📚 Uncertainty Calculations in IB Physics | IB物理：不确定度计算方法
In physics, no measurement is perfectly exact. Understanding uncertainty allows you to state how much confidence you have in your results and to compare them meaningfully with theoretical predictions.
在物理中，任何测量都不是绝对精确的。理解不确定度能够让你说明对结果的置信程度，并能有意义地将结果与理论预测进行比较。
1. Introduction to Uncertainty | 不确定度导论
Uncertainty is the range of values within which the true value of a measurement is expected to lie. It is not a mistake; it is an inevitable part of every experimental measurement.
不确定度是测量真值预期所在的数值范围。它不是错误，而是每一次实验测量中不可避免的一部分。
A complete measurement is always written as:
一个完整的测量结果总是写成：
x = x₀ ± Δx
where x₀ is the best estimate and Δx is the absolute uncertainty. For example, a length of 5.4 cm measured with a ruler marked in millimetres might be written as 5.4 ± 0.1 cm.
其中 x₀ 是最佳估计值，Δx 是绝对不确定度。例如，用毫米刻度尺测量的长度 5.4 cm 可写成 5.4 ± 0.1 cm。
2. Absolute, Fractional and Percentage Uncertainty | 绝对、分数和百分比不确定度
Absolute uncertainty has the same unit as the measurement itself. Fractional uncertainty is the ratio of the absolute uncertainty to the measured value. Percentage uncertainty is the fractional uncertainty multiplied by 100%.
绝对不确定度与测量值本身具有相同单位。分数不确定度是绝对不确定度与测量值的比值。百分比不确定度是分数不确定度乘以 100%。
If a current is measured as 2.50 ± 0.05 A, then:
如果电流测量值为 2.50 ± 0.05 A，则：
- Absolute uncertainty = 0.05 A
- Fractional uncertainty = 0.05 / 2.50 = 0.020
- Percentage uncertainty = 0.020 × 100% = 2.0%
- 绝对不确定度 = 0.05 A
- 分数不确定度 = 0.05 / 2.50 = 0.020
- 百分比不确定度 = 0.020 × 100% = 2.0%
Fractional uncertainty = Δx / x₀
Percentage uncertainty = (Δx / x₀) × 100%
3. Instrument Uncertainty | 仪器不确定度
The uncertainty due to the instrument itself is usually taken as half the smallest scale division for analogue instruments, or the smallest digital increment for digital instruments. Many IB exams state this explicitly.
由仪器本身引入的不确定度，通常取模拟仪器最小分度的一半，或数字仪器的最小数字增量。许多 IB 考试题目会明确给出这一规则。
- Ruler with 1 mm divisions: uncertainty ±0.5 mm (if measured from one end) but often ±1 mm when aligning ends.
- Digital stopwatch reading 12.34 s: uncertainty ±0.01 s.
- Analogue ammeter with scale divisions of 0.02 A: uncertainty ±0.01 A.
- 分度为 1 mm 的刻度尺：不确定度为 ±0.5 mm（如果从一端测量），但需对齐端部时常取 ±1 mm。
- 数字秒表显示 12.34 s：不确定度为 ±0.01 s。
- 分度为 0.02 A 的模拟电流表：不确定度为 ±0.01 A。
In practice, you should include both the instrument resolution and any reading estimation. For example, when taking multiple readings, the range of values may give a better estimate of uncertainty than the instrument alone.
实际中，你应该同时考虑仪器分辨率和读数估计。例如，当进行多次读数时，值域可能比单独仪器给出更好的不确定度估计。
4. Random and Systematic Errors | 随机误差与系统误差
Random errors cause measurements to scatter around the true value. They can be reduced by taking repeated measurements and averaging. Systematic errors cause measurements to be consistently too high or too low, often due to calibration problems or zero errors.
随机误差使测量值围绕真值散布。可以通过重复测量并取平均来减小。系统误差使测量值始终偏高或偏低，常见原因包括校准问题或零位误差。
Uncertainty analysis deals mainly with random errors, because systematic errors cannot be detected by repeating measurements. A precise measurement has small random uncertainty; an accurate measurement has small systematic error.
不确定度分析主要处理随机误差，因为重复测量无法发现系统误差。精确测量具有较小的随机不确定度；准确测量具有较小的系统误差。
5. Combining Uncertainties: Addition and Subtraction | 不确定度合并：加减法
When measurements are added or subtracted, the absolute uncertainties are added. This is because uncertainties always add constructively in the worst-case scenario.
当测量值相加或相减时，绝对不确定度相加。因为在最坏情况下不确定度总是同向叠加。
If A = a ± Δa and B = b ± Δb, then for P = A + B or P = A − B:
如果 A = a ± Δa，B = b ± Δb，则对于 P = A + B 或 P = A − B：
ΔP = Δa + Δb
Example: A mass of 150.0 ± 0.5 g is added to a container of mass 25.0 ± 0.1 g. The total mass is 175.0 ± 0.6 g.
示例：质量为 150.0 ± 0.5 g 的物体加到质量为 25.0 ± 0.1 g 的容器中。总质量为 175.0 ± 0.6 g。
For subtraction, for instance finding the mass of liquid, the same rule applies: if total mass is 175.0 ± 0.6 g and container mass is 25.0 ± 0.1 g, liquid mass = 150.0 ± 0.7 g (−0.6 and −0.1 combine to 0.7).
对于减法，例如求液体质量，同样规则适用：若总质量为 175.0 ± 0.6 g，容器质量为 25.0 ± 0.1 g，则液体质量 = 150.0 ± 0.7 g（0.6 与 0.1 合并为 0.7）。
6. Combining Uncertainties: Multiplication and Division | 不确定度合并：乘除法
When measurements are multiplied or divided, their fractional (or percentage) uncertainties are added.
当测量值相乘或相除时，其分数（或百分比）不确定度相加。
If P = A × B or P = A / B, then:
如果 P = A × B 或 P = A / B，则：
ΔP / |P| = Δa / |a| + Δb / |b|
Example: The area of a rectangle is length × width. If length = 5.0 ± 0.1 cm and width = 2.0 ± 0.1 cm, the area is 10.0 cm². The fractional uncertainty is 0.1/5.0 + 0.1/2.0 = 0.02 + 0.05 = 0.07, so ΔA = 0.07 × 10.0 = 0.7 cm². Thus area = 10.0 ± 0.7 cm².
示例：矩形面积为长 × 宽。如果长 = 5.0 ± 0.1 cm，宽 = 2.0 ± 0.1 cm，则面积为 10.0 cm²。分数不确定度为 0.1/5.0 + 0.1/2.0 = 0.02 + 0.05 = 0.07，所以 ΔA = 0.07 × 10.0 = 0.7 cm²。因此面积 = 10.0 ± 0.7 cm²。
7. Uncertainty for Powers and Roots | 幂函数与根式的不确定度
When a measurement is raised to a power n, the fractional uncertainty is multiplied by that power. This applies to roots too, since a square root is a power of ½.
当测量值取 n 次幂时，分数不确定度乘以该幂次。根式同样适用，因为平方根就是 ½ 次幂。
If P = Aⁿ, then:
如果 P = Aⁿ，则：
ΔP / |P| = |n| × (Δa / |a|)
Example: The volume of a cube is side³. If side = 2.0 ± 0.1 cm, fractional uncertainty in side is 0.1/2.0 = 0.05. Therefore fractional uncertainty in volume is 3 × 0.05 = 0.15. Volume = 8.0 cm³, so ΔV = 0.15 × 8.0 = 1.2 cm³. Volume = 8.0 ± 1.2 cm³.
示例：立方体体积为边长的立方。如果边长 = 2.0 ± 0.1 cm，边长的分数不确定度为 0.1/2.0 = 0.05。因此体积的分数不确定度为 3 × 0.05 = 0.15。体积 = 8.0 cm³，所以 ΔV = 0.15 × 8.0 = 1.2 cm³。体积 = 8.0 ± 1.2 cm³。
For other functions such as sin, cos, tan, and natural logarithms, use the maximum and minimum values to find the uncertainty.
对于其他函数如 sin、cos、tan 和自然对数，使用最大值和最小值来计算不确定度。
8. Uncertainties in Tables and Graphs | 数据表与图形中的不确定度
In IB physics, data tables must include uncertainties in the column headers or next to values. Typically each measured value is written as a central value with a ± uncertainty, and the unit is stated in the header.
在 IB 物理中，数据表必须在列标题或值旁边包含不确定度。通常每个测量值写成中心值 ± 不确定度，单位在标题中说明。
When plotting graphs, each data point must have vertical and horizontal error bars if both variables have uncertainties. The length of an error bar represents ± uncertainty in that direction.
绘图时，如果两个变量都有不确定度，每个数据点必须有垂直和水平误差线。误差线的长度代表该方向上的 ± 不确定度。
Example table format:
示例表格格式：
| Length / cm | Period / s |
| 20.0 ± 0.5 | 0.90 ± 0.02 |
| 40.0 ± 0.5 | 1.27 ± 0.02 |
| 60.0 ± 0.5 | 1.56 ± 0.02 |
9. Best-fit Lines and Error Bars | 最佳拟合直线与误差线
The best-fit line should pass within the error bars of as many points as possible and be smooth. It is used to determine the gradient and intercept of the relationship. To estimate the uncertainty in the gradient, draw the steepest and shallowest lines that are still consistent with the error bars.
最佳拟合直线应尽可能穿过多数点的误差线，并且平滑。它用于确定关系的斜率和截距。要估计斜率的不确定度，画出仍然与误差线一致的最陡和最缓的直线。
The uncertainty in the gradient is then:
斜率的不确定度为：
Δgradient = (max gradient − min gradient) / 2
Similarly, the uncertainty in the y-intercept is obtained from the spread of intercepts of the maximum and minimum gradient lines.
类似地，y 截距的不确定度由最大和最小斜率直线的截距范围获得。
10. Special Functions: Trigonometric and Logarithmic | 特殊函数：三角函数与对数
For a function y = f(x), the uncertainty in y is found by evaluating f at the maximum and minimum values of x:
对于函数 y = f(x)，y 的不确定度通过在 x 的最大值和最小值处计算 f 得到：
Δy = |f(x₀ + Δx) − f(x₀ − Δx)| / 2
Example: For θ = 30.0° ± 0.5°, sin θ = 0.5000. Evaluate sin(30.5°) = 0.5075 and sin(29.5°) = 0.4924. Half their difference is (0.5075 − 0.4924)/2 = 0.0076. So sin θ = 0.500 ± 0.008.
示例：对于 θ = 30.0° ± 0.5°，sin θ = 0.5000。计算 sin(30.5°) = 0.5075，sin(29.5°) = 0.4924。它们差值的一半为 (0.5075 − 0.4924)/2 = 0.0076。所以 sin θ = 0.500 ± 0.008。
For natural logarithms, the same maximum-minimum method works. This approach is safer than trying to memorise derived rules for every function.
对于自然对数，同样的最大-最小方法适用。这种方法比试图记住每个函数的推导规则更安全。
11. Calculations and Significant Figures | 计算与有效数字
The final uncertainty should be quoted to one significant figure unless it begins with 1 (then two significant figures may be used). The measured value is rounded to the same decimal place as the uncertainty.
最终不确定度通常只保留一位有效数字，除非它以 1 开头（此时可用两位有效数字）。测量值四舍五入到与不确定度相同的小数位。
Example: If a calculated resistance is 8.3471 Ω and its uncertainty is 0.426 Ω, the uncertainty rounds to 0.4 Ω (one significant figure), and the result is written as 8.3 ± 0.4 Ω. If the uncertainty were 0.142 Ω, you might write 8.35 ± 0.14 Ω.
示例：如果计算出的电阻为 8.3471 Ω，不确定度为 0.426 Ω，则不确定度四舍五入为 0.4 Ω（一位有效数字），结果写成 8.3 ± 0.4 Ω。如果不确定度为 0.142 Ω，则可以写成 8.35 ± 0.14 Ω。
In intermediate calculations, keep extra digits to avoid rounding errors, but never claim more precision than the least precise measurement allows.
在中间计算中，保留额外数字以避免舍入误差，但绝不能声称比最不精确的测量允许的精度更高。
12. Conclusion | 结论
Uncertainty calculations are a core skill in IB physics. They allow you to express the reliability of experimental data and to justify whether a result agrees with a theoretical prediction within experimental error. Practise combining uncertainties with addition, multiplication, and powers, and always represent uncertainties clearly in tables and graphs.
不确定度计算是 IB 物理的核心技能。它让你能够表达实验数据的可靠性，并判断结果是否在实验误差范围内与理论预测一致。多加练习加法、乘法和幂运算的不确定度合并，并始终在表格和图形中清楚表示不确定度。
By mastering these rules, you will be well prepared for both Paper 3 practical questions and your internal assessment.
掌握这些规则后，你将在 Paper 3 实验题和内部评估中游刃有余。
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