Uncertainty Calculations: Steps and Methods | 不确定度数值计算：步骤与方法

📚 Uncertainty Calculations: Steps and Methods | 不确定度数值计算：步骤与方法

In A-Level Physics, no measurement is perfectly exact. Every reading carries an uncertainty, and you are expected to calculate how these uncertainties affect your final results. This article explains the numerical steps for combining uncertainties in a clear, exam-focused way.

在 A-Level 物理中，任何测量都不可能是绝对精确的。每一次读数都带有不确定度，而你需要学会计算这些不确定度如何影响最终结果。本文将用清晰且紧扣考试的方式，讲解合成不确定度的数值计算步骤。

1. Absolute, Fractional and Percentage Uncertainty | 绝对、分数和百分比不确定度

Absolute uncertainty is the simplest form. It states the range within which the true value probably lies, and it carries the same unit as the measured quantity. For example, a length of 25.0 cm with an absolute uncertainty of 0.2 cm means the true length is likely between 24.8 cm and 25.2 cm.

绝对不确定度是最基本的形式。它表示真实值可能落在的范围，并且与测量量具有相同的单位。例如，长度为 25.0 cm，绝对不确定度为 0.2 cm，意味着真实长度很可能在 24.8 cm 到 25.2 cm 之间。

Fractional uncertainty is the ratio of the absolute uncertainty to the measured value. Percentage uncertainty is simply the fractional uncertainty multiplied by 100%.

分数不确定度是绝对不确定度与测量值之比。百分比不确定度则是分数不确定度乘以 100%。

Fractional uncertainty = Δx / x

Percentage uncertainty = (Δx / x) × 100%

When comparing two measurements, percentage uncertainty is especially useful because it shows which measurement is relatively more precise.

在比较两个测量值时，百分比不确定度特别有用，因为它能直接显示哪一个测量值相对更精确。

2. Determining Uncertainty from Instruments | 从仪器读数中确定不确定度

The uncertainty of a single measurement depends on the instrument. For analogue instruments such as a metre ruler or ammeter, the uncertainty is usually taken as half of the smallest division.

单次测量的不确定度取决于所用仪器。对于米尺、电流表等模拟式仪器，不确定度通常取最小刻度的一半。

For example, a ruler marked in millimetres has a smallest division of 1 mm, so the absolute uncertainty is ±0.5 mm. If the smallest division is 0.1 A on an ammeter, the uncertainty is ±0.05 A.

例如，最小刻度为 1 mm 的刻度尺，其绝对不确定度为 ±0.5 mm。如果电流表的最小刻度是 0.1 A，则不确定度为 ±0.05 A。

For digital instruments such as a digital stopwatch or electronic balance, the uncertainty is usually taken as the smallest digit shown, for example ±0.01 g on a balance or ±0.01 s on a stopwatch.

对于数字式仪器，如数字秒表或电子天平，不确定度通常取最后显示的最小数字位，例如天平的 ±0.01 g 或秒表的 ±0.01 s。

If the reading fluctuates, you should estimate the range of fluctuation and use half of that range as the uncertainty.

如果读数发生波动，你应该估计波动的范围，并取该范围的一半作为不确定度。

3. Repeated Measurements and Mean Uncertainty | 重复测量与平均不确定度

Repeating a measurement reduces the effect of random errors. When you have several readings, the best estimate of the true value is the mean. The absolute uncertainty can be estimated as half the range of the readings.

重复测量可以减小随机误差的影响。当你得到多个读数时，对真实值的最佳估计是平均值。绝对不确定度可以估计为读数极差的一半。

Δx = (maximum reading − minimum reading) / 2

For example, if five readings of the time period of a pendulum are 1.20 s, 1.24 s, 1.22 s, 1.19 s and 1.25 s, the mean is 1.22 s. The range is 0.06 s, so the absolute uncertainty is ±0.03 s.

例如，摆的周期五次读数分别为 1.20 s、1.24 s、1.22 s、1.19 s 和 1.25 s，平均值为 1.22 s。极差为 0.06 s，因此绝对不确定度为 ±0.03 s。

When you finally quote the mean value, round the uncertainty to one significant figure and give the mean to the same number of decimal places as the uncertainty.

最后给出平均值时，应将不确定度保留一位有效数字，并使平均值与不确定度保留相同的小数位数。

4. Combining Uncertainties: Addition and Subtraction | 合成不确定度：加减法

When measurements are added or subtracted, their absolute uncertainties always add together. This is true even when one quantity is subtracted from another.

当测量值相加或相减时，它们的绝对不确定度总是相加。即使一个量减去另一个量，这一规则也成立。

Δp = Δa + Δb

Suppose p = a + b or p = a − b. The final absolute uncertainty is simply the sum of the two absolute uncertainties.

假设 p = a + b 或 p = a − b，最终的绝对不确定度就是两个绝对不确定度之和。

Example: a = 12.0 ± 0.2 cm and b = 7.5 ± 0.3 cm. Then p = a + b = 19.5 cm, and Δp = 0.2 + 0.3 = 0.5 cm, so p = 19.5 ± 0.5 cm.

例如：a = 12.0 ± 0.2 cm，b = 7.5 ± 0.3 cm。则 p = a + b = 19.5 cm，Δp = 0.2 + 0.3 = 0.5 cm，因此 p = 19.5 ± 0.5 cm。

Similarly, if q = a − b = 4.5 cm, the uncertainty is still Δq = 0.5 cm. Subtraction does not reduce the uncertainty.

类似地，若 q = a − b = 4.5 cm，不确定度仍然是 Δq = 0.5 cm。减法并不会减小不确定度。

5. Combining Uncertainties: Multiplication and Division | 合成不确定度：乘除法

When measurements are multiplied or divided, you must combine fractional or percentage uncertainties. The fractional uncertainties add together.

当测量值相乘或相除时，必须使用分数不确定度或百分比不确定度进行合成。分数不确定度要相加。

Δp / p = Δa / a + Δb / b

For p = a × b or p = a / b, the final fractional uncertainty is the sum of the individual fractional uncertainties.

对于 p = a × b 或 p = a / b，最终分数不确定度等于各分数不确定度之和。

Example: a = 10.0 ± 0.1 cm and b = 4.0 ± 0.2 cm. Then p = a × b = 40 cm². The fractional uncertainty is 0.1/10.0 + 0.2/4.0 = 0.01 + 0.05 = 0.06. The percentage uncertainty is 6%, so Δp = 0.06 × 40 = 2.4 cm². Therefore p = 40.0 ± 2.4 cm².

例如：a = 10.0 ± 0.1 cm，b = 4.0 ± 0.2 cm。则 p = a × b = 40 cm²。分数不确定度为 0.1/10.0 + 0.2/4.0 = 0.01 + 0.05 = 0.06。百分比不确定度为 6%，因此 Δp = 0.06 × 40 = 2.4 cm²。所以 p = 40.0 ± 2.4 cm²。

This rule also applies when more than two quantities are multiplied or divided: simply add all fractional uncertainties.

当两个以上的量相乘或相除时，该规则同样适用：只需将所有分数不确定度相加。

6. Powers, Roots and More Complex Equations | 幂、根与更复杂的方程

When a measurement is raised to a power, the fractional uncertainty is multiplied by that power. This is the most common place where students forget to multiply.

当测量值有幂次时，分数不确定度要乘以该幂次。这是学生最常忘记乘上去的地方。

If p = xⁿ, then Δp / p = n × Δx / x

For example, if p = x³, then the fractional uncertainty in p is three times the fractional uncertainty in x. If p = √x = x^(1/2), then the fractional uncertainty is one-half of the fractional uncertainty in x.

例如，如果 p = x³，那么 p 的分数不确定度是 x 的分数不确定度的三倍。如果 p = √x = x^(1/2)，则分数不确定度是 x 的分数不确定度的一半。

In more complex equations, identify each measured variable. Then use the basic rules stage by stage. For example, if q = 2 × x² / y, then Δq/q = 2 × Δx/x + Δy/y. The constant 2 has no uncertainty because it is exact.

在更复杂的方程中，先找出每个被测量的变量。然后分阶段使用基本规则。例如，若 q = 2 × x² / y，则 Δq/q = 2 × Δx/x + Δy/y。常数 2 没有不确定度，因为它是精确值。

If an equation contains a sum or difference inside a product, first find the absolute uncertainty of the sum or difference, then convert it into a fractional uncertainty before combining with other terms.

如果方程中乘积内部含有加法或减法，先计算该加法或减法的绝对不确定度，然后转换为分数不确定度，再与其他项合成。

7. Constants and Final Reporting | 常数与最终结果报告

Mathematical constants such as π, 2, 4, and 1/3 are exact and contribute no uncertainty. Physical constants from data tables also normally have uncertainties much smaller than the measured values, so they can be treated as exact unless the question states otherwise.

π、2、4、1/3 等数学常数是精确的，不贡献不确定度。来自数据表的物理常数通常不确定度远小于测量值，因此在题目没有特别说明时可按精确值处理。

When reporting a final value, you should quote the uncertainty with one significant figure, and the value to the same decimal place as the uncertainty.

报告最终结果时，不确定度应保留一位有效数字，数值应与不确定度保留相同的小数位数。

Correct	V = 2260 ± 60 mm³
Incorrect	V = 2261.95 ± 60.4 mm³

The first example is better because a one-significant-figure uncertainty keeps the result honest. An uncertainty with many digits is misleadingly precise.

第一个例子更好，因为一位有效数字的不确定度让结果更真实。保留多位数字的不确定度会带来虚假的精确感。

8. Uncertainties in Graphs: Error Bars and Worst Lines | 不确定度的图形分析：误差棒与最斜线

Graphical analysis is a powerful way to estimate quantities and their uncertainties. On a graph, each data point can be drawn with an error bar in the vertical and horizontal directions.

图形分析是估计物理量及其不确定度的有力工具。在图上，每个数据点都可以画出垂直和水平方向的误差棒。

The line of best fit should be drawn so that it passes as close as possible to all points and their error bars. The gradient of this line is the best estimate of the gradient.

最佳拟合线应尽可能靠近所有数据点及其误差棒。这条线的斜率就是斜率的最佳估计值。

To find the uncertainty in the gradient, draw the steepest and shallowest lines that still pass through all error bars. These are called the worst acceptable lines.

为了求斜率的不确定度，画出仍然能穿过所有误差棒的最陡线和最缓线。这些线称为最可接受线。

Δgradient = (gradient_max − gradient_min) / 2

Use triangle coordinates from the graph to calculate each gradient. Do not confuse the uncertainty with the size of the graph; use actual scaled values.

计算每条线的斜率时，应从图上读取三角形对应的真实坐标值。不要把不确定度与图的大小混淆；要使用真实的比例数值。

The y-intercept of the worst lines can also be used to estimate the uncertainty in the intercept.

最斜线的 y 轴截距也可用来估计截距的不确定度。

9. Worked Example: Volume of a Cylinder | 完整例题：圆柱体体积

Problem: A cylinder has diameter d = 12.0 ± 0.1 mm and length L = 20.0 ± 0.2 mm. Calculate the volume V and its uncertainty.

题目：一个圆柱体的直径 d = 12.0 ± 0.1 mm，长度 L = 20.0 ± 0.2 mm。计算体积 V 及其不确定度。

Step 1: Write the equation for volume. The volume of a cylinder is

第一步：写出体积公式。圆柱体的体积为

V = π × (d/2)² × L = π × d² × L / 4

Step 2: Calculate the best value. V = π × (12.0)² × 20.0 / 4 = π × 144 × 5 = 720π ≈ 2262 mm³.

第二步：计算最佳值。V = π × (12.0)² × 20.0 / 4 = π × 144 × 5 = 720π ≈ 2262 mm³。

Step 3: Write the fractional uncertainty expression. Since V ∝ d² × L,

第三步：写出分数不确定度表达式。由于 V ∝ d² × L，

ΔV / V = 2 × Δd / d + ΔL / L

Step 4: Substitute the numbers.

第四步：代入数值。

ΔV / V = 2 × (0.1 / 12.0) + (0.2 / 20.0) = 0.0167 + 0.01 = 0.0267

The percentage uncertainty is about 2.7%.

百分比不确定度约为 2.7%。

Step 5: Find the absolute uncertainty. ΔV = 0.0267 × 2262 ≈ 60 mm³. Therefore V = 2260 ± 60 mm³.

第五步：求绝对不确定度。ΔV = 0.0267 × 2262 ≈ 60 mm³。因此 V = 2260 ± 60 mm³。

10. Common Mistakes and Examination Tips | 常见错误与考试技巧

One common mistake is adding percentage uncertainties when using addition or subtraction. For addition and subtraction, always add absolute uncertainties; for multiplication and division, use fractional or percentage uncertainties.

常见错误之一是在加减法中使用百分比不确定度相加。对于加减法，应使用绝对不确定度相加；对于乘除法，才使用分数或百分比不确定度。

Another mistake is ignoring a power. If a quantity is squared, the fractional uncertainty must be doubled. This is often tested in questions about area, kinetic energy and resistance.

另一个错误是忽略幂次。如果某个量是平方项，其分数不确定度必须翻倍。这在面积、动能和电阻相关题目中经常考查。

When drawing graphs, always plot the error bars before drawing the worst lines. Lines should be drawn using a sharp pencil and a clear ruler.

画图时，必须先画出误差棒，再画最斜线。直线应使用削尖的铅笔和透明的直尺绘制。

Finally, always show your working. Even if the final value is wrong, you can still earn method marks for correct fractional uncertainty equations and substitutions.

最后，一定要写出计算过程。即使最终数值错误，你仍然可以通过正确的分数不确定度表达式和代入过程获得步骤分。

11. Quick Checklist for Numerical Uncertainty Problems | 数值不确定度问题的快速检查清单

Use this checklist before you finish any uncertainty calculation.

在完成任何不确定度计算之前，请使用下面的检查清单。

Identify which quantities are measured and which are exact constants.

找出哪些量是测量值，哪些是精确常数。
For single readings, state the instrument uncertainty; for repeated readings, use half the range.

对于单次读数，说明仪器不确定度；对于重复读数，使用极差的一半。
Choose the correct combination rule: absolute for addition/subtraction, fractional for multiplication/division/powers.

选择正确的合成规则：加减法用绝对不确定度，乘除法、幂次用分数不确定度。
Multiply fractional uncertainty by the power for powers or roots.

对于幂或根，将分数不确定度乘以幂次。
Convert the final fractional uncertainty back to an absolute uncertainty if requested.

如果题目要求，将最终分数不确定度转换回绝对不确定度。
Quote your final answer with one significant figure in the uncertainty and the value matched to that precision.

报告最终答案时，不确定度保留一位有效数字，数值精度与之匹配。

Practising these steps consistently will help you avoid the most common traps and score full marks in uncertainty questions.

坚持练习这些步骤，将帮助你避开最常见的陷阱，并在不确定度相关题目中获得满分。

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Uncertainty Calculations: Steps and Methods | 不确定度数值计算：步骤与方法