Treatment of Uncertainties | 不确定度的处理

📚 Treatment of Uncertainties | 不确定度的处理

Every measured value in physics has an uncertainty. This is not a mistake; it reflects the limited precision of instruments and the variability in repeated readings. CIE A Level questions regularly ask you to estimate, combine and quote uncertainties correctly. This article sets out the core methods.

每个物理测量值都有不确定度。它不是错误，而是仪器精度有限和重复读数变化的表现。CIE A Level 经常要求正确估计、合成和表示不确定度。本文给出核心方法。

1. Why Uncertainty Matters | 为什么不确定度很重要

A measurement is useful only if its uncertainty is known. Writing ‘L = 1.25 m’ without an uncertainty claims that the length lies exactly at 1.25 m, which is never true. The accepted form is L = (1.25 ± 0.01) m.

只有知道不确定度，测量才有意义。只写 ‘L = 1.25 m’ 相当于声称长度恰为 1.25 m，这不可能。正确形式是 L = (1.25 ± 0.01) m。

The symbol ± gives the absolute uncertainty. It tells us the interval in which the true value is expected to lie: from 1.24 m to 1.26 m.

符号 ± 表示绝对不确定度，它给出真值预计所在区间：从 1.24 m 到 1.26 m。

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2. Random and Systematic Errors | 随机误差与系统误差

Random errors cause readings to scatter above and below the true value. They can be reduced by taking repeat readings and averaging.

随机误差使读数在真值上下波动。可通过多次读数取平均来减小。

Systematic errors shift all readings in one direction, for example a zero error on a meter or a badly calibrated scale. Averaging does not remove systematic errors; you must correct the instrument or subtract the offset.

系统误差使所有读数朝同一方向偏移，例如电表零误差或刻度校准不当。取平均不能消除系统误差；必须校正仪器或减去偏移量。

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3. Accuracy and Precision | 准确度与精确度

Accuracy describes how close a measurement is to the accepted true value. Precision describes how closely repeated readings agree with one another.

准确度描述测量值接近公认真值的程度；精确度描述重复读数彼此一致的程度。

A reading can be precise but inaccurate, for example if a digital balance has a zero offset: repeated readings may be very close, yet all are too high.

读数可能精确但不准确，例如电子天平有零位偏移：重复读数非常接近，但全部偏高。

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4. Absolute, Fractional and Percentage Uncertainty | 绝对、相对和百分比不确定度

The absolute uncertainty Δx has the same unit as the quantity. Fractional uncertainty is Δx / x, and percentage uncertainty is (Δx / x) × 100%.

绝对不确定度 Δx 与测量量单位相同。相对不确定度为 Δx / x，百分比不确定度为 (Δx / x) × 100%。

Example: length L = (0.48 ± 0.02) m. Absolute uncertainty = 0.02 m; percentage uncertainty = (0.02 / 0.48) × 100% = 4.2%.

例：长度 L = (0.48 ± 0.02) m。绝对不确定度 = 0.02 m；百分比不确定度 = (0.02 / 0.48) × 100% = 4.2%。

Percentage uncertainty is especially useful when comparing the quality of measurements with different sizes or units.

在比较不同大小或不同单位的测量质量时，百分比不确定度尤其有用。

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5. Reading Uncertainties for Analogue and Digital Instruments | 模拟和数字仪器的读数不确定度

For analogue instruments such as rulers, thermometers and moving-coil meters, the reading uncertainty is at least half of the smallest division. On a ruler with 1 mm divisions, Δ = 0.5 mm = 0.0005 m.

对于直尺、温度计、动圈式电表等模拟仪器，读数不确定度至少为最小分度值的一半。分度 1 mm 的直尺，Δ = 0.5 mm = 0.0005 m。

For digital instruments, the uncertainty is usually quoted as ± the last significant digit, for example a digital voltmeter reading 1.48 V has ΔV = ±0.01 V.

数字仪器的不确定度通常为末位数字 ±1，例如数字电压表读数为 1.48 V 时，ΔV = ±0.01 V。

Some instruments have a manufacturer’s percentage uncertainty or a quoted accuracy instead; always read the question carefully.

有些仪器给出制造商的百分比不确定度或标称准确度；务必仔细读题。

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6. Uncertainty from Repeated Measurements | 重复测量的不确定度

When several readings are taken, use the mean as the best estimate. A simple CIE method gives the absolute uncertainty as half the range:

多次读数时，以平均值为最佳估计值。CIE 简单的处理方法是绝对不确定度取极差的一半：

Δx = (x_max − x_min) ÷ 2

Example: repeated times 1.52 s, 1.60 s, 1.56 s, 1.54 s. Mean = 1.555 s; range = 0.08 s; Δt = 0.04 s. Write t = (1.56 ± 0.04) s if rounded sensibly.

例：重复时间为 1.52 s、1.60 s、1.56 s、1.54 s。平均值 = 1.555 s；极差 = 0.08 s；Δt = 0.04 s。合理修约后写 t = (1.56 ± 0.04) s。

The standard deviation is also acceptable, but CIE usually expects the half-range method unless the question states otherwise.

标准差也可接受，但除非题目另有说明，CIE 通常要求半极差法。

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7. Combining Uncertainties: Addition and Subtraction | 组合不确定度：加减运算

For quantities added or subtracted, add absolute uncertainties.

对于相加或相减的量，应相加绝对不确定度。

If Q = A + B or Q = A − B, ΔQ = ΔA + ΔB

Example: two masses m₁ = (100 ± 2) g and m₂ = (50 ± 1) g. Total m = 150 g; Δm = 2 + 1 = 3 g, so m = (150 ± 3) g.

例：两个质量为 m₁ = (100 ± 2) g、m₂ = (50 ± 1) g。总质量 m = 150 g；Δm = 2 + 1 = 3 g，所以 m = (150 ± 3) g。

For a difference, the uncertainties still add: the worst case may be high for one quantity and low for the other.

对于差值，不确定度仍然相加：最不利情况可能一个量偏高、另一个偏低。

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8. Combining Uncertainties: Multiplication, Division and Powers | 组合不确定度：乘除和幂运算

For products and quotients, add percentage or fractional uncertainties.

对于乘法或除法，应相加百分比或相对不确定度。

If Q = A × B or Q = A ÷ B, %ΔQ = %ΔA + %ΔB

Example: speed v = s / t with s = (100 ± 5) m, t = (20 ± 1) s. %Δs = 5%, %Δt = 5%; %Δv = 10%. v = 5.0 m s⁻¹; Δv = 10% × 5.0 = 0.5 m s⁻¹.

例：速度 v = s / t，其中 s = (100 ± 5) m，t = (20 ± 1) s。%Δs = 5%，%Δt = 5%；%Δv = 10%。v = 5.0 m

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