📚 Measurement Uncertainty in A-Level Physics: Assessment and Handling | A-Level 物理：不确定度的评定与处理方法
In experimental physics, no measurement is exact. Because of limited instrument precision, human reaction time and changing conditions, a measured value is always an estimate within a specific range. Uncertainty quantifies that range and tells us how much confidence we can place in the result. This article explains how to evaluate and process uncertainties in CIE A-Level Physics.
在实验物理中，没有任何测量是绝对精确的。由于仪器精度有限、人的反应时间以及条件变化，测量值始终是在一定范围内的估计值。不确定度量化了这一范围，并告诉我们结果的可信程度。本文讲解 CIE A-Level 物理中如何评定与处理不确定度。
1. What Is Measurement Uncertainty? | 什么是测量不确定度？
Uncertainty is a range around the best estimate within which the true value is expected to lie. It is not a mistake; it is an unavoidable feature of measurement.
不确定度是围绕最佳估计值的一个区间，真值预期落在这个区间内。它不是错误，而是测量中不可避免的特性。
The standard way to record a measured value is:
标准测量记录方式为：
value = best estimate ± absolute uncertainty
测量值 = 最佳估计值 ± 绝对不确定度
For example, a length written as (25.0 ± 0.2) cm means the best estimate is 25.0 cm and the true length is likely between 24.8 cm and 25.2 cm.
例如，长度写为 (25.0 ± 0.2) cm 表示最佳估计值为 25.0 cm，真实长度很可能在 24.8 cm 与 25.2 cm 之间。
2. Random and Systematic Errors | 随机误差与系统误差
Errors are not the same as uncertainties, but they are closely related. Random errors cause unpredictable fluctuations in readings, while systematic errors cause a consistent shift from the true value.
误差与不确定度不同，但二者密切相关。随机误差使读数产生不可预测的波动，系统误差则使测量结果始终偏离真值。
Repeating measurements helps reduce random errors because the average tends to cancel them. Systematic errors cannot be reduced by repetition; they must be removed by calibration, zeroing the instrument or improving the experimental technique.
重复测量有助于减小随机误差，因为取平均可以部分抵消它们的波动。系统误差不能通过重复测量来消除，而必须通过校准、调零或改进实验方法来修正。
| Type | Effect | Example | How to reduce |
| Random (随机) | Low precision (精密度低) | Air currents, vibration (气流、振动) | Repeat and average (重复取平均) |
| Systematic (系统) | Low accuracy (准确度低) | Zero error, uncalibrated instrument (零误差、未校准) | Calibrate or correct method (校准或修正方法) |
3. Representing Uncertainty | 不确定度的表示方法
Uncertainty can be expressed in three ways: absolute, fractional and percentage. Absolute uncertainty has the same unit as the measured quantity; fractional uncertainty is a ratio; percentage uncertainty is the fractional uncertainty multiplied by 100%.
不确定度可以用三种方式表示：绝对不确定度、分数不确定度和百分不确定度。绝对不确定度与测量量具有相同单位；分数不确定度是一个比值；百分不确定度是分数不确定度乘以100%。
fractional uncertainty = Δx / x
percentage uncertainty = (Δx / x) × 100%
For a current measured as 5.2 ± 0.1 A, the fractional uncertainty is 0.1 / 5.2 ≈ 0.019, and the percentage uncertainty is about 1.9%.
例如，电流测量值为 5.2 ± 0.1 A，其分数不确定度为 0.1 / 5.2 ≈ 0.019，百分不确定度约为 1.9%。
Always state which form you are using, because adding fractional uncertainties is valid only when quantities are multiplied or divided.
务必说明使用的是哪种表示形式，因为只有在乘除运算中才可以直接相加分数不确定度。
4. Estimating Uncertainty from One Reading | 单次读数的不确定度估计
For a single reading, the uncertainty is usually related to the resolution of the instrument. On a digital display, the last digit is often taken as the resolution, and the uncertainty is commonly taken as half of that digit.
对于单次读数，不确定度通常与仪器分辨率有关。数字显示中，最后一位通常视为分辨率，不确定度一般取该最小位的一半。
For an analogue scale, the smallest division is identified first. A common convention is that the uncertainty is half the smallest scale division.
对于模拟刻度，先确定最小分度值，通常取最小分度的一半作为不确定度。
Example: a ruler with millimetre markings has an uncertainty of ± 0.5 mm. A digital balance reading 25.45 g may carry an uncertainty of ± 0.005 g because the last digit is 0.01 g.
例如：分度值为毫米的刻度尺，其不确定度为 ± 0.5 mm；数字天平读数 25.45 g 的不确定度可取 ± 0.005 g，因为最小位是 0.01 g。
The actual uncertainty may be larger if the item cannot be positioned consistently, so judgement and experimental conditions must also be considered.
如果物体无法稳定放置，实际不确定度可能更大，因此还要考虑判断和实验条件的影响。
5. Best Estimate from Repeated Measurements | 重复测量的最佳估计值
When a measurement is repeated several times, the best estimate of the true value is the arithmetic mean.
当同一测量重复多次时，真值的最佳估计值是算术平均值。
mean = Σx / n
平均值 = Σx / n
For a small set of readings, the uncertainty is often taken as half the range:
对于少量读数，不确定度常取极差的一半：
Δx = (x_max − x_min) / 2
Δx = (x_max − x_min) / 2
Example: timings of 1.02 s, 1.05 s, 1.04 s and 1.06 s give a mean of 1.0425 s, a range of 0.04 s and an uncertainty of 0.02 s. The final result should be recorded as 1.04 ± 0.02 s.
例如：计时结果为 1.02 s、1.05 s、1.04 s 和 1.06 s，平均值为 1.0425 s，极差为 0.04 s，不确定度为 0.02 s。最终结果应记录为 1.04 ± 0.02 s。
If one reading is very different from the others, do not discard it without investigation; an error in recording or an external disturbance may explain it.
如果某个读数与其他数值相差很大，不要未经调查就舍去，可能的原因包括记录错误或外界干扰。
6. Combining Uncertainties: Addition and Subtraction | 不确定度的合成：加减法
When two measured quantities are added or subtracted, the absolute uncertainties must be added.
当两个测量量相加或相减时，绝对不确定度必须相加。
If y = a + b or y = a − b, then Δy = Δa + Δb
若 y = a + b 或 y = a − b，则 Δy = Δa + Δb
Example: (2.0 ± 0.1) m added to (3.0 ± 0.2) m gives 5.0 ± 0.3 m. The same rule applies to subtraction: (5.0 ± 0.1) m − (3.0 ± 0.2) m = 2.0 ± 0.3 m.
例如：(2.0 ± 0.1) m 加上 (3.0 ± 0.2) m 得到 5.0 ± 0.3 m。减法也同理：(5.0 ± 0.1) m − (3.0 ± 0.2) m = 2.0 ± 0.3 m。
Do not add percentage uncertainties for addition or subtraction; the absolute uncertainty is the meaningful quantity here.
注意，加减法中不要将百分不确定度相加，此时绝对不确定度才是有效的量。
7. Combining Uncertainties: Multiplication, Division and Powers | 不确定度的合成：乘法、除法与幂
When quantities are multiplied or divided, the fractional or percentage uncertainties are added.
当两个量相乘或相除时，分数不确定度或百分不确定度相加。
If y = a × b or y = a / b, then Δy / y = Δa / a + Δb / b
若 y = a × b 或 y = a / b，则 Δy / y = Δa / a + Δb / b
For a power, the fractional uncertainty is multiplied by the power.
对于幂函数，分数不确定度要乘以幂指数。
If y = aⁿ, then Δy / y = n(Δa / a)
若 y = aⁿ，则 Δy / y = n(Δa / a)
Worked example: mass m = (2.00 ± 0.02) kg and volume V = (0.50 ± 0.01) m³. Density ρ = m / V = 4.00 kg m⁻³. The fractional uncertainty in m is 0.02 / 2.00 = 0.01; in V it is 0.01 / 0.50 = 0.02. Total fractional uncertainty is 0.03, so the absolute uncertainty is 0.03 × 4.00 = 0.12 kg m⁻³. Therefore ρ = (4.0 ± 0.1) kg m⁻³.
示例：质量 m = (2.00 ± 0.02) kg，体积 V = (0.50 ± 0.01) m³。密度 ρ = m / V = 4.00 kg m⁻³。m 的分数不确定度为 0.02 / 2.00 = 0.01；V 的分数不确定度为 0.01 / 0.50 = 0.02。总分数不确定度为 0.03，所以绝对不确定度为 0.03 × 4.00 = 0.12 kg m⁻³。因此 ρ = (4.0 ± 0.1) kg m⁻³。
Constants such as 2, π or 1000 do not contribute fractional uncertainty unless they are also measured quantities.
像 2、π 或 1000 这样的常数不贡献分数不确定度，除非它们本身也是测量量。
8. Uncertainties in More Complex Functions | 复杂函数中的不确定度
If a measured quantity is used inside a function such as sin, cos, log or an exponential, the fractional uncertainty rules do not apply directly. A reliable method is to calculate the result using the maximum and minimum possible input values.
如果测量量出现在 sin、cos、log 或指数函数中，分数不确定度规则不能直接套用。可靠的方法是分别用输入量的最大值和最小值计算结果。
The uncertainty in the final result is then half the difference between the maximum and minimum
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