📚 Calculating Uncertainties | 不确定度的计算
All experimental measurements in physics carry some degree of uncertainty. Knowing how to quantify and combine these uncertainties is a vital skill in IB Physics, enabling you to assess the reliability of your data and draw meaningful conclusions. This article guides you through the essential methods for calculating uncertainties, from single readings to complex derived quantities.
物理学中的所有实验测量都存在一定的不确定度。掌握如何量化和组合这些不确定度是IB物理的关键技能，它使你能够评估数据的可靠性并得出有意义的结论。本文将指导你掌握从单次读数到复杂导出量不确定度计算的基本方法。
1. Introduction to Uncertainties | 不确定度简介
s of our instruments and our senses. A measured value should therefore be reported as: best estimate ± absolute uncertainty.
每当你测量一个物理量，结果永远不是精确的。仪表上的读数、指针的偏转或直尺上的长度都包含固有的不确定度。这种不确定度量化了我们对测量结果的疑虑；它并不意味着犯了错误，而是反映了仪器和感官的极限。因此，测量值应报告为：最佳估计值 ± 绝对不确定度。
2. Types of Errors: Systematic and Random | 误差类型：系统误差与随机误差
Systematic errors shift all readings in a consistent direction. They arise from imperfect calibration, a zero offset (e.g. a balance that does not read zero when empty), or a drift in an instrument during an experiment. Random errors, on the other hand, cause readings to be scattered unpredictably on either side of the true value. They result from fluctuations in reaction time, unpredictable vibrations, or small variations in the measured quantity itself.
系统误差使所有读数朝同一个方向偏移。它们产生于校准不当、零点偏移（例如空载时天平不指零）或实验过程中仪器的漂移。另一方面，随机误差导致读数不可预测地分散在真值两侧。它们来源于反应时间的波动、不可预测的振动或被测物理量本身的微小变化。
Systematic errors cannot be reduced by repeated readings; instead they must be identified and corrected. Random errors, however, can be reduced by taking many readings and calculating a mean. In IB investigations, you are expected to distinguish between the two and account for them in your uncertainty analysis.
系统误差无法通过重复读数来减小；相反，必须识别并纠正它们。然而，随机误差可以通过多次读数并计算平均值来减小。在IB探究中，你应当区分这两种误差并在不确定度分析中加以考虑。
3. Accuracy and Precision | 准确度与精密度
Accuracy describes how close a measurement is to the true value. Precision describes the spread of repeated measurements, without reference to the true value. A set of measurements can be precise (small scatter) but inaccurate (far from the true value) if a systematic error is present. Conversely, measurements with poor precision will have a large random uncertainty, making it difficult to judge accuracy.
准确度描述测量值与真值的接近程度。精密度描述重复测量结果的分散程度，而不涉及真值。如果存在系统误差，一组测量可以是精密的（离散小）但不准确（远离真值）。相反，精密度差的测量会具有较大的随机不确定度，从而难以判断准确度。
Think of a target: precise but inaccurate shots cluster closely together but away from the bullseye; accurate and precise shots hit the bullseye with little scatter. In lab work, always comment on both aspects when evaluating your data.
试想一个靶子：精密度高但准确度低的射击紧密成群但偏离靶心；准确且精密的射击则击中靶心且离散度小。在实验工作中，评估数据时始终要从这两方面进行评论。
4. Absolute, Relative and Percentage Uncertainty | 绝对、相对和百分比不确定度
is the ratio Δx / x, a dimensionless number. The percentage uncertainty is (Δx / x) × 100%.
绝对不确定度（Δx）是以与测量值相同单位表示的误差范围。例如，测量长度为15.3 ± 0.2 cm，此时ΔL = 0.2 cm。相对不确定度是比值 Δx / x，是一个无量纲数。百分比不确定度则是 (Δx / x) × 100%。
the above length, relative uncertainty = 0.2/15.3 ≈ 0.0131; percentage uncertainty ≈ 1.31% asier to work with relative or percentage uncertainties for multiplications and divisions.
这些形式可以互相换算。对于上述长度，相对不确定度 = 0.2/15.3 ≈ 0.0131；百分比不确定度 ≈ 1.31%。在多步计算中组合不确定度时，乘除法使用相对或百分比不确定度要方便得多。
Percentage uncertainty = (Δx / x) × 100%
百分比不确定度 = (Δx / x) × 100%
5. Determining Uncertainty from a Single Measurement | 从单次测量确定不确定度
gth measurement is ±0.5 mm. This accounts for the difficulty in reading between the marks.
如果你使用模拟刻度（例如直尺或弹簧秤）只读取一个读数，绝对不确定度通常估计为最小刻度值的一半。对于具有毫米刻度的米尺，最小刻度为1 mm，因此单次长度测量的不确定度为 ±0.5 mm。这反映了读准刻度之间位置的难度。
e.g. ±0.01 g for a balance that reads to two decimal places. Always check the manufacturer' mes the uncertainty is given as a percentage of the reading plus a fixed number of digits.
对于数字仪器，情况更简单：不确定度通常取显示最末位的一个数字，例如一台可读到小数点后两位的天平，其不确定度为 ±0.01 g。务必查阅制造商手册；有时不确定度以读数的百分比加固定数字位数给出。
6. Uncertainty in Repeated Measurements | 重复测量的不确定度
at a measurement several times, the best estimate of the quantity is the arithmetic mean x̄ ty can be estimated by the half-range method, which is recommended for IB Physics: Δx = (xmax - xmin) / 2. This simple method gives a conservative uncertainty that covers the spread of your data.
当你多次重复某一测量时，物理量的最佳估计值是算术平均值 x̄。随机不确定度可以用半极差法估算，这是IB物理推荐的方法：Δx = (xmax - xmin) / 2。这种简单方法给出一个保守的不确定度，覆盖了数据的离散范围。
Δx = (xmax − xmin) / 2
Δx = (xmax − xmin) / 2
You may also encounter the use of standard deviation to estimate uncertainty, but for most IB internal assessments, the half-range is sufficient and clearly shows the source of the uncertainty. Always record the raw readings and the calculation steps.
你还可能遇到使用标准偏差估算不确定度的情况，但在大多数IB内部评估中，半极差就足够了，并能清晰地展示不确定度的来源。务必记录原始读数和计算步骤。
7. Combining Uncertainties: Addition and Subtraction | 组合不确定度：加减法
st-case scenario would be for the errors to reinforce each other. Therefore, ΔR = ΔA + ΔB.
当一个物理量 R 是测量值 A 与 B 的和或差（R = A + B 或 R = A − B）时，绝对不确定度简单地相加。之所以采用这一规则，是因为最不利的情况就是各个误差相互加强。因此，ΔR = ΔA + ΔB。
ΔR = ΔA + ΔB
ΔR = ΔA + ΔB
Example: L₂ = 80.0 cm. The absolute uncertainty is ΔL = 0.2 + 0.3 = 0.5 cm. Result: 80.0 ± 0.5 cm.
示例：你测量了两个长度：L₁ = 50.0 ± 0.2 cm，L₂ = 30.0 ± 0.3 cm。总长 L = L₁ + L₂ = 80.0 cm。绝对不确定度 ΔL = 0.2 + 0.3 = 0.5 cm。结果：80.0 ± 0.5 cm。
R = ΔA + ΔB. Never subtract uncertainties, as this would underestimate the possible range.
即使对于减法（R = A − B），不确定度仍然相加：ΔR = ΔA + ΔB。切勿将不确定度相减，因为这样会低估可能的误差范围。
8. Combining Uncertainties: Multiplication and Division | 组合不确定度：乘除法
hen the fractional uncertainty in R is the sum of the fractional uncertainties in A and B:
对于涉及乘法或除法的计算，相对不确定度（或百分比不确定度）相加。如果 R = A × B 或 R = A ÷ B，那么 R 的相对不确定度为 A 和 B 的相对不确定度之和：
ΔR/R = ΔA/A + ΔB/B
ΔR/R = ΔA/A + ΔB/B
Example: So ΔW/W = 0.05 + 0.05 = 0.10. Absolute ΔW = 0.10 × 20.0 = 2.0 J. Result: W = 20.0 ± 2.0 J.
示例：一个力 F = 10.0 ± 0.5 N 作用在距离 d = 2.0 ± 0.1 m 上。功 W = F × d = 20.0 J。F 的相对不确定度：0.5/10.0 = 0.05；d 的相对不确定度：0.1/2.0 = 0.05。因此 ΔW/W = 0.05 + 0.05 = 0.10。绝对 ΔW = 0.10 × 20.0 = 2.0 J。结果：W = 20.0 ± 2.0 J。
ive uncertainty in speed = 0.01 + 0.0182 = 0.0282, giving Δv ≈ 0.0282 × 9.09 = 0.26 m·s⁻¹.
该规则也适用于除法：如果一位跑者以11.0 ± 0.2 s跑完100.0 ± 1.0 m，速度 v = d / t。相对不确定度分别为 1.0/100.0 = 0.01 和 0.2/11.0 ≈ 0.0182，因此速度的相对不确定度 = 0.01 + 0.0182 = 0.0282，由此得到 Δv ≈ 0.0282 × 9.09 = 0.26 m·s⁻¹。
9. Combining Uncertainties: Power Functions | 组合不确定度：幂函数
wer, the relative uncertainty is multiplied by the modulus of that power. If R = Aⁿ, then:
当一个量进行幂运算时，相对不确定度乘以该幂的绝对值。如果 R = Aⁿ，那么：
ΔR/R = |n| × ΔA/A
ΔR/R = |n| × ΔA/A
Example: e calculated volume V ≈ 33.5 cm³, so ΔV = 0.15 × 33.5 ≈ 5.0 cm³. Hence V = 33.5 ± 5.0 cm³.
示例：球体体积 V = (4/3)πr³。如果半径 r = 2.0 ± 0.1 cm，r 的相对不确定度为 0.1/2.0 = 0.05。对于幂 3，ΔV/V = 3 × 0.05 = 0.15。计算出体积 V ≈ 33.5 cm³，因此 ΔV = 0.15 × 33.5 ≈ 5.0 cm³。于是 V = 33.5 ± 5.0 cm³。
. For instance, for g = 4π²l/T², the relative uncertainty in g is: Δg/g = Δl/l + 2 × ΔT/T.
如果公式包含多个变量且某些带有幂，先分别将各相对不确定度乘以相应指数绝对值，然后运用乘除法规则相加。例如，对于 g = 4π²l/T²，g 的相对不确定度为：Δg/g = Δl/l + 2 × ΔT/T。
Below is a summary table of the rules:
下面是规则总结表：
| Operation (运算) | Uncertainty rule (不确定度规则) |
|---|---|
| R = A ± B | ΔR = ΔA + ΔB |
| R = A × B or A ÷ B | ΔR/R = ΔA/A + ΔB/B |
| R = Aⁿ | ΔR/R = |n| × ΔA/A |
10. Uncertainty of a Gradient | 斜率
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