Combining Uncertainties for Edexcel A-Level Physics (1.25) | Edexcel A-Level 物理：不确定度的合成 (1.25)

📚 Combining Uncertainties for Edexcel A-Level Physics (1.25) | Edexcel A-Level 物理：不确定度的合成 (1.25)

In Edexcel A-level Physics, Topic 1.2.5 requires you to combine uncertainties when processing experimental data. No measurement is perfectly exact; an uncertainty shows the range within which the true value is likely to lie. Combining uncertainties correctly allows you to estimate the total uncertainty in a calculated quantity, such as resistance, density or acceleration, and is a key practical skill in Unit 1 and Unit 2.

在 Edexcel A-level 物理中，主题 1.2.5 要求在数据处理时合成不确定度。任何测量都不可能绝对精确；不确定度表示真值可能落入的范围。正确合成不确定度可以估算计算量（如电阻、密度或加速度）的总不确定度，这是第一、第二单元实验技能的核心内容。

1. Why Uncertainties Matter | 为什么不确定度很重要

Every experimental measurement has an uncertainty because instruments have limited resolution, human reaction times vary, and environmental conditions fluctuate. Reporting a value without an uncertainty is scientifically incomplete. In Edexcel practical assessments, you are expected to calculate uncertainties and use them to judge whether two results agree.

每一次实验测量都带有不确定度，因为仪器分辨率有限、人的反应时间不同且环境条件会波动。报告一个不带不确定度的数值在科学上是不完整的。在 Edexcel 实验评估中，你需要计算不确定度并用它判断两个结果是否一致。

2. Absolute vs Fractional vs Percentage Uncertainty | 绝对、相对与百分不确定度

An absolute uncertainty has the same unit as the measurement and is written as ±. For example, a length measured as 5.0 ± 0.1 cm has an absolute uncertainty of 0.1 cm. Fractional uncertainty is the absolute uncertainty divided by the measured value. Percentage uncertainty is the fractional uncertainty multiplied by 100%. These forms are used in different combination rules, so you must be able to convert among them quickly.

绝对不确定度与被测量具有相同单位，写作 ±。例如长度测量为 5.0 ± 0.1 cm，其绝对不确定度为 0.1 cm。相对不确定度是绝对不确定度除以测量值。百分不确定度是相对不确定度乘以 100%。这些形式在不同的合成规则中使用，因此你必须能快速相互转换。

percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%

For the length above, the fractional uncertainty is 0.1 ÷ 5.0 = 0.02 and the percentage uncertainty is 0.02 × 100% = 2%.

以上长度为例，相对不确定度为 0.1 ÷ 5.0 = 0.02，百分不确定度为 0.02 × 100% = 2%。

3. Reading Instruments: Resolution and Reading Uncertainty | 仪器读数：分辨率与读数不确定度

Instrument resolution is the smallest scale division. For an analogue scale such as a metre rule, the reading uncertainty is usually ± half the resolution. For a digital instrument, the reading uncertainty is often taken as ± the last digit. When repeated readings are taken, the uncertainty in the mean can be estimated using the half range: (maximum reading − minimum reading) ÷ 2.

仪器分辨率是最小刻度间隔。对于米尺等模拟刻度，读数不确定度通常取分辨率的一半。对于数字仪器，读数不确定度常取最后一位数字的 ±1。若进行多次重复测量，平均值的绝对不确定度可用半极差估算：（最大读数−最小读数）÷ 2。

Δx = (maximum reading − minimum reading) ÷ 2

This half-range method is simple and is commonly accepted in A-level practical work, especially when fewer than six repeats are carried out.

半极差法简单，是 A-level 实验工作中常用且被接受的方法，尤其当重复次数少于六次时。

4. Combining Uncertainties: Addition and Subtraction | 加减法运算中不确定度的合成

If a quantity is the sum or difference of two measurements, you add the absolute uncertainties. Suppose length L = L₁ − L₂, where L₁ = 12.0 ± 0.1 cm and L₂ = 7.0 ± 0.1 cm. The best estimate of L is 5.0 cm and the absolute uncertainty is 0.1 + 0.1 = 0.2 cm, so L = 5.0 ± 0.2 cm. This rule gives the maximum possible uncertainty and is the standard Edexcel A-level method for combined readings that are added or subtracted.

若某物理量是两个测量值的和或差，应将绝对不确定度相加。例如长度 L = L₁ − L₂，其中 L₁ = 12.0 ± 0.1 cm、L₂ = 7.0 ± 0.1 cm。L 的最佳估计值为 5.0 cm，绝对不确定度为 0.1 + 0.1 = 0.2 cm，因此 L = 5.0 ± 0.2 cm。该规则给出最大可能不确定度，是对相加或相减的合成读数使用的标准 Edexcel A-level 方法。

For a = b + c or a = b − c, Δa = Δb + Δc

5. Combining Uncertainties: Multiplication and Division | 乘除法运算中不确定度的合成

When quantities are multiplied or divided, you add the percentage uncertainties. If R = V / I, and V = 6.0 ± 0.2 V, I = 2.0 ± 0.1 A, then the best estimate is R = 3.0 Ω. The percentage uncertainty in V is (0.2 / 6.0) × 100% = 3.3%, and in I it is (0.1 / 2.0) × 100% = 5.0%. The total percentage uncertainty is 3.3% + 5.0% = 8.3%. The absolute uncertainty in R is

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