Measurements and their errors | 测量与误差

📚 Measurements and their errors | 测量与误差

Measurement is the foundation of all experimental physics. Every law, every equation, and every prediction ultimately rests on data obtained from measurements. Understanding how to take measurements, how to express them with appropriate precision, and how to quantify the associated uncertainties is therefore not merely a procedural skill — it is a fundamental part of doing physics correctly.

测量是一切实验物理学的基础。每一条定律、每一个方程、每一项预测，最终都建立在测量所获得的数据之上。因此，理解如何进行测量、如何以恰当的精度表达测量结果、以及如何量化相关的不确定度，不仅仅是一项操作技能——这是正确进行物理研究的基本素养。

1. SI Units and Base Quantities | SI单位与基本量

The International System of Units (SI) defines seven base quantities from which all other physical quantities are derived. In A-level physics, you are expected to know the base units and to be able to express derived units in terms of them. The seven base quantities are: mass (kilogram, kg), length (metre, m), time (second, s), current (ampere, A), temperature (kelvin, K), amount of substance (mole, mol), and luminous intensity (candela, cd).

国际单位制（SI）定义了七个基本量，所有其他物理量都由它们导出。在A-level物理中，你需要熟记这些基本单位，并能够用它们表示导出单位。七个基本量分别是：质量（千克，kg）、长度（米，m）、时间（秒，s）、电流（安培，A）、温度（开尔文，K）、物质的量（摩尔，mol）和发光强度（坎德拉，cd）。

Derived units are combinations of base units. For example, velocity is measured in metres per second (m s⁻¹), acceleration in metres per second squared (m s⁻²), and force in newtons (N), where 1 N = 1 kg m s⁻². When solving problems, always check that your final units are consistent — this is known as dimensional analysis and is a powerful tool for spotting errors.

导出单位是基本单位的组合。例如，速度的单位为米每秒（m s⁻¹），加速度的单位为米每二次方秒（m s⁻²），力的单位为牛顿（N），其中 1 N = 1 kg m s⁻²。解题时，务必检查最终单位是否一致——这被称为量纲分析，是发现错误的有力工具。

2. Prefixes and Standard Form | 词头与科学计数法

Physical quantities span an enormous range of magnitudes — from the size of a nucleus (about 10⁻¹⁵ m) to the distance to distant galaxies (about 10²⁶ m). To manage this range, SI prefixes are used to denote multiples and submultiples of units. You should be familiar with the common prefixes: pico (p, 10⁻¹²), nano (n, 10⁻⁹), micro (μ, 10⁻⁶), milli (m, 10⁻³), centi (c, 10⁻²), kilo (k, 10³), mega (M, 10⁶), giga (G, 10⁹), and tera (T, 10¹²).

物理量所跨越的数量级范围极大——从原子核的尺寸（约10⁻¹⁵ m）到遥远星系的距离（约10²⁶ m）。为了应对如此宽广的范围，SI词头被用来表示单位的倍数和分数。你需要熟悉常用词头：皮（p, 10⁻¹²）、纳（n, 10⁻⁹）、微（μ, 10⁻⁶）、毫（m, 10⁻³）、厘（c, 10⁻²）、千（k, 10³）、兆（M, 10⁶）、吉（G, 10⁹）和太（T, 10¹²）。

When performing calculations, always convert quantities to base units first. For example, a measurement of 2.5 ms should be rewritten as 2.5 × 10⁻³ s before substituting into an equation. Scientific notation with powers of ten ensures consistency and reduces the risk of arithmetic errors. This is particularly important when dealing with equations that involve several quantities raised to powers, such as kinetic energy Eₖ = ½mv².

在进行计算时，务必先将所有量转换为基本单位。例如，2.5 ms 应改写为 2.5 × 10⁻³ s，然后才能代入方程。使用以10的幂表示的科学计数法可以保证一致性，并降低算术错误的风险。这在处理涉及多个量的幂次方程时尤为重要，例如动能 Eₖ = ½mv²。

3. Order of Magnitude Estimates | 数量级估算

An order of magnitude is a factor of ten. When making an order-of-magnitude estimate, you round every quantity to the nearest power of ten and then perform the calculation. This technique is extremely useful for checking whether an answer is physically reasonable and for making quick approximations in unfamiliar situations.

一个数量级即一个十的因子。在进行数量级估算时，将每个量四舍五入到最接近的十的幂，然后进行计算。这项技术在检验答案在物理上是否合理、以及在陌生情境中做快速近似时极为有用。

For example, to estimate the mass of air in a typical classroom of dimensions 10 m × 8 m × 3 m, take the volume as 240 m³ ≈ 2 × 10² m³ and the density of air as about 1 kg m⁻³. The mass is therefore of the order of 2 × 10² kg, i.e. a few hundred kilograms. Practice making such estimates regularly — AQA examination questions often ask you to select a reasonable estimate from a list of options.

例如，要估算典型教室（10 m × 8 m × 3 m）中空气的质量，取体积为240 m³ ≈ 2 × 10² m³，空气密度约为1 kg m⁻³。因此质量的数量级为 2 × 10² kg，即几百千克。平时要经常练习这类估算——AQA考试题目常常要求你从选项中选出合理的估算值。

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

Every measurement is subject to error. It is crucial to distinguish between two types: random errors and systematic errors. A random error causes readings to fluctuate unpredictably around the true value. These arise from unpredictable variations in the environment, the instrument, or the observer. Random errors can be reduced by taking multiple readings and calculating the mean.

每一次测量都会受到误差的影响。区分两类误差至关重要：随机误差和系统误差。随机误差使读数围绕真值不可预测地波动，它们源于环境、仪器或观测者的不可预测变化。随机误差可以通过多次读数并计算平均值来减小。

A systematic error causes readings to be consistently offset from the true value in one direction. Common sources include a poorly calibrated instrument, zero error, or a faulty experimental technique. Systematic errors cannot be reduced by repetition — they must be identified and corrected by recalibrating the equipment or redesigning the procedure. For example, if a balance reads 0.5 g when empty, every mass measurement will be 0.5 g too high.

系统误差使读数持续地朝一个方向偏离真值。常见来源包括仪器校准不当、零位误差或实验方法缺陷。系统误差无法通过重复测量来减小——必须通过重新校准设备或改进实验流程来识别和纠正。例如，如果一台天平在空载时读数为0.5 g，那么每一次质量测量都会偏高0.5 g。

Accuracy describes how close a measurement is to the true value, and is affected primarily by systematic errors. Precision describes how closely repeated measurements agree with each other, and is affected by random errors. A measurement can be precise but inaccurate — for example, a stopwatch that runs consistently fast gives precise timings that are nevertheless systematically wrong.

准确度描述测量值接近真值的程度，主要受系统误差影响。精密度描述重复测量结果之间的一致性，主要受随机误差影响。一次测量可能精密度高但准确度低——例如，一块稳定偏快的手表给出的时间测量很精密，但系统性地不准确。

5. Uncertainty and Resolution | 不确定度与分辨率

The uncertainty of a measurement is an interval within which the true value is expected to lie. It is not the same as an error — uncertainty is a quantitative statement about the quality of a measurement, whereas error is the difference between the measured value and the true value. The uncertainty of a single reading is typically taken as half the resolution of the instrument. For a ruler marked in millimetres, the resolution is 1 mm, so the uncertainty of a single reading is ±0.5 mm.

测量不确定度是真值预期所在的区间。它不等同于误差——不确定度是对测量质量的定量表述，而误差是测量值与真值之间的差值。单次读数的不确定度通常取仪器分辨率的一半。对于以毫米为刻度的直尺，分辨率是1 mm，因此单次读数的不确定度为 ±0.5 mm。

When several readings are taken, the uncertainty can be estimated using the range: if readings are spread between x_min and x_max, the uncertainty is approximately (x_max − x_min)/2. The mean of the readings is quoted as the best estimate, and the uncertainty as the spread. For example, if four readings of the diameter of a wire are 0.82 mm, 0.84 mm, 0.83 mm and 0.85 mm, the mean is 0.835 mm and the uncertainty is (0.85 − 0.82)/2 = 0.015 mm. The final answer should be quoted as 0.835 ± 0.015 mm.

当进行多次读数时，可以使用极差来估计不确定度：如果读数分布在 x_min 和 x_max 之间，则不确定度约为 (x_max − x_min)/2。取读数的平均值作为最佳估计值，并以散布范围作为不确定度。例如，四次测量金属丝直径的结果为0.82 mm、0.84 mm、0.83 mm和0.85 mm，则平均值为0.835 mm，不确定度为 (0.85 − 0.82)/2 = 0.015 mm。最终结果应表示为 0.835 ± 0.015 mm。

6. Absolute, Fractional and Percentage Uncertainty | 绝对、分数与百分比不确定度

Uncertainty can be expressed in three equivalent ways. The absolute uncertainty is the actual interval quoted with the measurement, for example 5.0 ± 0.1 cm. The fractional uncertainty is the ratio of the absolute uncertainty to the measured value:

不确定度可以用三种等价的方式表达。绝对不确定度是与测量值一起给出的实际区间，例如 5.0 ± 0.1 cm。分数不确定度是绝对不确定度与测量值的比值：

fractional uncertainty = absolute uncertainty ÷ measured value

For the example above, the fractional uncertainty is 0.1 ÷ 5.0 = 0.02. The percentage uncertainty is the fractional uncertainty multiplied by 100%, giving 2%. Percentage uncertainty is the most commonly used form in examination questions, as it allows the uncertainties of different quantities to be compared directly.

对上述例子，分数不确定度为 0.1 ÷ 5.0 = 0.02。百分比不确定度是分数不确定度乘以100%，即2%。百分比不确定度是考试题目中最常用的形式，因为它允许直接比较不同量的不确定度。

When quoting a final result, the uncertainty must be rounded to one significant figure, and the measured value must be rounded to the same decimal place. For example, if a calculation gives 3.4567 ± 0.123 s, the correct quotation is 3.46 ± 0.12 s. Quoting more decimal places than the uncertainty justifies is misleading and will be penalised in examinations.

在给出最终结果时，不确定度应保留一位有效数字，测量值应与不确定度对齐到相同的小数位。例如，如果计算得到 3.4567 ± 0.123 s，正确的表达应为 3.46 ± 0.12 s。保留比不确定度所允许的更多小数位会产生误导，在考试中会被扣分。

7. Combining Uncertainties | 不确定度的合成

When a result is calculated from several measured quantities, the uncertainties of the inputs must be combined to give the uncertainty of the final result. Two simple rules cover most situations encountered at A-level.

当一个结果由多个测量量计算得出时，必须对各输入量的不确定度进行合成，以得到最终结果的不确定度。两条简单规则覆盖了A-level阶段遇到的大部分情况。

Rule 1 — Addition and subtraction: If y = a + b or y = a − b, the absolute uncertainties add. For example, if a = 12.3 ± 0.1 and b = 4.56 ± 0.02, then a − b = 7.74 ± 0.12. Note that uncertainties always add — even when the quantities are subtracted, the uncertainties do not cancel.

规则一——加法与减法：如果 y = a + b 或 y = a − b，则绝对不确定度相加。例如，若 a = 12.3 ± 0.1，b = 4.56 ± 0.02，则 a − b = 7.74 ± 0.12。注意不确定度总是相加——即使量是相减的，不确定度也不会抵消。

Rule 2 — Multiplication and division: If y = a × b or y = a ÷ b, the percentage uncertainties add. For example, if the current is measured as 2.0 ± 0.1 A (5%) and the resistance as 10.0 ± 0.2 Ω (2%), then the power P = I²R requires special treatment. Since I is squared, its percentage uncertainty is doubled. The total percentage uncertainty is therefore 2 × 5% + 2% = 12%.

规则二——乘法与除法：如果 y = a × b 或 y = a ÷ b，则百分比不确定度相加。例如，若电流测量值为 2.0 ± 0.1 A（5%），电阻为 10.0 ± 0.2 Ω（2%），则功率 P = I²R 需要特殊处理。由于 I 被平方，其百分比不确定度需加倍。因此总百分比不确定度为 2 × 5% + 2% = 12%。

When a quantity is raised to a power n, its percentage uncertainty is multiplied by n. For a quantity multiplied by a constant, the fractional uncertainty is unchanged. These rules allow you to propagate uncertainties through almost any equation encountered in the AQA specification.

当一个量被提升到n次幂时，其百分比不确定度乘以n。当一个量乘以常数时，分数不确定度不变。这些规则让你能够在AQA考纲中几乎任何方程中传播不确定度。

8. Significant Figures | 有效数字

The number of significant figures in a measurement reflects the precision of the instrument used. A value of 3.50 g has three significant figures, whereas 3.5 g has two. The trailing zero in 3.50 is significant because it indicates that the measurement was made to the nearest 0.01 g. Zeros at the beginning of a number, such as 0.0035, are not significant — this value has two significant figures.

一个测量结果的有效数字位数反映了所用仪器的精密度。3.50 g 有三位有效数字，而 3.5 g 有两位。3.50 末尾的零是有效的，因为它表明测量精确到0.01 g。数字开头的零，如0.0035，不是有效数字——该数值有两位有效数字。

As a general rule, the final result of a calculation should be quoted to the same number of significant figures as the least precise piece of data used. If you multiply 2.5 (2 s.f.) by 3.75 (3 s.f.), the answer should be quoted to 2 significant figures: 9.4, not 9.375. During intermediate steps, keep extra digits to avoid rounding errors, but round only at the very end.

一般规则是：计算的最终结果应保留与所用数据中有效数字位数最少的那个相同的位数。如果用 2.5（2位有效数字）乘以 3.75（3位有效数字），答案应保留2位有效数字：9.4，而不是9.375。在中间步骤中保留额外数字以避免舍入误差，但最终才进行四舍五入。

When combining a measured value with its uncertainty, the number of decimal places of the value should match that of the uncertainty. A final answer of v = 12.3456 ± 0.5 m s⁻¹ should be corrected to v = 12.3 ± 0.5 m s⁻¹. This consistency ensures that the quoted result honestly reflects the quality of the measurement.

当将测量值与其不确定度一起给出时，测量值的小数位数应与不确定度一致。最终答案 v = 12.3456 ± 0.5 m s⁻¹ 应修正为 v = 12.3 ± 0.5 m s⁻¹。这种一致性确保所引用的结果真实地反映测量的质量。

9. Graphs and Error Bars | 图表与误差棒

Graphs are powerful tools for displaying experimental data and for determining relationships between variables. When plotting data, the independent variable is placed on the x-axis and the dependent variable on the y-axis. Axes should be labelled with the quantity and its unit, for example “time / s” rather than just “time”.

图表是展示实验数据、确定变量之间关系的强大工具。在绘图时，自变量放在x轴，因变量放在y轴。坐标轴应标注量与单位，例如”时间 / s”而不是仅仅”时间”。

Each data point on a graph should be accompanied by an error bar showing the uncertainty of the measurement. The length of the bar represents ±1 absolute uncertainty in the vertical direction for an uncertainty in the dependent variable, and in the horizontal direction for the independent variable. A best-fit line should then be drawn so that it passes through all error bars, if possible, with roughly equal numbers of points above and below the line.

图上的每个数据点都应有误差棒来显示该测量的不确定度。误差棒的长度表示垂直方向（因变量的不确定度）或水平方向（自变量的不确定度）的 ±1 绝对不确定度。然后绘制最佳拟合线，使其尽可能穿过所有误差棒，且线上下方的数据点数量大致相等。

The gradient of the best-fit line and its intercept with the y-axis are often the quantities of interest. The uncertainty in the gradient can be estimated by drawing lines of maximum and minimum slope that still pass through the error bars; the gradient of the best-fit line is then quoted as the mean of these two extreme gradients, with half the difference taken as the uncertainty.

最佳拟合线的斜率及其与y轴的截距通常是要寻找的目标量。斜率的不确定度可以通过绘制仍能穿过所有误差棒的最大和最小斜率线来估算；最佳拟合线的斜率取这两条极端斜率的平均值，而不确定度取二者差值的一半。

10. The Relationship Between Variables | 变量之间的关系

A straight-line graph through the origin indicates direct proportionality: y ∝ x. However, many physical relationships are not linear. The power equation P = I²R gives a parabolic curve when P is plotted against I. To establish a linear relationship, one can plot P against I², which yields a straight line of gradient R.

通过原点的直线图表示正比关系：y ∝ x。然而，许多物理关系不是线性的。功率方程 P = I²R 在 P 对 I 绘图时给出抛物线。为了建立线性关系，可以绘制 P 对 I² 的图，这将得到一条斜率为 R 的直线。

This technique of linearisation is extremely important in practical work. For an exponential decay, N = N₀e^(−λt), taking the natural logarithm of both sides gives ln N = ln N₀ − λt, which is linear with gradient −λ when ln N is plotted against t. You should be comfortable rearranging equations to achieve a linear form, extracting gradient and intercept, and interpreting these in physical terms.

这种线性化技巧在实验工作中极为重要。对于指数衰变 N = N₀e^(−λt)，对两边取自然对数得到 ln N = ln N₀ − λt，当绘制 ln N 对 t 的图时，这是一条斜率为 −λ 的直线。你应该熟练掌握重新整理方程以获得线性形式、提取斜率和截距、并从物理意义上解释它们。

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

Examiners consistently report the same misconceptions. First, students confuse accuracy with precision — remember that accuracy refers to the closeness to the true value, while precision refers to the spread of repeated readings. Second, students forget to convert units before calculation, particularly when mixing mm and m or g and kg in the same equation. Third, students quote too many significant figures in final answers.

考官反复报告同样的误解。第一，学生混淆准确度与精密度——记住准确度是指接近真值的程度，而精密度是指重复读数之间的散布程度。第二，学生在计算前忘记换算单位，特别在同一方程中混用mm和m或g和kg时。第三，学生在最终答案中保留了过多的有效数字。

A fourth common error is adding absolute uncertainties when multiplying quantities. Remember: for addition and subtraction, add absolute uncertainties; for multiplication and division, add percentage uncertainties; for powers, multiply the percentage uncertainty by the power. A fifth error is ignoring the uncertainty of instruments when drawing graphs — always include error bars.

第四个常见错误是在乘法运算时相加绝对不确定度。记住：加减法时相加绝对不确定度；乘除法时相加百分比不确定度；幂次时将百分比不确定度乘以幂次。第五个错误是在绘图时忽略仪器的不确定度——务必包含误差棒。

Finally, when quoting a final measured value with its uncertainty, ensure the precision is consistent. The uncertainty should always be stated to one significant figure, and the measurement rounded to match. Practise these habits early so they become automatic in the examination room.

最后，在给出最终测量值及其不确定度时，确保精度一致。不确定度应始终保留一位有效数字，测量值四舍五入与之匹配。尽早养成这些习惯，使它们在考场上成为自然反应。

12. Summary | 本章总结

Measurements and their errors form the bedrock of experimental physics. The key ideas to master are: the SI system of base and derived units; the use of prefixes and scientific notation; the distinction between random and systematic errors; the expression of uncertainty as absolute, fractional or percentage values; the rules for combining uncertainties; the correct use of significant figures; and the graphical treatment of data with error bars and best-fit lines.

测量与误差构成实验物理学的基石。需要掌握的核心要点包括：SI基本单位与导出单位体系；词头与科学计数法的使用；随机误差与系统误差的区别；以绝对、分数或百分比形式表达不确定度；不确定度的合成规则；有效数字的正确使用；以及带误差棒和最佳拟合线的数据图形处理方法。

These skills are assessed both in written examinations and in the practical endorsement at A-level. Mastery of this topic not only secures marks directly but also improves your performance across every other topic, because correct measurements and sound analysis underpin all experimental physics. Revise these concepts regularly and practise past-paper questions to build confidence.

这些技能在笔试和A-level实践考核中都会被评估。掌握这一主题不仅能直接获得分数，还能提高你在其他所有主题中的表现，因为正确的测量和可靠的分析支撑着所有实验物理学。定期复习这些概念并练习历年真题，以建立信心。

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Measurements and their errors | 测量与误差