📚 Finding the value of an uncertainty | 计算不确定度的值
In experimental physics, every measurement comes with a level of doubt. No instrument is perfect, and no reading can be repeated infinitely. The value of an uncertainty quantifies this doubt, allowing us to express how reliable a measurement or calculated result truly is. This article explains how to find the value of an uncertainty from raw data, from instrumental limits, and through propagation when measurements are combined.
在实验物理中，每一次测量都伴随着一定程度的不确定性。没有任何仪器是完美的，也没有任何读数可以无限次重复。不确定度的值量化了这种不确定性，使我们能够表达一个测量或计算结果到底有多可靠。本文将解释如何从原始数据、从仪器限制出发，以及在组合测量时通过误差传递来求得不确定度的数值。
1. Understanding Uncertainty in Measurements | 理解测量中的不确定度
t to an infinite number of decimal places. The uncertainty gives a range within which the true value is expected to lie. For example, a reading of 12.3 cm with an uncertainty of ±0.1 cm means the true length is likely between 12.2 cm and 12.4 cm. Finding this range is essential before using the measurement in calculations.
不确定度不是错误，而是对任何测量精度有限性的反映。当你用尺子测量一段长度时，你无法读出无穷多位小数。不确定度给出了一个真值预期会落入的范围。例如，读数为 12.3 cm，不确定度为 ±0.1 cm，意味着真实长度很可能在 12.2 cm 到 12.4 cm 之间。在将测量值用于计算之前，确定这个范围至关重要。
2. Instrumental Uncertainty | 仪器不确定度
y in a single reading is taken as half the smallest scale division. If the smallest division is 1 mm, the instrumental uncertainty is ±0.5 mm. For a digital instrument, like a digital voltmeter or balance, the uncertainty is usually given by the manufacturer as ±1 in the last displayed digit, or a percentage of the reading plus a number of digits. In examinations, unless stated otherwise, use ± the smallest scale division for analogue instruments and ±1 in the last digit for digital instruments.
最简单的不确定度来源是仪器本身。对于模拟刻度，例如尺子或动圈式仪表，单次读数的绝对不确定度取最小刻度分度值的一半。如果最小分度值是 1 mm，仪器不确定度就是 ±0.5 mm。对于数字仪器，例如数字电压表或天平，不确定度通常由制造商给出，形式为最后一位显示数字的 ±1，或读数的百分之几加若干数字。在考试中，除非另有说明，对模拟仪器使用 ± 最小分度值，对数字仪器使用 ± 最后一位数字的 1。
3. Uncertainty from Repeated Readings | 重复读数的不确定度
tainty in the mean, a simple method is to use half the range of the readings. For example, if five readings of a length are 15.1, 15.3, 15.0, 15.2, 15.1 cm, the mean is 15.14 cm, the range is 0.3 cm, so the uncertainty is ±0.15 cm (half the range). This method is widely accepted at A Level. More advanced statistics use the standard deviation, but the half‑range method is straightforward and sufficient for most practical work.
随机误差会导致读数波动。进行多次读数并计算平均值可以减少随机误差的影响。求平均值的不确定度的一个简单方法是使用读数范围的一半。例如，如果一段长度的五次读数是 15.1, 15.3, 15.0, 15.2, 15.1 cm，平均值是 15.14 cm，范围是 0.3 cm，因此不确定度为 ±0.15 cm（范围的一半）。这种方法在 A Level 中被广泛接受。更高级的统计方法会使用标准偏差，但半范围法简单明了，足以满足大多数实验工作的要求。
4. Absolute, Fractional and Percentage Uncertainty | 绝对不确定度、相对不确定度和百分比不确定度
fractional uncertainty is Δx/x (no units), and the percentage uncertainty is (Δx/x) × 100% e percentage uncertainty is most useful. A mass of 1.00 ± 0.01 kg has a percentage uncertainty of 1%, whereas a mass of 0.10 ± 0.01 kg has 10% – clearly less precise.
绝对不确定度 Δx 具有与测量值相同的单位，写作 x ± Δx。相对不确定度是 Δx/x（无单位），百分比不确定度是 (Δx/x) × 100%。理解这些形式至关重要，因为不同的运算需要不同的形式。在比较两次测量的精确度时，百分比不确定度最为有用。1.00 ± 0.01 kg 的质量百分比不确定度是 1%，而 0.10 ± 0.01 kg 的百分比不确定度是 10%，显然后者精密度较低。
5. Combining Uncertainties: Addition and Subtraction | 不确定度的组合：加法和减法
uncertainties add. If a quantity R is given by R = A + B or R = A − B, the rule is simple: ΔR = ΔA + ΔB m and its absolute uncertainty is 0.2 + 0.1 = 0.3 cm, so we write 8.0 ± 0.3 cm. Note that even for subtraction the uncertainties add; they never cancel.
当测量值相加或相减时，绝对不确定度直接相加。如果一个量 R = A + B 或 R = A − B，规则很简单：ΔR = ΔA + ΔB。这给出了最大可能的不确定度。例如，如果将两段长度 5.0 ± 0.2 cm 和 3.0 ± 0.1 cm 首尾相接，总长度为 8.0 cm，其绝对不确定度为 0.2 + 0.1 = 0.3 cm，因此我们写作 8.0 ± 0.3 cm。注意即使是减法，不确定度也是相加的，它们永远不会相互抵消。
6. Combining Uncertainties: Multiplication and Division | 不确定度的组合：乘法和除法
ivision, the fractional (or percentage) uncertainties add. If R = A × B or R = A / B, then ΔR/R = ΔA/A + ΔB/B ± 0.2 V and a current of 2.0 ± 0.1 A are used to find resistance R = V/I = 2.5 Ω, the percentage uncertainties are (0.2/5.0)×100% = 4% and (0.1/2.0)×100% = 5%, giving a total percentage uncertainty of 9%. Hence, ΔR = 0.09 × 2.5 ≈ 0.225 Ω, so R = 2.5 ± 0.2 Ω (to 1 s.f. of uncertainty).
对于乘法和除法，相对（或百分比）不确定度相加。如果 R = A × B 或 R = A / B，那么 ΔR/R = ΔA/A + ΔB/B。这意味着结果的百分比不确定度是各百分比不确定度之和。例如，用 5.0 ± 0.2 V 的电压和 2.0 ± 0.1 A 的电流来计算电阻 R = V/I = 2.5 Ω，百分比不确定度分别为 (0.2/5.0)×100% = 4% 和 (0.1/2.0)×100% = 5%，总的百分比不确定度为 9%。因此，ΔR = 0.09 × 2.5 ≈ 0.225 Ω，所以 R = 2.5 ± 0.2 Ω（不确定度保留一位有效数字）。
7. Combining Uncertainties: Powers and Roots | 不确定度的组合：幂和方根
fractional uncertainty is multiplied by the absolute value of that power. If R = Aⁿ, then ΔR/R = |n| × (ΔA/A) 3.00 ± 0.06 cm, the volume V = (4/3)πr³ involves r³. The percentage uncertainty in r is 2%. The power is 3, so the percentage uncertainty in r³ is 3 × 2% = 6%. Thus the volume will have a 6% uncertainty (plus any constant factor, which does not affect percentage uncertainty).
当测量值被乘方时，相对不确定度要乘以该指数的绝对值。如果 R = Aⁿ，则 ΔR/R = |n| × (ΔA/A)。这同样适用于方根，因为平方根就是 ½ 次方。例如，一个球的半径是 r = 3.00 ± 0.06 cm，体积 V = (4/3)πr³ 涉及 r³。半径 r 的百分比不确定度为 2%，指数为 3，因此 r³ 的百分比不确定度为 3 × 2% = 6%。因此体积也会有 6% 的不确定度（常数因子不影响百分比不确定度）。
8. Combining Uncertainties: Multiplication by a Constant | 不确定度的组合：乘以常数
mber. If R = k A, where k is a precise constant (e.g., 2, π, or a conversion factor), then ΔR = |k| ΔA us of 2.0 ± 0.1 cm is doubled to find the diameter, d = 2r = 4.0 cm and Δd = 2 × 0.1 = 0.2 cm, giving 4.0 ± 0.2 cm. The percentage uncertainty stays at 5%.
当测量值乘以一个纯数字（无不确定度的常数）时，绝对不确定度也会乘以该数字。如果 R = k A，其中 k 是一个精确的常数（例如 2、π 或某个换算因子），那么 ΔR = |k| ΔA。相对不确定度保持不变，因为测量值和绝对不确定度按相同比例缩放。例如，将半径 2.0 ± 0.1 cm 乘以 2 得到直径，d = 2r = 4.0 cm，Δd = 2 × 0.1 = 0.2 cm，因此直径为 4.0 ± 0.2 cm。百分比不确定度仍为 5%。
9. Determining Uncertainty in the Gradient of a Graph | 确定图中斜率的不确定度
Many A Level experiments involve finding the gradient of a straight-line graph. To find the uncertainty in the gradient, first draw the best-fit line through the data points. Then draw the steepest and the shallowest reasonable straight lines that still fit the data (often called worst-fit lines). Calculate the gradients of these three lines: mbest, mmax, and mmin. The absolute uncertainty in the gradient can be estimated as Δm = (mmax − mmin) / 2. This method requires careful judgement, but it is standard for practical assessments.
许多 A Level 实验都涉及求直线图的斜率。要找出斜率的不确定度，首先通过数据点画出最佳拟合线。然后分别画出仍然与数据吻合的最陡和最平缓的合理直线（通常称为最差拟合线）。计算这三条线的斜率：mbest、mmax 和 mmin。斜率的绝对不确定度可以估算为 Δm = (mmax − mmin) / 2。这种方法需要仔细判断，但它是实验考核中的标准做法。
10. Worked Examples: Finding Uncertainty in a Derived Quantity | 计算实例：推导量的不确定度求取
late the volume. Radius r = d/2 = 1.00 cm, but d has uncertainty. The percentage uncertainty in d is (0.01/2.00)×100% = 0.5% e V = π(d/2)²h = (π/4)d²h. The power on d is 2, so percentage uncertainty in d² is 2 × 0.5% = 1.0%. The height h has percentage uncertainty (0.01/5.00)×100% = 0.2%. For multiplication, add percentage uncertainties: total in V is 1.0% + 0.2% = 1.2% . Density ρ = m/V = 50.0/15.7 ≈ 3.18 g cm⁻³. Percentage uncertainty in m is (0.1/50.0)×100% = 0.2%, in V is 1.2%. Total percentage uncertainty in ρ = 0.2% + 1.2% = 1.4%. Absolute Δρ = 0.014 × 3.18 ≈ 0.0445 g cm⁻³. Thus ρ = 3.18 ± 0.04 g cm⁻³.
考虑求一个金属圆柱体的密度。测量值：质量 m = 50.0 ± 0.1 g，直径 d = 2.00 ± 0.01 cm，高度 h = 5.00 ± 0.01 cm。第 1 步：计算体积。半径 r = d/2 = 1.00 cm，但 d 存在不确定度。d 的百分比不确定度为 (0.01/2.00)×100% = 0.5%。体积 V = π(d/2)²h = (π/4)d²h。d 的指数是 2，因此 d² 的百分比不确定度为 2 × 0.5% = 1.0%。高度 h 的百分比不确定度为 (0.01/5.00)×100% = 0.2%。对于乘法，百分比不确定度相加：V 的总百分比不确定度为 1.0% + 0.2% = 1.2%。V = (π/4)×(2.00)²×5.00 ≈ 15.7 cm³。绝对不确定度 ΔV = 0.012 × 15.7 ≈ 0.19 cm³，因此 V = 15.7 ± 0.2 cm³（不确定度保留一位有效数字）。密度 ρ = m/V = 50.0/15.7 ≈ 3.18 g cm⁻³。m 的百分比不确定度为 (0.1/50.0)×100% = 0.2%，V 的为 1.2%。ρ 的总百分比不确定度 = 0.2% + 1.2% = 1.4%。绝对 Δρ = 0.014 × 3.18 ≈ 0.0445 g cm⁻³。因此 ρ = 3.18 ± 0.04 g cm⁻³。
11. Tips for Practical Examinations | 实验考试技巧
learly show the mean, the range, and the half-range uncertainty. Quote final uncertainties to 1 significant figure (occasionally 2 if the first digit is 1) and round the measured value to the same decimal place. Never overstate precision: a calculated value of 3.182456 g cm⁻³ with an uncertainty of 0.04 g cm⁻³ should be written as 3.18 ± 0.04 g cm⁻³. Label axes on graphs and include error bars where required.
始终记录原始读数及其仪器不确定度，然后使用适当的传递规则。进行重复读数时，清楚写出平均值、范围和半范围不确定度。最终的不确定度通常保留一位有效数字（如果首位是 1，有时保留两位），并将测量值四舍五入到相同的小数位。切勿夸大精密度：若计算值为 3.182456 g cm⁻³，而不确定度为 0.04 g cm⁻³，则应写作 3.18 ± 0.04 g cm⁻³。在图表上标注坐标轴，并根据需要画出误差棒。
12. Summary | 总结
Finding the value of an uncertainty is a step‑by‑step process. Identify the source of uncertainty: instrumental or random. Quantify it as an absolute uncertainty. When combining measurements, apply the correct propagation rule: add absolute uncertainties for addition/subtraction, add percentage uncertainties for multiplication/division, and multiply the percentage uncertainty by the power for functions of the form Aⁿ. For graphical work, use the max‑min method to find the uncertainty in the gradient. Mastering these techniques will not only improve your practical skills but also help you achieve high marks in the CIE A Level Physics examinations.
计算不确定度的值是一个逐步的过程。先确定不确定度的来源：是仪器引起的还是随机误差。将其量化为绝对不确定度。在组合测量值时，应用正确的传递规则：加减法时绝对不确定度相加，乘除法时百分比不确定度相加，对于 Aⁿ 形式的函数，将百分比不确定度乘以指数。对于图形工作，使用最大－最小法来求斜率的不确定度。掌握这些技巧不仅可以提高你的实验技能，还能帮助你在 CIE A Level 物理考试中取得高分。
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