📚 Precision, accuracy, errors and uncertainties | 精度、准确度、误差与不确定度
In experimental physics, the terms precision, accuracy, error, and uncertainty describe the quality and trustworthiness of measurements. These concepts are fundamental to all practical work and appear in almost every A-level physics question that involves data analysis. This article explains each concept in detail, shows how to calculate and combine uncertainties, and demonstrates how to present results with error bars and graphical interpretation. Understanding these ideas will help you design better experiments, judge the reliability of data, and write clear conclusions in your CIE examination papers.
在实验物理中，精度、准确度、误差与不确定度这些术语描述了测量结果的质量和可信度。这些概念是所有实验工作的基础，几乎出现在每一道涉及数据分析的 A-level 物理试题中。本文详细解释每个概念，展示如何计算和合成不确定度，并说明如何用误差棒呈现结果和解读图表。理解这些理念将帮助你设计更好的实验、判断数据的可靠性，并在 CIE 考试中写出清晰的结论。
1. Defining Precision and Accuracy | 定义精度和准确度
Precision refers to how closely repeated measurements agree with one another, without any reference to the true value. If you measure the length of a wire five times and obtain 1.23 m, 1.24 m, 1.23 m, 1.22 m, and 1.24 m, your measurements are precise because the spread is small. Accuracy, on the other hand, is the closeness of a measured value to the true or accepted value. Suppose the true length is 1.30 m; although your readings are precise, they are not accurate because they consistently fall below the true value. Precision tells you about random variation; accuracy tells you about systematic bias.
精度指的是重复测量值之间彼此接近的程度，与真值无关。如果你对一根导线的长度测量五次，得到 1.23 m、1.24 m、1.23 m、1.22 m 和 1.24 m，这些测量值是精密的，因为数据散布很小。而准确度是测量值接近真实值或公认值的程度。假如真实长度为 1.30 m，虽然你的读数很精密，但并不准确，因为它们始终低于真值。精度反映随机变异；准确度反映系统偏差。
A useful analogy is shooting arrows at a target. Tightly grouped arrows far from the bullseye show high precision but low accuracy. Arrows spread around the bullseye show high accuracy but low precision. An ideal experiment aims for both: a tight cluster centred on the true value. In practical physics, you must estimate both precision and accuracy from the spread of data and from comparisons with known constants or accepted results.
一个常用的比喻是射箭靶。密集地射在远离靶心的位置表示高精度低准确度；分散在靶心周围则高准确度低精度。理想实验追求两者兼备：一组紧密围绕真值的数据。在实际物理中，你必须从数据的散布以及与已知常数或公认结果的对比中，同时评估精度和准确度。
2. Random Errors and Systematic Errors | 随机误差和系统误差
Random errors cause measurements to fluctuate unpredictably about a mean value. They arise from unpredictable variations in the measurement process, such as reading a scale from slightly different angles (parallax), fluctuations in temperature, or electronic noise. Random errors reduce precision but can be averaged out by taking many readings. Their magnitude is often estimated by the spread of results, such as the half-range or standard deviation.
随机误差导致测量值在平均值附近发生不可预测的波动。它们来自测量过程中无法预见的变动，比如从稍许不同的角度读取刻度（视差）、温度起伏或电子噪声。随机误差会降低精度，但可以通过多次读数取平均来削弱其影响。其大小通常用结果分布来估算，例如半范围或标准偏差。
Systematic errors produce a consistent offset in the same direction every time a measurement is made. They can stem from a poorly calibrated instrument, a zero error, or an incorrect experimental technique that introduces a fixed bias. Systematic errors affect accuracy but do not affect the spread of readings. They cannot be reduced by averaging; they must be identified and corrected through calibration or by adjusting technique.
系统误差每次测量都会在同一个方向上产生固定的偏移。它们可能源于仪器校准不良、零误差，或者引入固定偏差的不当实验方法。系统误差影响准确度，但不影响读数的散布。取平均无法减小系统误差；必须通过校准或调整方法来识别和修正。
3. Sources of Errors in Experiments | 实验中的误差来源
nclude the least count of a ruler or the resolution of a digital meter. For example, a metre rule graduated in millimetres can be read to no better than ±0.5 mm. Environmental factors like drafts, temperature changes, and vibrations can introduce random fluctuations. Observer errors include parallax when a pointer does not align with the scale and the eye is not perpendicular, or reaction time in starting and stopping a stopwatch.
在实验室中，误差可源自测量仪器、环境、观察者和方法本身。仪器的限制包括直尺的最小分度或数字仪表的读数分辨率。例如，刻度为毫米的米尺读数，其不确定度最优为 ±0.5 mm。环境因素如气流、温度变化和震动会引入随机波动。观察者误差包括指针与刻度不重合时的视差，当眼睛未正视刻度时发生，或用秒表时启动和停止的反应时间。
Method errors arise from experimental design. For instance, measuring the period of a pendulum by timing only one swing introduces a large relative uncertainty from reaction time. Timing 20 swings and dividing by 20 reduces the percentage uncertainty dramatically. Similarly, assuming a wire's resistance is measured with a perfect voltmeter ignores the voltmeter's own resistance, which may cause a systematic shift. Careful identification of all possible error sources is the first step in reducing them.
方法误差来自实验设计。例如，只测量一次摆动来得到单摆周期，会因为反应时间而引入较大的相对不确定度。改为计时 20 次摆动并除以 20，会显著降低百分比不确定度。同样，假设施加在导线上的电压是用理想电压表测得，忽略了电压表自身的内阻，这可能导致系统偏移。仔细识别所有可能的误差来源，是减小误差的第一步。
4. Understanding Uncertainty | 理解不确定度
the inherent limitations of the measurement. Every measured value should be reported with its uncertainty, for example, (5.30 ± 0.05) cm. The ± 0.05 cm tells us that the true length is likely between 5.25 cm and 5.35 cm. This interval is sometimes called the confidence interval.
不确定度是对真值预期所在范围的一个定量指示。它不意味着错误；而是反映了测量固有的局限性。每一个测量值都应附有不确定度来报告，例如 (5.30 ± 0.05) cm。± 0.05 cm 表明真实长度可能介于 5.25 cm 和 5.35 cm 之间。这个区间有时称为置信区间。
There are two main types of uncertainty estimates: Type A, evaluated by statistical analysis of repeated readings (e.g., standard deviation of the mean), and Type B, evaluated by other means such as instrument resolution, calibration certificates, or manufacturer's specifications. In A-level physics, the simpler approach usually combines the reading uncertainty (half the smallest scale division) and the spread of repeated measurements, taking the larger of the two or using the half-range after several readings.
不确定度评定主要有两类：A 类，通过对重复读数的统计分析得出（例如平均值的标准偏差）；B 类，通过其他方法评定，如仪器分辨率、校准证书或制造商技术规格。在 A-level 物理中，较简单的方法通常是把读数不确定度（最小刻度的一半）和重复测量的分布结合起来，取两者中较大的，或在多次读数后使用半范围。
5. Absolute, Fractional and Percentage Uncertainty | 绝对、相对和百分比不确定度
The absolute uncertainty h is L = 1.24 m and the absolute uncertainty is ΔL = 0.01 m, we write 1.24 m ± 0.01 m. The fractional uncertainty is ΔL / L = 0.01 / 1.24 ≈ 0.0081, and the percentage uncertainty is (ΔL / L) × 100% ≈ 0.81% forms are dimensionless and useful when combining uncertainties from different quantities.
绝对不确定度是以与被测量相同单位表示的不确定度。若长度 L = 1.24 m，绝对不确定度 ΔL = 0.01 m，则记为 1.24 m ± 0.01 m。相对不确定度为 ΔL / L = 0.01 / 1.24 ≈ 0.0081，百分比不确定度为 (ΔL / L) × 100% ≈ 0.81%。相对和百分比形式无量纲，在合成不同物理量的不确定度时非常有用。
t, the uncertainty is at least the last significant digit, e.g., ±0.01 V for a voltmeter reading 2.34 V. When repeated measurements are available, the absolute uncertainty can be taken as half the range: (max − min)/2, provided this is larger than the instrument' y. A common rule in CIE is to use the larger of the instrument uncertainty and the spread.
对于用模拟刻度读取的单次读数，绝对不确定度通常取最小分度值的一半。对于数字式仪表，不确定度至少是最末一位有效数字，如电压表读数为 2.34 V 时，不确定度为 ±0.01 V。当有重复测量时，绝对不确定度可取范围的一半：(最大值 − 最小值)/2，前提是它大于仪器读数的不确定度。CIE 中常见规则是取仪器不确定度与数据散布两者中较大的那个。
6. Combining Uncertainties: Addition and Subtraction | 组合不确定度：加法和减法
b, then the absolute uncertainty in Q is the sum of the individual absolute uncertainties:
ΔQ = Δa + Δb
re the initial and final temperatures of water as T₁ = (20.0 ± 0.5) °C and T₂ = (45.0 ± 0.5) °C, the temperature rise ΔT = T₂ − T₁ = 25.0 °C, and its absolute uncertainty is Δ(ΔT) = 0.5 + 0.5 = 1.0 °C. Thus ΔT = (25.0 ± 1.0) °C.
当两个测量值相加或相减时，绝对不确定度相加。若物理量 Q = a + b 或 Q = a − b，则 Q 的绝对不确定度等于各绝对不确定度之和：ΔQ = Δa + Δb。这条规则容易记忆：通过加减组合数值时，绝对不确定度直接传播。例如，测得水的初温和末温为 T₁ = (20.0 ± 0.5) °C 和 T₂ = (45.0 ± 0.5) °C，温升 ΔT = T₂ − T₁ = 25.0 °C，其绝对不确定度为 Δ(ΔT) = 0.5 + 0.5 = 1.0 °C，因此 ΔT = (25.0 ± 1.0) °C。
7. Combining Uncertainties: Multiplication and Division | 组合不确定度：乘法和除法
tional (or percentage) uncertainties add. For Q = a × b or Q = a / b, the relationship is:
ΔQ/Q = Δa/a + Δb/b
ate has length L = (10.0 ± 0.2) cm and width W = (5.0 ± 0.1) cm. The area A = L × W = 50.0 cm². Fractional uncertainty in L is 0.2/10.0 = 0.02, and in W is 0.1/5.0 = 0.02. Therefore ΔA/A = 0.02 + 0.02 = 0.04, giving ΔA = 0.04 × 50.0 = 2.0 cm². The area is reported as (50.0 ± 2.0) cm².
当物理量相乘或相除时，它们的相对（或百分比）不确定度相加。对于 Q = a × b 或 Q = a / b，关系为：ΔQ/Q = Δa/a + Δb/b。换言之，积或商的相对不确定度等于各因子相对不确定度之和。假设一块矩形板长 L = (10.0 ± 0.2) cm，宽 W = (5.0 ± 0.1) cm，面积 A = L × W = 50.0 cm²。L 的相对不确定度为 0.2/10.0 = 0.02，W 为 0.1/5.0 = 0.02。于是 ΔA/A = 0.02 + 0.02 = 0.04，得 ΔA = 0.04 × 50.0 = 2.0 cm²。面积应报告为 (50.0 ± 2.0) cm²。
8. Combining Uncertainties: Powers and Roots | 组合不确定度：幂与方根
the power. For Q = aⁿ, where n is a constant (could be a fraction for roots), the rule is:
ΔQ/Q = |n| × (Δa/a)
nsider measuring the diameter d of a sphere as (4.00 ± 0.08) cm. The radius r = d/2 = 2.00 cm, and the fractional uncertainty in r is the same as in d: 0.08/4.00 = 0.02. To find the volume V = (4/3)πr³, the power on r is 3, so fractional uncertainty in V is 3 × 0.02 = 0.06. The absolute uncertainty in V can then be calculated from ΔV = 0.06 × V.
如果一个量被乘方，相对不确定度要乘以该幂次的绝对值。对于 Q = aⁿ，其中 n 是常数（对于根号，n 为分数），规则是：ΔQ/Q = |n| × (Δa/a)。因此，若一个量平方（n = 2），百分比不确定度加倍；若开平方（n = ½），百分比不确定度减半。考虑测量球的直径 d = (4.00 ± 0.08) cm，半径 r = d/2 = 2.00 cm，r 的相对不确定度与 d 相同：0.08/4.00 = 0.02。为求体积 V = (4/3)πr³，r 的幂次为 3，所以 V 的相对不确定度为 3 × 0.02 = 0.06。然后由 ΔV = 0.06 × V 计算其绝对不确定度。
9. Uncertainty from a Set of Repeated Readings | 重复测量读数的不确定度
n A-level work), the uncertainty is often taken as half the range, i.e., (maximum − minimum)/2. For example, five timings of 20 pendulum swings give periods: 24.2 s, 24.5 s, 23.9 s, 24.1 s, 24.3 s. The mean is (24.2+24.5+23.9+24.1+24.3)/5 = 24.20 s. The range is 24.5 − 23.9 = 0.6 s, so the uncertainty is 0.3 s. The result is reported as (24.2 ± 0.3) s.
对同一物理量进行多次独立测量时，真值的最佳估计就是平均值。对于小数据集（在 A-level 实验中常见），不确定度常取范围的一半，即 (最大值 − 最小值)/2。例如，对单摆摆动 20 次计时五次，得到：24.2 s, 24.5 s, 23.9 s, 24.1 s, 24.3 s。平均值为 24.20 s，范围为 0.6 s，故不确定度为 0.3 s。结果可报告为 (24.2 ± 0.3) s。
ment limit. A stopwatch typically reads to 0.01 s, but human reaction time introduces an uncertainty of about 0.1– stopwatch) is larger than the half-range, use the instrument limit. In CIE practical assessments, you should always justify your choice of uncertainty.
将此基于分布的确定度与仪器限制进行比较很重要。秒表通常读数精度为 0.01 s，但人的反应时间会引入约 0.1–0.2 s 的不确定度。若仪器不确定度（例如手动秒表为 ±0.2 s）大于半范围，则应采用仪器限度。在 CIE 实验评估中，你应始终说明选择不确定度的理由。
10. Using Error Bars to Represent Uncertainty | 使用误差棒表示不确定度
When plotting experimental data on a graph, each point is accompanied by an error bar that shows the range of possible values. The vertical error bar extends from y − Δy to y + Δy, and the horizontal error bar extends from x − Δx to x + Δx, if both variables have uncertainties. The length of each bar is twice the absolute uncertainty. Error bars give a visual representation of precision: shorter bars indicate more precise measurements.
在图表上绘制实验数据时，每个点都附有一个显示可能值范围的误差棒。纵轴误差棒从 y − Δy 延伸到 y + Δy，横轴误差棒从 x − Δx 延伸到 x + Δx，如果两个变量都有不确定度的话。每个误差棒的长度是绝对不确定度的两倍。误差棒提供了精度的可视化表示：越短的棒表明测量越精密。
Error bars are crucial when drawing a best-fit line. The line should pass through as many error bars as possible, or at least lie within them. If a point lies far from the line with a small error bar, it indicates a possible mistake or an underestimate of uncertainty. When calculating the gradient, you can also use the extreme lines (steepest and shallowest lines that still fit within the error bars) to determine the uncertainty in the gradient.
在绘制最佳拟合线时，误差棒至关重要。线应尽可能穿过所有误差棒，或至少位于误差棒之内。若某点带有极小的误差棒却远离拟合线，则表明可能存在错误或低估了不确定度。计算斜率时，你还可以利用极端线（仍处于误差棒内的最陡和最缓的两条线）来确定斜率的不确定度。
11. Precision vs Accuracy in Graphical Analysis | 图形分析中的精度与准确度
A graph can reveal both precision and accuracy. Small error bars and points closely clustered around a smooth curve indicate high precision. If the best-fit line passes through or near the origin when theory predicts it should, the experiment shows good accuracy. If the line has a systematic intercept away from zero, it suggests a systematic error, such as an unaccounted offset or a zero error in a meter.
图形能同时揭示精度和准确度。误差棒小、数据点紧密分布在光滑曲线周围，表明精度高。如果根据理论最佳拟合线应通过原点附近且实际也如此，说明实验准确度好。如果拟合线存在偏离零点的系统截距，就暗示了系统误差，比如未考虑的偏移量或仪表零位误差。
Consider an experiment to verify Ohm's law by measuring voltage across a resistor for different currents. A systematic error arising from an ammeter with a constant zero offset will shift all current readings by the same amount, leaving the slope (resistance) largely unaffected but introducing a non-zero intercept. Analysing the intercept helps diagnose and correct systematic errors. Graphical methods thus serve as a powerful diagnostic tool for both random and systematic effects.
设想一个用测量不同电流下电阻两端电压来验证欧姆定律的实验。由于安培表固定零位偏移引起的系统误差，会使所有电流读数偏移同一数值，斜率（电阻）基本不受影响，但会产生非零截距。分析截距有助于诊断和修正系统误差。因此，图像方法是同时诊断随机和系统效应的有力工具。
12. Best Practices to Reduce Errors and Improve Reliability | 减少误差与提高可靠性的最佳做法
To minimise random errors, take multiple readings and use mean values. Increase the sample size and, where appropriate, repeat entire experimental runs. For timing experiments, measure a large number of cycles instead of a single cycle to reduce the impact of reaction time. Always estimate the uncertainty in your measurements and propagate it correctly through calculations. In a written report, state the uncertainty explicitly with every result and discuss the main sources of error.
要减小随机误差，应多次读数并使用平均值。增大样本量，并在合适时重复整个实验过程。在计时实验中，测量多个周期而非单个周期以减少反应时间的影响。始终估算测量中的不确定度，并在计算中正确传递。在书面报告中，明确注明每个结果的不确定度，并讨论主要误差来源。
To tackle systematic errors, calibrate instruments before use, check zero settings, and compare results with accepted values where possible. Use control experiments to isolate and identify biases. When designing an experiment, think critically about whether a chosen method inherently overestimates or underestimates a quantity. For example, measuring the acceleration of free fall with a simple pendulum can suffer from a systematic error if the amplitude is too large; keeping oscillations small helps match the theoretical model.
为解决系统误差，使用前应校准仪器，检查零点设置，并在可能时与公认值对比。利用对照实验来隔离和识别偏差。在设计实验时，要批判性地思考所选方法是否会天生高估或低估某个量。例如，用单摆测量自由落体加速度时，若振幅过大就会产生系统误差；保持小角度摆动有助于匹配理论模型。
ar table with headings that include units and uncertainties. Plot graphs with labelled axes, sensible scales, and error bars. When calculating a final quantity, quote the result to the same decimal place or significant figures as the uncertainty. Never give a result with a precision unjustified by the uncertainty — nonsensical because a high-precision number is meaningless when the uncertainty is large.
最后，始终用清晰的表格呈现数据，表头应包含单位和不确定度。作图时要标注坐标轴、选择合适的刻度并加上误差棒。在计算最终结果时，将数值表达为与不确定度相同的小数位或有效数字。切忌给出与不确定度不匹配的高精度数值——例如 1.23456 ± 0.1 是无意义的，因为当不确定度很大时高精度数值毫无意义。
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