A-Level Physics: Measurement and Error Analysis – Deriving Uncertainty Formulas | A-Level 物理：测量与误差分析——不确定度公式推导

📚 A-Level Physics: Measurement and Error Analysis – Deriving Uncertainty Formulas | A-Level 物理：测量与误差分析——不确定度公式推导

Every experimental measurement in physics carries some degree of uncertainty. To make valid scientific conclusions, we need to understand how these uncertainties propagate through calculations. This article walks you through the derivations of the key uncertainty formulas used in A-Level physics, from simple addition to complex power functions, allowing you to handle error analysis with confidence.

在物理学实验中，每一次测量都带有一定的不确定度。为了得出有效的科学结论，我们必须理解这些不确定度如何在计算中传播。本文将带你逐一推导 A-Level 物理中使用的重要不确定度公式，从简单的相加到复杂的幂函数，让你能够自信地处理误差分析。

1. What is Uncertainty and Why Do We Derive It? | 什么是不确定度？为什么要推导它？

Uncertainty quantifies the range within which the true value of a measurement is expected to lie. It arises from instrument resolution, environmental fluctuations, and human judgement. When we compute a new quantity from several measured values, the uncertainty in the result depends on the uncertainties of the inputs. Deriving these propagation rules ensures that we do not underestimate or overestimate the final error.

不确定度量化了测量值真实值可能存在的范围。它来源于仪器分辨率、环境波动和人为判断。当我们通过几个测量值计算一个新物理量时，结果的误差取决于各输入量的误差。推导这些传播规则可以确保我们既不会低估也不会高估最终误差。

2. Absolute and Relative Uncertainty | 绝对不确定度与相对不确定度

Absolute uncertainty Δx is the range around the measured value, expressed in the same units. If a length is measured as (10.0 ± 0.2) cm, then ΔL = 0.2 cm. Relative uncertainty is the ratio Δx/x, often expressed as a percentage. Understanding both is essential because the propagation rules differ for each form.

绝对不确定度 Δx 是围绕测量值的范围，使用相同单位。若测量长度为 (10.0 ± 0.2) cm，则 ΔL = 0.2 cm。相对不确定度是比值 Δx/x，通常以百分比表示。理解这两种形式至关重要，因为传播规则对每一种形式都不同。

3. Deriving Uncertainty for Addition and Subtraction | 推导加减法的不确定度公式

Suppose we have two measured quantities A ± ΔA and B ± ΔB. We want the uncertainty in Q = A + B or Q = A − B. The extreme possible values of Q occur when both errors add constructively. Therefore, the absolute uncertainty in Q is simply the sum of the absolute uncertainties: ΔQ = ΔA + ΔB. This is the worst-case scenario derivation.

假设我们有两个测量量 A ± ΔA 和 B ± ΔB。我们需要求 Q = A + B 或 Q = A − B 的不确定度。Q 的极端可能值出现在两个误差同向叠加时。因此，Q 的绝对不确定度就是两个绝对不确定度之和：ΔQ = ΔA + ΔB。这是最坏情况的推导。

In a more statistically sound approach, if errors are independent and random, we combine them in quadrature: ΔQ = √(ΔA² + ΔB²). However, at A-Level, the simple addition method is usually taught for addition and subtraction, which safely overestimates the uncertainty.

在更符合统计理论的处理中，如果误差是独立且随机的，我们使用平方和的平方根进行合成：ΔQ = √(ΔA² + ΔB²)。但在 A-Level 阶段，加减法通常教授简单相加的方法，这样可以安全地高估不确定度。

4. Deriving Uncertainty for Multiplication and Quotient | 推导乘除法的不确定度公式

When Q = A × B, the largest possible Q occurs when both A and B are at their maximum values: (A + ΔA)(B + ΔB) ≈ AB + AΔB + BΔA, ignoring the small term ΔAΔB. Similarly, the smallest Q is approximately AB − AΔB − BΔA. The absolute uncertainty in Q is therefore roughly AΔB + BΔA. Dividing by Q = AB gives the relative uncertainty: ΔQ/Q = ΔA/A + ΔB/B.

当 Q = A × B 时，最大可能的 Q 出现在 A 和 B 均取最大值时：(A + ΔA)(B + ΔB) ≈ AB + AΔB + BΔA，忽略了小项 ΔAΔB。类似地，最小 Q 约为 AB − AΔB − BΔA。因此 Q 的绝对不确定度大约为 AΔB + BΔA。除以 Q = AB 得到相对不确定度：ΔQ/Q = ΔA/A + ΔB/B。

For division Q = A/B, the maximum Q arises from maximum A and minimum B: (A + ΔA)/(B − ΔB). Using the binomial approximation 1/(B−ΔB) ≈ (1/B)(1 + ΔB/B), we again find relative uncertainty ΔQ/Q = ΔA/A + ΔB/B. Thus, for both multiplication and division, relative uncertainties add.

对于除法 Q = A/B，最大 Q 来自最大 A 和最小 B：(A + ΔA)/(B − ΔB)。利用二项式近似 1/(B−ΔB) ≈ (1/B)(1 + ΔB/B)，我们再次得到相对不确定度 ΔQ/Q = ΔA/A + ΔB/B。因此，对于乘法和除法，相对不确定度都是相加的。

5. Uncertainty Derivation for a Power Law | 幂函数的不确定度推导

If Q = Aⁿ, where n is a constant, the relative uncertainty in Q is |n| times the relative uncertainty in A. This can be derived by writing Q ± ΔQ = (A ± ΔA)ⁿ ≈ Aⁿ(1 ± n(ΔA/A)). Expanding gives ΔQ/Q = |n|(ΔA/A). This rule is extremely useful in physics, for example when calculating the kinetic energy ½mv², where the velocity term appears squared.

如果 Q = Aⁿ，其中 n 为常数，则 Q 的相对不确定度是 A 的相对不确定度的 |n| 倍。这可以通过写出 Q ± ΔQ = (A ± ΔA)ⁿ ≈ Aⁿ(1 ± n(ΔA/A)) 来推导。展开后得到 ΔQ/Q = |n|(ΔA/A)。这种规则在物理学中非常有用，例如计算动能 ½mv²，其中速度项带有平方。

For a more general case Q = kAᵐBⁿ, where k is a constant, the total relative uncertainty is ΔQ/Q = |m|(ΔA/A) + |n|(ΔB/B). Each input’s relative uncertainty is multiplied by the absolute value of its exponent and then summed.

对于更一般的情况 Q = kAᵐBⁿ，其中 k 为常数，总相对不确定度为 ΔQ/Q = |m|(ΔA/A) + |n|(ΔB/B)。每个输入量的相对不确定度乘以其指数的绝对值后再求和。

6. Deriving the Uncertainty in the Mean of Repeated Measurements | 推导多次测量平均值的不确定度

When you take N repeated measurements x₁, x₂, …, xₙ, the mean x̄ = (Σxᵢ)/N. The standard deviation σ characterises the spread. The uncertainty in the mean, often called the standard error, is σ/√N. This formula comes from the fact that the variance of the sum is Nσ², so the variance of the mean is σ²/N, and taking the square root gives the standard error.

当你进行 N 次重复测量 x₁, x₂, …, xₙ 时，平均值 x̄ = (Σxᵢ)/N。标准差 σ 表征了数据的离散程度。平均值的不确定度，通常称为标准误差，为 σ/√N。这个公式源于总和的方差为 Nσ²，因此平均值的方差为 σ²/N，开平方后得到标准误差。

This derivation shows that increasing the number of measurements reduces the random uncertainty in the average, which is why multiple trials are essential in good experimental design.

这一推导表明，增加测量次数可以减小平均值的随机不确定度，这就是为什么在良好的实验设计中重复多次是必不可少的。

7. Combining Uncertainties in a Density Calculation | 密度计算中的不确定度合成

A classic practical is measuring the density ρ of a material. We measure mass m ± Δm and dimensions to find volume V ± ΔV, then use ρ = m/V. The relative uncertainty follows: Δρ/ρ = Δm/m + ΔV/V. If V is calculated from a cube’s side length L as V = L³, then ΔV/V = 3(ΔL/L). Overall, Δρ/ρ = Δm/m + 3ΔL/L.

一个经典的实验是测量材料的密度 ρ。我们测得质量 m ± Δm，并通过尺寸测量得到体积 V ± ΔV，然后使用 ρ = m/V。相对不确定度遵循：Δρ/ρ = Δm/m + ΔV/V。如果体积通过立方体边长 L 计算为 V = L³，则 ΔV/V = 3(ΔL/L)。总体而言，Δρ/ρ = Δm/m + 3ΔL/L。

This example illustrates how multiple uncertainty contributions are combined systematically, and how a power in a formula magnifies the influence of that variable’s measurement error.

这个例子说明了如何系统地合成多个不确定度来源，以及公式中的幂如何放大该变量测量误差的影响。

8. Derivation of Percentage Uncertainty in Trigonometric Functions | 三角函数中百分比不确定度的推导

If an angle θ is measured with uncertainty Δθ (in radians), the uncertainty in sin θ is approximately |cos θ| Δθ. This arises from the small-angle approximation or differentiation: Δ(sin θ) ≈ |d(sin θ)/dθ| Δθ = |cos θ| Δθ. The absolute uncertainty in sin θ thus depends on the angle itself, becoming largest near θ = 0.

For a logarithmic function Q = ln x, the uncertainty is ΔQ = Δx/x. This is another useful derivative-based rule that appears when processing data with log scales.

对于对数函数 Q = ln x，其不确定度为 ΔQ = Δx/x。这是另一种基于求导的有用规则，当处理对数刻度的数据时经常出现。

9. Uncertainty in a Linear Regression Slope | 线性回归斜率的不确定度推导

When you plot a straight-line graph and determine the slope m by the method of least squares, the uncertainty Δm can be derived from the spread of data points around the line. The formula involves the standard error of the residuals and the range of x-values. A simplified derivation shows Δm is proportional to the standard deviation of the y-residuals divided by the spread in x.

当你绘制直线图并通过最小二乘法确定斜率 m 时，不确定度 Δm 可以从数据点围绕直线的分散程度推导出来。公式涉及残差的标准误差和 x 值的范围。简化推导表明，Δm 与 y 残差的标准差除以 x 的分布宽度成正比。

At A-Level, you are often asked to estimate this uncertainty from the “worst-fit” lines: Δm = |m_best − m_worst|. Understanding the mathematical basis reinforces why that graphical method works.

在 A-Level 阶段，通常要求你通过“最差拟合”线来估计这一不确定度：Δm = |最佳斜率 − 最差斜率|。理解其数学基础可以强化为什么这种图形方法是有效的。

10. Practical Steps for Deriving Formulaic Uncertainty | 推导公式不确定度的实际步骤

To derive uncertainty formulas for any given equation: (1) Identify the mathematical operation between variables. (2) Apply the rule: for addition/subtraction use absolute uncertainties; for multiplication/division/power use relative uncertainties. (3) If a variable appears more than once, handle it carefully, as the same uncertainty may be correlated. (4) Always express the final uncertainty with the correct number of significant figures.

要为任意给定方程推导不确定度公式：(1) 识别变量之间的数学运算。(2) 应用规则：加减法使用绝对不确定度；乘法和除法和幂函数使用相对不确定度。(3) 如果某个变量出现多次，要谨慎处理，因为同一不确定度可能存在相关性。(4) 始终用正确的有效数字表示最终不确定度。

These derivations are not just theoretical exercises; they are directly tested in practical skills papers, where you must calculate and justify percentage differences and discuss experimental reliability.

这些推导不仅仅是理论练习，它们直接出现在实验技能考核中，你必须计算并论证百分比差异，并讨论实验的可靠性。

11. Common Pitfalls in Deriving and Applying Uncertainty | 推导和应用不确定度时的常见错误

One common mistake is adding percentage uncertainties for addition of quantities instead of absolute uncertainties. Another is forgetting the absolute value of the exponent in power-law formulas. Students also often report final uncertainties with too many decimal places, ignoring the convention that uncertainty should typically be given to one significant figure, or two if the leading digit is 1.

一个常见错误是用百分比不确定度相加来处理加减法，而不是使用绝对不确定度。另一个是在幂函数公式中忘记对指数取绝对值。学生也经常在最终报告中用过多的小数位，忽视了不确定度通常保留一位有效数字，或者在前导数字为 1 时保留两位的惯例。

Understanding the derivation helps you avoid these pitfalls because you internalise why the rules are structured as they are, rather than just memorising them.

理解推导过程有助于你避免这些错误，因为你内化了这些规则之所以如此构建的原因，而不仅仅是死记硬背。

12. Conclusion: Mastering Error Analysis Through Derivation | 总结：通过推导掌握误差分析

The uncertainty formulas used in A-Level physics are rooted in simple calculus and algebraic approximations. By deriving them step by step, you build a robust framework for any experimental calculation. This skill not only boosts your exam performance but also prepares you for university-level laboratory work where rigorous error propagation is standard practice.

A-Level 物理中使用的不确定度公式根植于简单的微积分和代数近似。通过逐步推导，你为任何实验计算构建了一个稳健的框架。这项技能不仅能提升你的考试成绩，还能为你未来大学实验室中严格进行误差传播的标准实践做好准备。

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A-Level Physics: Measurement and Error Analysis – Deriving Uncertainty Formulas | A-Level 物理：测量与误差分析——不确定度公式推导