Accuracy in Physics Measurements | 物理测量中的准确性

📚 Accuracy in Physics Measurements | 物理测量中的准确性

In A-Level Physics, the term ‘accuracy’ carries a precise technical meaning that extends far beyond everyday usage. It describes how closely a measured value agrees with the true or accepted value of a physical quantity. This distinction is fundamental to practical assessment in AQA specifications, where examiners expect candidates to evaluate and improve the reliability of their experimental data.

在 A-Level 物理中，”准确性”一词具有远超日常用法的精确技术含义。它描述的是测量值与被测物理量的真实值或公认值之间的接近程度。这一区分是 AQA 考纲实践评估的基础，考官期望考生能够评估并改进其实验数据的可靠性。

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1. Defining Accuracy in Physics | 物理学中的准确性定义

Accuracy is defined as the degree of closeness between a measured value and the true value of the quantity being measured. If an experiment yields results that cluster tightly around the accepted value — for example, measuring the acceleration due to gravity as 9.81 m s⁻², 9.82 m s⁻² and 9.80 m s⁻² when the true value is 9.81 m s⁻² — the measurements are said to be highly accurate.

准确性的定义是测量值与待测物理量真实值之间的接近程度。如果实验产生的结果紧密围绕公认值分布——例如测得重力加速度为 9.81 m s⁻²、9.82 m s⁻² 和 9.80 m s⁻²，而真实值为 9.81 m s⁻²——那么这些测量就被认为是高度准确的。

It is critical to recognise that accuracy is not an intrinsic property of a single reading; it is a judgement made by comparing measurements against an external standard. The true value can never be known perfectly in practice — it is an idealised concept. Instead, we rely on accepted values published by scientific bodies, such as the CODATA recommended values, or on calibrations traceable to national standards.

必须认识到，准确性并不是单次读数的固有属性；它是通过将测量结果与外部标准进行比较而做出的判断。在实践当中，真实值永远无法完美获知——它是一个理想化的概念。相反，我们依赖科学机构发布的公认值（如 CODATA 推荐值），或者依赖可溯源至国家标准的校准结果。

For AQA examinations, accuracy is assessed through your ability to identify whether systematic errors are present, to state how close results are to the true value, and to justify whether an experimental method is fit for purpose. A method that measures the resistivity of a wire to within 2% of the textbook value is accurate; one that consistently delivers values 15% too high is not, regardless of how consistent the readings appear.

在 AQA 考试中，准确性通过以下能力来评估：识别是否存在系统误差、说明结果与真实值的接近程度，以及论证实验方法是否适合其用途。一种能将金属丝电阻率测量值控制在教科书值 2% 以内的方法是准确的；而一种始终高出真实值 15% 的方法则是不准确的，无论读数看起来多么一致。

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2. Accuracy vs Precision | 准确性与精密度

Precision describes the degree of agreement between repeated measurements of the same quantity. If you repeat a measurement five times and obtain values that differ by tiny amounts, your data set is precise. However, precise data is not necessarily accurate — a badly calibrated balance may give extremely consistent readings that are all shifted away from the true mass by the same margin.

精密度描述的是对同一物理量进行重复测量时各结果之间的一致程度。如果你重复测量同一量五次，得到的数值差异极小，那么你的数据集就是精密的。然而，精密的数据不一定准确——一台校准不良的天平可能给出极其一致的读数，但所有读数都偏离真实质量相同的幅度。

Feature | 特征	Accuracy | 准确性	Precision | 精密度
Definition | 定义	Closeness to the true value | 与真实值的接近程度	Closeness of repeated readings to each other | 重复读数之间的接近程度
Affected by | 受何种因素影响	Systematic errors | 系统误差	Random errors | 随机误差
Improvement | 如何提高	Calibration, better method | 校准、改进方法	Repetition, averaging | 重复、取平均

Consider a target analogy. Precision is about the grouping of arrow holes; accuracy is about whether the holes are at the bullseye. You can achieve: (a) precise and accurate — tight grouping at the centre; (b) precise but inaccurate — tight grouping away from the centre; (c) imprecise but accurate on average — scattered holes centred around the bullseye; and (d) imprecise and inaccurate — scattered holes away from the centre.

考虑一个靶心类比。精密度涉及箭孔的聚集程度；准确性涉及箭孔是否位于靶心。可能出现：(a) 精密且准确——紧密地聚集在中心；(b) 精密但不准确——紧密聚集但不位于中心；(c) 不精密但平均准确——箭孔散乱但中心位于靶心；以及 (d) 不精密且不准确——箭孔散乱且偏离中心。

In AQA practical work, you must report both aspects. When describing a set of results, state that they are precise if the range of repeated readings is small, and state that they are accurate if the mean value is close to the accepted value. Examiners frequently award marks for explicitly contrasting these two terms rather than using them interchangeably.

在 AQA 实验操作中，你必须同时报告这两个方面。描述一组结果时，如果重复读数的极差很小，应说明它们是精密的；如果平均值接近公认值，应说明它们是准确的。考官通常会奖励明确区分这两个术语、而不是将它们混用的答案。

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3. Systematic and Random Errors | 系统误差与随机误差

An understanding of error types is essential for judging accuracy. A systematic error is a consistent, reproducible shift in readings caused by a flaw in the apparatus, the method, or the observer. Examples include a zero error on a balance, a stopwatch that runs slow, a ruler with a worn end, or heat losses in a calorimetry experiment that always reduce the measured temperature rise.

理解误差类型对于判断准确性至关重要。系统误差是由仪器、方法或观察者自身的缺陷引起的读数一致且可重现的偏移。例子包括天平的零点误差、走时偏慢的秒表、端部磨损的尺子，或量热实验中始终降低测得温升的热损失。

Systematic errors affect accuracy directly because they push every measurement away from the true value in the same direction. They cannot be eliminated by repeating readings or by averaging; repeating a flawed measurement simply reproduces the same flaw. The only remedies are calibration against a known standard, improving the experimental design, or applying a calculated correction factor.

系统误差直接影响准确性，因为它将每次测量都推向偏离真实值的同一方向。它无法通过重复读数或取平均来消除；重复一个有缺陷的测量只会重现同样的缺陷。唯一的补救方法是对照已知标准进行校准、改进实验设计，或应用计算出的修正因子。

A random error is a fluctuation in readings caused by unpredictable factors — slight variations in reaction time when using a stopwatch, electrical noise in a voltmeter, air currents disturbing a balance, or difficulty judging the exact endpoint of a pointer. Random errors affect precision, but their effect on the mean value is reduced by taking many readings and calculating the average.

随机误差是由不可预测因素引起的读数波动——使用秒表时反应时间上的细微波动、电压表中的电噪声、气流扰动天平，或难以判断指针的精确位置。随机误差影响精密度，但通过多次读数并计算平均值，其对平均值的影响会被减小。

Identifying the type of error in an experiment is a high-level skill in AQA. A clue to a systematic error is that the same wrong answer appears every time, even when the experiment is repeated under the same conditions. A clue to a random error is that repeated readings scatter around a central value. When asked to suggest improvements, link the treatment to the cause: control variables for systematic effects, repeat and average for random effects.

在实验中识别误差类型是 AQA 要求的高级技能。系统误差的线索是每次都出现相同的错误答案，即使在相同条件下重复实验也是如此。随机误差的线索是重复读数围绕中心值散乱分布。当被要求提出改进建议时，应将处理措施与原因联系起来：控制变量以应对系统效应，重复并取平均以应对随机效应。

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4. Measurement Uncertainty | 测量不确定度

Uncertainty is the quantitative expression of doubt about a measurement. Because no measurement is perfect, every reading should be quoted with an interval that is likely to contain the true value. If you measure a length as 25.4 mm with an uncertainty of ±0.2 mm, you are stating that the true value lies between 25.2 mm and 25.6 mm with reasonable confidence.

不确定度是对测量结果存疑程度的定量表达。由于任何测量都不完美，每次读数都应附带一个可能包含真实值的区间。如果你测得长度为 25.4 mm，不确定度为 ±0.2 mm，你就是在说明真实值以合理置信度位于 25.2 mm 到 25.6 mm 之间。

In AQA physics, the uncertainty of a single reading is typically taken as half the resolution of the instrument. A digital balance that displays mass to 0.1 g, for instance, gives an uncertainty of ±0.05 g per reading. An analogue voltmeter with a scale division of 0.5 V gives an uncertainty of ±0.25 V, though in practice you must also account for your ability to interpolate between divisions.

在 AQA 物理中，单次读数的不确定度通常取仪器分辨率的一半。例如，显示精度为 0.1 g 的数字天平，每次读数的不确定度为 ±0.05 g。刻度分度为 0.5 V 的模拟电压表，不确定度为 ±0.25 V，尽管在实际中还必须考虑你在分度之间进行内插的能力。

When repeated readings are taken, a better estimate of uncertainty is half the range of the readings. If you measure the time period of a pendulum as 1.02 s, 1.04 s, 1.03 s and 1.05 s, the mean is 1.035 s and half the range is (1.05 – 1.02)/2 = 0.015 s. The uncertainty is therefore quoted as ±0.015 s, which is often smaller than the uncertainty of a single reading.

当进行重复读数时，不确定度的更好估计量是读数极差的一半。如果你测得摆的周期为 1.02 s、1.04 s、1.03 s 和 1.05 s，平均值为 1.035 s，极差的一半为 (1.05 – 1.02)/2 = 0.015 s。因此不确定度应表示为 ±0.015 s，这通常小于单次读数的不确定度。

Always quote uncertainties to one significant figure, and round the measured value so that its last significant figure matches the decimal place of the uncertainty. A result written as 1.035 ± 0.05 s is poorly presented; the correct form is 1.04 ± 0.05 s. This convention ensures that the uncertainty is not hidden by excessive decimal places.

始终将不确定度保留一位有效数字，并将测量值四舍五入，使末位有效数字与不确定度的小数位对齐。写成 1.035 ± 0.05 s 的结果呈现不佳；正确形式应为 1.04 ± 0.05 s。这一惯例确保不确定度不会被过多的小数位所掩盖。

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5. Resolution of Instruments | 仪器分辨率

The resolution of an instrument is the smallest change in the quantity being measured that produces a detectable change in the reading. A ruler marked in millimetres has a resolution of 1 mm; a micrometer screw gauge typically has a resolution of 0.01 mm. Higher resolution does not automatically guarantee higher accuracy, but it is a necessary condition for precise measurement.

仪器的分辨率是待测量发生可检测变化的最小改变量。以毫米为刻度的尺子分辨率为 1 mm；千分尺通常分辨率为 0.01 mm。更高的分辨率并不自动保证更高的准确性，但它是精密测量的必要条件。

For digital instruments, the resolution is the value of the last displayed digit. A digital multimeter showing 5.32 V has a resolution of 0.01 V. For analogue instruments, the resolution is the smallest scale division, but the effective resolution may be finer if the user can interpolate between divisions. A metre rule with millimetre divisions can often be read to 0.5 mm by estimation, although examiners generally accept half a division as the uncertainty.

对于数字仪器，分辨率即最后一位显示数字所对应的值。显示 5.32 V 的数字万用表分辨率为 0.01 V。对于模拟仪器，分辨率是最小的刻度分度，但如果使用者能进行内插，有效分辨率可能更细。带毫米分度的米尺通常可以通过估读读到 0.5 mm，尽管考官通常接受半个分度作为不确定度。

Choosing the right instrument is a matter of matching resolution to the magnitude of the effect you are measuring. When measuring a 2 cm extension of a spring, a metre rule gives a 5% percentage uncertainty of ±0.5 mm on a 20 mm reading, which is acceptable. But when measuring a 0.5 mm extension, the same ruler gives a 100% uncertainty — a travelling microscope or a digital vernier calliper is required instead.

选择正确的仪器需要将分辨率与所测量效应的大小相匹配。测量弹簧 2 cm 的伸长量时，米尺在 20 mm 读数上给出 ±0.5 mm 的 5% 百分不确定度，这是可以接受的。但测量 0.5 mm 的伸长量时，同一把尺子给出 100% 的不确定度——此时需要使用读数显微镜或数字游标卡尺。

AQA questions frequently ask you to choose an appropriate measuring device for a given experiment. Your justification should refer to both the resolution of the instrument and the size of the quantity being measured. State the percentage uncertainty that each candidate instrument would produce, and select the one that gives an acceptably small uncertainty — typically below 5% — for the context of the experiment.

AQA 问题经常要求你为给定实验选择合适的测量设备。你的论证应同时涉及仪器的分辨率和待测量的量级。说明每个候选仪器将产生的百分不确定度，并选择在实验背景下给出可接受的小不确定度（通常低于 5%）的仪器。

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6. Absolute and Percentage Uncertainty | 绝对不确定度与百分不确定度

Absolute uncertainty is the raw uncertainty quoted in the same units as the measurement, such as ±0.05 g or ±0.5 °C. Percentage uncertainty expresses this as a fraction of the measured value multiplied by 100. For a mass of 50.00 g with an absolute uncertainty of ±0.05 g, the percentage uncertainty is (0.05 ÷ 50.00) × 100% = 0.1%.

绝对不确定度是以与测量值相同单位给出的原始不确定度，如 ±0.05 g 或 ±0.5 °C。百分不确定度将其表示为测量值的百分比：乘以 100。对于绝对不确定度为 ±0.05 g 的 50.00 g 质量，百分不确定度为 (0.05 ÷ 50.00) × 100% = 0.1%。

Percentage uncertainty is far more informative than absolute uncertainty when comparing measurements of different sizes. A ±1 mm uncertainty on a 1000 mm measurement is only 0.1%, but the same ±1 mm on a 10 mm measurement is 10%. Reporting both forms allows the reader to judge the quality of the measurement relative to its magnitude.

在比较不同大小的测量值时，百分不确定度比绝对不确定度更具信息量。1000 mm 测量值上的 ±1 mm 不确定度仅为 0.1%，但同样的 ±1 mm 在 10 mm 测量值上就是 10%。同时报告两种形式可以使读者相对于量值判断测量的质量。

There is an important mathematical relationship: percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%. Rearranging this allows you to calculate the absolute uncertainty from a known percentage, or to determine how large a measurement must be to achieve a target percentage uncertainty. For AQA calculations, always show your working and quote both forms when requested.

有一个重要的数学关系：百分不确定度 = (绝对不确定度 ÷ 测量值) × 100%。重新整理此式可以从已知百分比计算绝对不确定度，或确定为达到目标百分不确定度所需的最小测量量。对于 AQA 计算题，始终展示你的计算过程，并在要求时同时引用两种形式。

When timing many oscillations of a pendulum instead of one, the absolute uncertainty in the total time remains the same (limited by the stopwatch), but it is divided across more oscillations for the period calculation. Timing 20 oscillations with a total uncertainty of ±0.2 s gives a period uncertainty of ±0.01 s — a substantial improvement in percentage uncertainty compared with timing a single oscillation.

当测量摆的多次摆动而不是单次摆动时，总时间的绝对不确定度保持不变（受秒表限制），但在计算周期时分摊到更多次摆动上。测量 20 次摆动，总不确定度为 ±0.2 s，则周期不确定度为 ±0.01 s——与测量单次摆动相比，百分不确定度得到了显著改善。

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7. Combining Uncertainties | 不确定度的合成

Most experimental results are derived from several measured quantities combined by arithmetic operations. When uncertainties must be propagated through these calculations, AQA specifies simple rules. For addition and subtraction, add the absolute uncertainties. For example, if A = 12.3 ± 0.1 cm and B = 4.7 ± 0.2 cm, then A + B = 17.0 ± 0.3 cm and A − B = 7.6 ± 0.3 cm.

大多数实验结果是由若干测量量通过算术运算组合而成的。当不确定度必须通过这些计算进行传播时，AQA 规定了简单规则。对于加法和减法，将绝对不确定度相加。例如，如果 A = 12.3 ± 0.1 cm 且 B = 4.7 ± 0.2 cm，则 A + B = 17.0 ± 0.3 cm，A − B = 7.6 ± 0.3 cm。

For multiplication and division, add the percentage uncertainties. If V = 5.0 V ± 2% and I = 0.50 A ± 3%, then the resistance R = V ÷ I = 10.0 Ω, and the percentage uncertainty in R is 2% + 3% = 5%. The absolute uncertainty in R is therefore 5% of 10.0 Ω = ±0.5 Ω, giving R = 10.0 ± 0.5 Ω.

对于乘法和除法，将百分不确定度相加。如果 V = 5.0 V ± 2%，I = 0.50 A ± 3%，则电阻 R = V ÷ I = 10.0 Ω，R 的百分不确定度为 2% + 3% = 5%。因此 R 的绝对不确定度为 10.0 Ω 的 5% = ±0.5 Ω，即 R = 10.0 ± 0.5 Ω。

For powers, multiply the percentage uncertainty by the exponent. If r = 2.0 cm ± 2%, then r² = 4.0 cm² with percentage uncertainty 2 × 2% = 4%, and r³ = 8.0 cm³ with percentage uncertainty 3 × 2% = 6%. This rule is particularly important for calculations involving area, volume, and inverse-square relationships.

对于幂运算，将百分不确定度乘以指数。如果 r = 2.0 cm ± 2%，则 r² = 4.0 cm²，百分不确定度为 2 × 2% = 4%；r³ = 8.0 cm³，百分不确定度为 3 × 2% = 6%。这条规则对于涉及面积、体积和平方反比关系的计算尤其重要。

These rules should be applied step by step for complex expressions. For a quantity like k = (m ÷ t²), first calculate the percentage uncertainty in t² by doubling the percentage uncertainty in t, then add the percentage uncertainty in m. AQA mark schemes consistently award method marks for showing the uncertainty propagation clearly at each stage.

对于复杂表达式，应逐步应用这些规则。对于 k = (m ÷ t²) 这样的量，首先将 t 的百分不确定度加倍得到 t² 的百分不确定度，然后加上 m 的百分不确定度。AQA 评分标准始终奖励在每个阶段清晰展示不确定度传播过程的方法分。

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8. Significant Figures and Rounding | 有效数字与舍入

Significant figures are the digits of a number that carry meaning about its precision. The number of significant figures in a measurement should reflect the uncertainty of the instrument. A reading of 25.4 mm on a ruler with 1 mm divisions claims three significant figures, but the last digit is uncertain because of interpolation. Quoting more digits than justified is misleading.

有效数字是数字中承载精度信息的数位。测量值的有效数字位数应反映仪器的不确定度。在 1 mm 分度的尺子上读得 25.4 mm 声称三位有效数字，但由于内插，最后一位是可疑的。引用超出合理范围的更多位数会产生误导。

In AQA calculations, the final result should be given to the same number of significant figures as the least precise piece of data used in the calculation. If you multiply a length of 12.4 cm (3 s.f.) by a width of 5.2 cm (2 s.f.), the area should be rounded to 2 significant figures: 64 cm². Writing 64.48 cm² would falsely imply greater precision than the data allows.

在 AQA 计算中，最终结果应保留与计算中使用的精度最低的数据相同的有效数字位数。如果你将 12.4 cm（3 位有效数字）的长度乘以 5.2 cm（2 位有效数字）的宽度，面积应四舍五入到 2 位有效数字：64 cm²。写成 64.48 cm² 会虚假地暗示比数据所允许的更高的精度。

Be careful with zeros in significant figures. Leading zeros are never significant: 0.025 m has two significant figures. Trailing zeros after a decimal point are significant: 2.50 m has three significant figures. Trailing zeros without a decimal point are ambiguous: 2500 m could have two, three, or four significant figures, which is why scientific notation — 2.50 × 10³ m — is preferred in physics.

关于零的有效数字需特别小心。前导零不算有效数字：0.025 m 有两位有效数字。小数点后的尾随零算有效数字：2.50 m 有三位有效数字。没有小数点时尾随零是有歧义的：2500 m 可能有两位、三位或四位有效数字，因此物理中更倾向使用科学记数法——2.50 × 10³ m。

When rounding, apply standard rules: if the digit after the last significant figure is 5 or greater, round up; otherwise leave unchanged. Intermediate steps in multi-stage calculations should carry extra digits to avoid rounding errors, with rounding applied only to the final answer. Examiners penalise premature rounding because it systematically distorts the final result.

舍入时应用标准规则：如果末位有效数字之后的数字大于或等于 5，则进位；否则保持不变。多阶段计算的中间步骤应保留额外位数以避免舍入误差，仅在最终答案处进行舍入。考官会因过早舍入扣分，因为它会系统性地扭曲最终结果。

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9. Graphical Methods to Improve Accuracy | 提高准确性的图解方法

Graphical analysis is one of the most powerful tools in A-Level physics for extracting accurate results from imperfect data. Plotting a graph and calculating the gradient of a line of best fit averages out many random errors, because the line represents the overall trend rather than any single data point. This is far more reliable than using one pair of readings.

图解分析是 A-Level 物理中从不完美数据中提取准确结果的最有力工具之一。绘制图形并计算最佳拟合直线的斜率可以平均掉许多随机误差，因为直线代表整体趋势而非任何单个数据点。这比使用单一读数对要可靠得多。

The true gradient of a physical relationship is often a quantity with direct physical meaning — the gradient of a force–extension graph gives the spring constant, the gradient of a voltage–current graph gives resistance, and the gradient of an s–t² graph for falling objects gives half the acceleration due to gravity. To obtain the most accurate gradient, choose a line of best fit that has as many points above it as below it, and use two widely separated points on the line itself, not data points.

物理关系的真实斜率通常具有直接的物理意义——力–伸长量图的斜率给出劲度系数，电压–电流图的斜率给出电阻，下落物体的 s–t² 图的斜率给出重力加速度的一半。为获得最准确的斜率，应选择上下数据点数量大致均衡的最佳拟合直线，并使用直线上相距较远的两个点，而不是数据点本身。

Determining whether the line passes through the origin is another accuracy check. If theory predicts proportionality (y = kx) and your best-fit line has a significant nonzero intercept, a systematic error is likely present — perhaps a zero error in the apparatus or a background effect that was not subtracted. Examining the intercept is therefore diagnostic of systematic problems.

判断直线是否过原点也是准确性检查之一。如果理论预测正比关系 (y = kx)，而你的最佳拟合直线有明显的非零截距，那么很可能存在系统误差——也许是仪器的零点误差或未扣除的背景效应。因此检查截距是诊断系统问题的有效手段。

AQA also expects you to construct graphs with the independent variable on the x-axis and the dependent variable on the y-axis, using appropriate scales so that the plotted data occupy at least half of each axis. Plotting points with their error bars, where practical, allows you to judge whether a straight line fits within the uncertainty of every measurement — if it does, the line is a valid representation of the data.

AQA 还期望你将自变量放在 x 轴上、因变量放在 y 轴上，并使用合适的比例使绘制数据在每条轴上至少占一半。在可行的情况下绘制带有误差线的数据点，可以判断一条直线是否适合每个测量的不确定度范围——如果适合，这条直线就是数据的有效表示。

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10. Reducing Errors in the Laboratory | 实验室中的误差控制

Improving accuracy in practical work requires a systematic approach. First, calibrate all instruments against known standards before use. Zero the balance with the container on it, check the ruler for wear, and verify that the thermometer reads 0 °C in melting ice and 100 °C in boiling water. Calibration eliminates many systematic errors at their source.

在实验操作中提高准确性需要系统性的方法。首先，使用前用已知标准校准所有仪器。将容器置于天平上时清零、检查尺子磨损情况，并验证温度计在冰水混合物中读数为 0 °C、在沸水中读数为 100 °C。校正在源头消除了许多系统误差。

Second, take repeated readings and calculate the mean. The mean of N readings has a random uncertainty that decreases with the square root of N, so doubling the number of readings reduces the random component of uncertainty by about 30%. For time measurements, measuring many oscillations or many periods of rotation is particularly effective, as the timing uncertainty is spread across a larger total measurement.

其次，进行重复读数并计算平均值。N 次读数的平均值其随机不确定度随 N 的平方根而减小，因此将读数次数加倍可将随机不确定度分量减小约 30%。对于时间测量，测量多次摆动或多个旋转周期尤其有效，因为计时不确定度分摊到了更大的总测量量上。

Third, control the environmental variables. Keep temperatures stable by insulating apparatus or working away from direct sunlight. Ensure electrical circuits are in a steady state before recording readings — for example, wait for a thermistor circuit to reach thermal equilibrium. Stabilise the conditions so that the quantity you intend to measure is the only variable changing.

第三，控制环境变量。通过隔热或远离阳光直射来保持温度稳定。确保电路在记录读数前处于稳定状态——例如，等待热敏电阻电路达到热平衡。稳定条件，使你想要测量的量是唯一变化的量。

Fourth, use the correct technique. Read analogue scales perpendicularly to avoid parallax error — place your eye directly above the pointer and use a mirror scale if available. Read the bottom of a meniscus at eye level for liquid volumes. Consistent technique between readings reduces the random scatter that arises from inconsistent observation methods.

第四，使用正确的技术。垂直读取模拟刻度以避免视差误差——将眼睛置于指针正上方，如有反射镜刻度则加以利用。读取液体体积时在视线水平线上读取弯月面底部。一致的读取技术减小了因观察方法不一致而产生的随机散布。

Finally, consider whether the apparatus itself is appropriate for the scale of the measurement. A light gate is far more accurate than a manual stopwatch for measuring the period of a fast oscillation; a micrometer is essential for measuring the diameter of a wire that affects resistivity to the fourth power through r⁴. Matching instrument resolution to the sensitivity required by the formula is an examiner’s favourite improvement.

最后，考虑仪器本身是否适合测量的尺度。光门比手动秒表测量快速振荡周期要准确得多；测量导线直径时必须使用千分尺，因为其通过 r⁴ 影响电阻率。将仪器分辨率与公式所要求的灵敏度相匹配，是考官最青睐的改进建议。

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11. Worked Example: Evaluating Accuracy | 实例分析：评估准确性

A student determines the acceleration due to gravity using a simple pendulum. The length is measured as 1.000 m with a metre rule (±0.5 mm). The time for 20 oscillations is measured as 40.2 s with a stopwatch of resolution 0.1 s. The period T = 40.2 ÷ 20 = 2.01 s. Using g = 4π²L ÷ T², the student calculates g = 9.77 m s⁻².

一位学生使用单摆测定重力加速度。长度用米尺测量为 1.000 m（±0.5 mm）。20 次摆动的时间用分辨率为 0.1 s 的秒表测得为 40.2 s。周期 T = 40.2 ÷ 20 = 2.01 s。利用 g = 4π²L ÷ T²，学生计算得 g = 9.77 m s⁻²。

The accepted value of g is 9.81 m s⁻², so the experimental result is 0.04 m s⁻² or about 0.4% lower than the true value. The percentage uncertainty in length is (0.0005 ÷ 1.000) × 100% = 0.05%. The percentage uncertainty in the period

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