Mastering Combined Uncertainties in Edexcel A Level Science | 掌握 Edexcel A Level 科学中的合成不确定度

📚 Mastering Combined Uncertainties in Edexcel A Level Science | 掌握 Edexcel A Level 科学中的合成不确定度

In Edexcel A Level Biology, Chemistry and Physics, practical and investigative skills include quantifying uncertainty. You need to combine uncertainties from measured quantities to estimate the reliability of a derived result. This article explains the core rules with worked examples.

在 Edexcel A Level 生物、化学和物理课程中，实验与探究技能要求对不确定度进行量化处理。你需要合成各测量量的不确定度，从而估计导出结果的可靠性。本文结合实例说明核心规则。

1. What Are Uncertainties and Why They Matter | 什么是测量不确定度及其重要性

Every measurement has a limit set by instrument resolution, human judgement or random variation. When you combine measurements into a calculated result, those uncertainties propagate through the calculation. Edexcel practical assessment criteria expect you to report uncertainty in a consistent and meaningful way.

任何测量都受仪器分辨率、人为判断或随机变化限制。当你把多个测量量代入公式计算结果时，这些不确定度会传递。Edexcel 实验评估标准要求你以一致且有意义的方式报告不确定度。

2. Absolute, Fractional and Percentage Uncertainty | 绝对不确定度、相对不确定度与百分不确定度

Absolute uncertainty is the uncertainty expressed in the same unit as the measurement, such as ±0.1 cm on a ruler. Fractional uncertainty is the absolute uncertainty divided by the measured value. Percentage uncertainty is the fractional uncertainty multiplied by 100%. Different combining rules use different forms.

绝对不确定度是与测量值同单位的不确定度，例如直尺的 ±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%。不同的合成规则需要使用不同形式。

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

3. Combining Uncertainties for Addition and Subtraction | 加减运算中不确定度的合成

When a result is found by adding or subtracting measured quantities, add the absolute uncertainties. For example, if length = (5.0 ± 0.1) cm and width = (3.0 ± 0.1) cm, then the perimeter of the rectangle is 16.0 cm. The absolute uncertainty is 0.1 + 0.1 + 0.1 + 0.1 = 0.4 cm, so the result is (16.0 ± 0.4) cm.

当结果由测量量的加减得到时，应把各绝对不确定度相加。例如长度 = (5.0 ± 0.1) cm，宽度 = (3.0 ± 0.1) cm，则矩形周长为 16.0 cm。绝对不确定度为 0.1 + 0.1 + 0.1 + 0.1 = 0.4 cm，因此写作 (16.0 ± 0.4) cm。

For R = A + B or R = A − B: ΔR = ΔA + ΔB

4. Combining Uncertainties for Multiplication and Division | 乘除运算中不确定度的合成

For multiplication or division, add the percentage uncertainties. Suppose speed = distance ÷ time, with distance = (100 ± 1) m and time = (20.0 ± 0.5) s. The percentage uncertainties are 1.0% for distance and 2.5% for time. The total percentage uncertainty is 3.5%. Speed = 100 ÷ 20.0 = 5.0 m/s, so the absolute uncertainty is 3.5% × 5.0 = 0.175 ≈ 0.2 m/s. The final answer is (5.0 ± 0.2) m/s.

乘法或除法应使用百分不确定度相加。例如速度 = 路程 ÷ 时间，路程 = (100 ± 1) m，时间 = (20.0 ± 0.5) s。路程的百分不确定度为 1.0%，时间为 2.5%，总百分不确定度为 3.5%。速度 = 100 ÷ 20.0 = 5.0 m/s，因此绝对不确定度 = 3.5% × 5.0 = 0.175 ≈ 0.2 m/s。最终答案为 (5.0 ± 0.2) m/s。

5. Combining Uncertainties for Powers and Roots | 幂与方根运算中不确定度的合成

When a measured quantity is raised to a power n, multiply the percentage uncertainty by |n|. For the volume of a cube from side length L, volume = L³, so the percentage uncertainty in volume is three times the percentage uncertainty in L. For a square root, n = ½, so the percentage uncertainty is multiplied by ½.

当测量量被乘方 n 次时，百分不确定度要乘以 |n|。例如立方体边长 L 的体积 = L³，因此体积的百分不确定度 = 3 × L 的百分不确定度。平方根时 n = ½，即乘以 ½。

For V = L³: %u(V) = 3 × %u(L)

6. Standard Deviation and Repeated Measurements | 标准差与重复测量

If you repeat

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