝对不确定度相加。如果 z = a + b 或 z = a − b，则 Δz = Δa + Δb。即使其中一个量被减去，该规则同样适用，因为不确定度不会相互抵消；相反，在最不利情况下，与真值的可能偏差会累加起来。
For example, suppose two lengths are measured as a = 8.0 ± 0.1 cm and b = 3.0 ± 0.2 cm. The difference is z = 8.0 − 3.0 = 5.0 cm. The combined absolute uncertainty is Δz = 0.1 + 0.2 = 0.3 cm, so z = 5.0 ± 0.3 cm. Notice that the fractional uncertainty in the difference is 0.3 / 5.0 = 0.06, which is much larger than the individual fractional uncertainties.
例如，假设两个长度测量为 a = 8.0 ± 0.1 cm 和 b = 3.0 ± 0.2 cm。差值为 z = 8.0 − 3.0 = 5.0 cm。合成的绝对不确定度为 Δz = 0.1 + 0.2 = 0.3 cm，因此 z = 5.0 ± 0.3 cm。注意差值的分数不确定度为 0.3 / 5.0 = 0.06，远大于各自单独的分数不确定度。
Δz = Δa + Δb
4. Rule 2: Multiplying and Dividing Quantities | 规则二：乘除法
When two quantities are multiplied or divided, their fractional or percentage uncertainties add. If z = a b or z = a / b, then the fractional uncertainty in z is (Δz / z) = (Δa / a) + (Δb / b). Multiplying both sides by 100% gives the equivalent rule for percentage uncertainties: %U(z) = %U(a) + %U(b).
当两个量相乘或相除时，它们的分数或百分不确定度相加。如果 z = a b 或 z = a / b，那么 z 的分数不确定度为 (Δz / z) = (Δa / a) + (Δb / b)。两边同时乘以 100% 就得到等价的百分不确定度规则：%U(z) = %U(a) + %U(b)。
For example, a car travels a distance s = 200 ± 5 m in a time t = 10.0 ± 0.2 s. The average speed is v = s / t = 200 / 10.0 = 20 m s⁻¹. The percentage uncertainty in s is (5 / 200) × 100% = 2.5%, and in t is (0.2 / 10.0) × 100% = 2.0%. Therefore the percentage uncertainty in v is 2.5% + 2.0% = 4.5%. The absolute uncertainty in speed is 20 × 4.5% = 0.9 m s⁻¹, so v = 20.0 ± 0.9 m s⁻¹.
例如，一辆汽车行驶距离 s = 200 ± 5 m，所用时间 t = 10.0 ± 0.2 s。平均速度为 v = s / t = 200 / 10.0 = 20 m s⁻¹。s 的百分不确定度为 (5 / 200) × 100% = 2.5%，t 的百分不确定度为 (0.2 / 10.0) × 100% = 2.0%。因此 v 的百分不确定度为 2.5% + 2.0% = 4.5%。速度的绝对不确定度为 20 × 4.5% = 0.9 m s⁻¹，所以 v = 20.0 ± 0.9 m s⁻¹。
%U(z) = %U(a) + %U(b)
5. Rule 3: Powers and Roots | 规则三：乘方与开方
When a quantity is raised to a power n, the percentage uncertainty in the result is n times the percentage uncertainty in the original quantity. If z = aⁿ, then %U(z) = |n| × %U(a). The absolute value of n is used because uncertainty is always positive. This rule also covers roots, since a square root is the power ½ and a cube root is the power ⅓.
当一个量被乘方 n 次时，结果的百分不确定度是原量百分不确定度的 n 倍。如果 z = aⁿ，那么 %U(z) = |n| × %U(a)。使用 n 的绝对值是因为不确定度总是正值。该规则也适用于开方，因为平方根就是 ½ 次幂，立方根就是 ⅓ 次幂。
For example, the radius of a sphere is measured as r = 2.00 ± 0.05 cm. The percentage uncertainty in r is (0.05 / 2.00) × 100% = 2.5%. The volume is V = (4/3) π r³. Since the constant factor (4/3) π has no uncertainty, the percentage uncertainty in V is 3 × 2.5% = 7.5%. If the calculated volume is 33.5 cm³, then its absolute uncertainty is 33.5 × 7.5% ≈ 2.5 cm³.
例如，一个球体的半径测量为 r = 2.00 ± 0.05 cm。r 的百分不确定度为 (0.05 / 2.00) × 100% = 2.5%。体积为 V = (4/3) π r³。由于常数因子 (4/3) π 没有不确定度，V 的百分不确定度为 3 × 2.5% = 7.5%。如果计算出的体积为 33.5 cm³，那么它的绝对不确定度为 33.5 × 7.5% ≈ 2.5 cm³。
%U(z) = |n| × %U(a)
6. Constants and Scaling | 常数与比例缩放
Multiplying a measured quantity by an exact constant scales the absolute uncertainty but leaves the percentage uncertainty unchanged. If z = k a, where k is a known constant with no uncertainty, then Δz = k Δa. The fractional uncertainty is Δz / z = k Δa / (k a) = Δa / a, so the percentage uncertainty in z is the same as the percentage uncertainty in a.
用一个精确常数去乘一个被测量，会按比例缩放绝对不确定度，但百分不确定度保持不变。如果 z = k a，其中 k 是一个已知的没有不确定度的常数，那么 Δz = k Δa。分数不确定度为 Δz / z = k Δa / (k a) = Δa / a，因此 z 的百分不确定度与 a 的百分不确定度相同。
This situation commonly arises with π, the gravitational field strength g, or fixed coefficients in an equation. For example, if a pendulum length L is measured as 50.0 ± 0.2 cm, then the percentage uncertainty is 0.4%. When calculating the period from T = 2π √(L / g), the factor 2π and g act as constants; they scale the absolute value but do not add to the percentage uncertainty in L. The percentage uncertainty in L feeds into the square root rule from Section 5.
这种情况经常出现在 π、重力场强 g 或方程中的固定系数上。例如，如果单摆长度 L 测量为 50.0 ± 0.2 cm，那么百分不确定度为 0.4%。当用 T = 2π √(L / g) 计算周期时，因子 2π 和 g 作为常数；它们会缩放绝对值，但不会给 L 的百分不确定度增加任何量。L 的百分不确定度按照第 5 节的平方根规则传递。
7. Repeated Readings and Mean Values | 重复读数与平均值
When you take several readings of the same quantity under the same conditions, the best estimate of the true value is the arithmetic mean. A simple estimate of the absolute uncertainty is half the range of the readings, calculated as (maximum − minimum) / 2. This method is acceptable in many A-Level practical tasks and gives a realistic spread without requiring statistical tools.
当你在相同条件下对同一个量进行多次读数时，真值的最佳估计是算术平均值。绝对不确定度的一个简单估计是读数范围的一半，即 (最大值 − 最小值) / 2。这种方法在许多 A-Level 实验任务中是可接受的，并且无需统计工具就能给出符合实际的离散程度。
For example, five measurements of the time for a ball to fall give 0.42 s, 0.45 s, 0.44 s, 0.43 s and 0.46 s. The mean is 0.44 s. The range is 0.46 − 0.42 = 0.04 s, so the uncertainty is ±0.02 s. The result is reported as t = 0.44 ± 0.02 s. When this mean is used in further calculations, the percentage uncertainty is (0.02 / 0.44) × 100% ≈ 4.5%.
例如，一个球下落时间的五次测量为 0.42 s、0.45 s、0.44 s、0.43 s 和 0.46 s。平均值为 0.44 s。范围是 0.46 − 0.42 = 0.04 s，因此不确定度为 ±0.02 s。结果报告为 t = 0.44 ± 0.02 s。当该平均值用于进一步计算时，百分不确定度为 (0.02 / 0.44) × 100% ≈ 4.5%。
8. Uncertainties from Graphs | 由图线确定不确定度
Many Edexcel practical questions require you to determine the gradient of a straight-line graph and then find its uncertainty. After plotting data points with error bars, draw a line of best fit. Then draw the steepest or shallowest reasonable line that still passes through the error bars, often called the worst fit line. The uncertainty in the gradient is half the difference between the maximum and minimum gradients, or simply the difference between the best-fit gradient and one worst-fit gradient.
许多 Edexcel 实验题要求你确定直线图的斜率，然后求出它的不确定度。在绘制带误差棒的数据点后，画一条最佳拟合线。然后再画一条仍然穿过误差棒的最陡或最浅的合理直线，通常称为最坏拟合线。斜率的不确定度是最大和最小斜率之差的一半，或者简单地说，是最佳拟合斜率与一条最坏拟合斜率之差。
For example, if the best-fit gradient is 1.85 m s⁻² and the worst-fit gradient is 1.70 m s⁻², then the uncertainty is |1.85 − 1.70| = 0.15 m s⁻². Therefore the gradient is reported as 1.85 ± 0.15 m s⁻². Avoid drawing worst-fit lines that ignore most error bars, because the uncertainty would then be unrealistically small or large.
例如，如果最佳拟合斜率为 1.85 m s⁻²，最坏拟合斜率为 1.70 m s⁻²，那么不确定度为 |1.85 − 1.70| = 0.15 m s⁻²。因此斜率报告为 1.85 ± 0.15 m s⁻²。避免画出忽略大多数误差棒的最坏拟合线，因为那样得到的不确定度会过小或过大而不真实。
9. Worked Example: Resistivity Calculation | 工作示例：电阻率计算
A student measures the resistance R of a wire, its length L, and its diameter d, then calculates resistivity from ρ = R π d² / (4L). The measurements are R = 10.0 ± 0.2 Ω, L = 1.000 ± 0.005 m, and d = 0.50 ± 0.01 mm. Calculate the percentage uncertainty in ρ and state the final result with its absolute uncertainty.
一名学生测量了一根导线的电阻 R、长度 L 和直径 d，然后用 ρ = R π d² / (4L) 计算电阻率。测量值为 R = 10.0 ± 0.2 Ω，L = 1.000 ± 0.005 m，d = 0.50 ± 0.01 mm。计算 ρ 的百分不确定度，并写出带绝对不确定度的最终结果。
First find the percentage uncertainty in each measured quantity: %U(R) = (0.2 / 10.0) × 100% = 2.0%; %U(L) = (0.005 / 1.000) × 100% = 0.5%; %U(d) = (0.01 / 0.50) × 100% = 2.0%. In the formula, d is squared, so its percentage uncertainty is doubled: 2 × 2.0% = 4.0%. The overall percentage uncertainty in ρ is therefore 2.0% + 4.0% + 0.5% = 6.5%.
首先求每个被测量的百分不确定度：%U(R) = (0.2 / 10.0) × 100% = 2.0%；%U(L) = (0.005 / 1.000) × 100% = 0.5%；%U(d) = (0.01 / 0.50) × 100% = 2.0%。在公式中，d 被平方，因此它的百分不确定度要加倍：2 × 2.0% = 4.0%。所以 ρ 的总体百分不确定度为 2.0% + 4.0% + 0.5% = 6.5%。
Using the measured values, ρ = (10.0 Ω × π × (0.50 × 10⁻³ m)²) / (4 × 1.000 m) ≈ 1.96 × 10⁻⁶ Ω m. The absolute uncertainty is 1.96 × 10⁻⁶ × 6.5% ≈ 0.13 × 10⁻⁶ Ω m. Hence the final result is ρ = 1.96 × 10⁻⁶ ± 0.13 × 10⁻⁶ Ω m, often written as (1.96 ± 0.13) × 10⁻⁶ Ω m.
代入测量值，ρ = (10.0 Ω × π × (0.50 × 10⁻³ m)²) / (4 × 1.000 m) ≈ 1.96 × 10⁻⁶ Ω m。绝对不确定度为 1.96 × 10⁻⁶ × 6.5% ≈ 0.13 × 10⁻⁶ Ω m。因此最终结果为 ρ = 1.96 × 10⁻⁶ ± 0.13 × 10⁻⁶ Ω m，常写作 (1.96 ± 0.13) × 10⁻⁶ Ω m。
10. Common Exam Mistakes and Tips | 常见考试错误与技巧
One common mistake is adding absolute uncertainties for multiplication and division. Remember: for × and ÷, you must add percentage or fractional uncertainties, not absolute ones. Another mistake is forgetting to double or treble the percentage uncertainty when a quantity is squared or cubed. Always check the power before combining uncertainties.
一个常见错误是在乘法和除法时把绝对不确定度相加。请记住：对于 × 和 ÷，你必须把百分或分数不确定度相加，而不是绝对不确定度。另一个错误是当某个量被平方或立方时，忘记把百分不确定度加倍或增加三倍。在合成不确定度之前，一定要检查幂次。
Also, do not confuse precision with accuracy. A digital meter may show many digits, but that only reflects resolution, not the true spread of repeated readings. When a question gives a measurement as a single value plus a tolerance, use the tolerance as the absolute uncertainty. When you take repeated readings yourself, use half the range as a simple uncertainty estimate.
此外，不要把精度与准确度混淆。数字仪表可能显示很多位数字，但这只反映分辨率，不反映重复读数的真实离散范围。当题目给出的测量值是单一数值加一个允差时，就把该允差作为绝对不确定度。当自己进行重复读数时，可以用范围的一半作为简单的不确定度估计。
- For z = a ± b: add absolute uncertainties.
- 对于 z = a ± b：绝对不确定度相加。
- For z = a b or z = a / b: add percentage uncertainties.
- 对于 z = a b 或 z = a / b：百分不确定度相加。
- For z = aⁿ: multiply percentage uncertainty by |n|.
- 对于 z = aⁿ：百分不确定度乘以 |n|。
- Always state the final uncertainty to one or two significant figures.
- 最终不确定度总是保留一位或两位有效数字。
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