📚 Random Errors in A-Level Physics | A-Level物理中的随机误差
In A-Level Physics, measuring any quantity involves some degree of uncertainty, and understanding the difference between random and systematic errors is essential for designing experiments, analysing data, and evaluating results. Random errors cause readings to scatter unpredictably around the true value, and although they cannot be eliminated completely, their effect can be reduced by careful technique and repeated measurements.
在A-Level物理中，测量任何物理量都伴随着一定程度的不确定性。理解随机误差与系统误差的区别，对于设计实验、分析数据以及评估结果至关重要。随机误差会使读数围绕真实值不可预测地散布，虽然无法完全消除，但通过细致的操作和重复测量，可以降低其影响。
1. What Are Random Errors? | 什么是随机误差
Random errors are unpredictable fluctuations in measured quantities that arise from uncontrolled variables or limitations in the observation process. Unlike systematic errors, which shift every reading in the same direction by the same amount, random errors cause some readings to be too high and others too low, with no consistent pattern. When plotted on a graph or listed in a table, these errors produce a scatter of data points around the true value.
随机误差是测量过程中由不可控变量或观测手段的限制所导致的不可预测的波动。与系统误差不同——系统误差会使每次读数向同一方向偏移相同的大小——随机误差会使一些读数偏高、另一些偏低，没有一致的规律。当数据点绘制在图表上或列在表格中时，这些误差会使数据点围绕真实值产生散布。
For example, when timing the swing of a pendulum using a stopwatch, the reaction time of the experimenter varies slightly from trial to trial. One reading might be 1.24 s, the next 1.19 s, then 1.22 s. These differences are not caused by a fault in the stopwatch but by random variations in human response and other minor environmental factors.
例如，使用秒表测量单摆摆动周期时，实验者的反应时间每次都会有微小差异。一次读数可能是1.24 s，下一次是1.19 s，接着是1.22 s。这些差异并非秒表故障所致，而是人的反应速度以及其他细微环境因素的随机变化造成的。
2. Sources of Random Errors | 随机误差的来源
Random errors can originate from many sources in a physics laboratory. Common sources include:
在物理实验室中，随机误差可能源于许多方面。常见的来源包括：
Inconsistent judgment when reading a scale, such as estimating the last digit on a ruler or a digital meter.
读取刻度时判断不一致，例如估计直尺或数字仪表最后一位读数。
Fluctuations in experimental conditions, such as temperature, pressure, or mains voltage, that affect the quantity being measured.
实验条件波动，如温度、气压或电源电压变化，会影响被测量。
Reaction time errors in timing experiments, where the observer presses the stopwatch too early or too late.
计时实验中的反应时间误差，观察者过早或过晚按下秒表。
Vibration or movement of equipment that causes the measuring instrument to give slightly different readings.
设备振动或移动导致测量仪器给出略有不同的读数。
Electrical noise in sensors and digital instruments, especially when measuring very small signals.
传感器和数字仪器中的电噪声，特别是在测量微弱信号时。
In each case, the size and direction of the error are unpredictable for a single reading. However, if many readings are taken, the random errors tend to average out, with the mean value becoming closer to the true value.
在每种情况下，单次读数的误差大小和方向都是不可预测的。然而，如果进行多次测量，随机误差往往会相互平均，使平均值更接近真实值。
3. Random Errors vs Systematic Errors | 随机误差与系统误差
It is crucial to distinguish random errors from systematic errors. A systematic error is a reproducible inaccuracy that consistently affects readings in the same direction, such as a zero error on a balance or an incorrectly calibrated thermometer. Systematic errors affect the accuracy of a measurement but not its precision, and they cannot be reduced by repetition.
区分随机误差与系统误差至关重要。系统误差是一种可重现的失准，会持续使所有读数向相同方向偏移，例如天平未调零或温度计校准错误。系统误差影响测量准确度，但不影响精密度，而且重复测量无法减少系统误差。
| Feature | Random Error | Systematic Error |
| Effect on readings | Scatter above and below true value | Consistently shift all readings one way |
| Affects precision? | Yes, reduces precision | No, readings may still be precise |
| Affects accuracy? | On average, little effect if many readings | Yes, makes results inaccurate |
| Can repetition reduce it? | Yes, by averaging | No, repetition does not help |
In an experiment, both types of error may be present. For example, when measuring the density of a metal cylinder with a micrometer, a worn micrometer zero might cause a systematic error, while the difficulty of applying consistent pressure between measurements produces random errors.
在一个实验中，两类误差可能同时存在。例如，用千分尺测量金属圆柱体的密度时，千分尺零点磨损会产生系统误差，而每次测量时施加的压力难以保持一致则会产生随机误差。
4. Precision and Accuracy | 精密度与准确度
Random errors are closely related to the concept of precision. Precision describes how close repeated measurements are to each other. If random errors are small, the measurements are precise, meaning that the spread of readings is narrow. Accuracy, on the other hand, describes how close a measurement is to the true value. A set of readings can be precise but inaccurate if a systematic error is present.
随机误差与精密度概念密切相关。精密度描述多次重复测量结果之间的接近程度。如果随机误差很小，测量精密度就高，即读数散布范围窄。准确度则描述测量值与真实值之间的接近程度。如果存在系统误差，一组读数可能精密度很高但准确度较低。
For example, suppose an object has a true mass of 2.50 kg. An experimenter obtains four readings: 2.54 kg, 2.53 kg, 2.55 kg, and 2.54 kg. These readings are highly precise because they are all close to each other, but they are not accurate because their mean is about 2.54 kg, not 2.50 kg. This indicates that a systematic error is present. In contrast, readings of 2.45 kg, 2.52 kg, 2.58 kg, and 2.49 kg would show a larger random spread, giving lower precision but better average accuracy.
例如，假设某物体的真实质量为2.50 kg，实验者获得四次读数：2.54 kg、2.53 kg、2.55 kg和2.54 kg。这些读数精密度很高，因为它们彼此接近，但准确度不高，因为平均值约为2.54 kg，不是2.50 kg，这表明存在系统误差。相比之下，读数2.45 kg、2.52 kg、2.58 kg和2.49 kg的随机散布更大，精密度较低，但平均值更接近真实值。
In A-Level practical assessments, you should be able to comment on both precision and accuracy, and to suggest whether observed discrepancies are due to random or systematic effects.
在A-Level实验考核中，你应该能够对精密度和准确度进行评价，并判断观察到的偏差是来自随机效应还是系统效应。
5. Reducing Random Errors | 减少随机误差
Although random errors cannot be completely eliminated, their effect can be minimised using appropriate techniques:
尽管随机误差无法完全消除，但可以通过合适的方法将其影响降至最低：
Repeat measurements and calculate the mean. Averaging reduces the effect of random fluctuations and gives a more reliable estimate of the true value.
重复测量并计算平均值。取平均可以减少随机波动的影响，给出更可靠的真实值估计。
Use instruments with higher resolution. For example, a digital caliper with resolution 0.01 mm is less affected by judgement errors than a ruler with 1 mm divisions.
使用分辨率更高的仪器。例如，分辨率为0.01 mm的数显卡尺比刻度为1 mm的直尺受判断误差的影响更小。
Control the environment, such as steady temperature and lighting, to reduce fluctuations in experimental conditions.
控制环境，如保持温度和光照稳定，以减少实验条件的波动。
Use automated data logging sensors instead of manual timing or reading, which eliminates human reaction time and reading errors.
使用自动数据记录传感器代替手动计时或读数，从而消除人体反应时间和读数误差。
Take more readings within the same time and use the range to identify anomalous results before averaging.
在相同时间内增加读数次数，并在取平均之前利用极差识别异常结果。
It is important to note that repeating measurements does not reduce systematic errors. If the instrument is mis-calibrated, taking more readings will simply return a more precise but still inaccurate mean.
需要注意的是，重复测量并不能减少系统误差。如果仪器校准错误，增加读数只会得到一个更精密但仍然不准确的平均值。
6. Uncertainty and Random Errors | 不确定度与随机误差
Random errors are quantified through the uncertainty of a measurement. The absolute uncertainty gives the range within which the true value is likely to lie. For a set of repeated readings, the uncertainty is often estimated from the range:
随机误差通过测量的不确定度来量化。绝对不确定度给出了真实值可能落入的范围。对于一组重复读数，不确定度通常由极差估计：
If only one reading is taken, the absolute uncertainty is typically half the smallest division of the instrument.
如果只取一次读数，绝对不确定度通常为仪器最小刻度的一半。
If several readings are taken, the uncertainty can be estimated as half the range: Δx = (xmax − xmin)/2.
若读取多个读数，不确定度可估计为极差的一半：Δx = (xmax − xmin)/2。
As an alternative, the uncertainty may be taken as the resolution of the instrument if random scatter is smaller than the resolution.
另一种做法是，如果随机散布小于仪器分辨率，则不确定度可取仪器分辨率。
uncertainty is ±0.03 s. The result is reported as (1.28 ± 0.03) s, rounded appropriately.
例如，四次单摆周期测量为1.25 s、1.31 s、1.28 s和1.26 s，平均值为1.275 s，极差为1.31 s − 1.25 s = 0.06 s，因此绝对不确定度为±0.03 s。结果应记录为(1.28 ± 0.03) s，并合理修约。
In AQA A-Level Physics, you should be familiar with both absolute uncertainties and percentage uncertainties, and you must be able to convert between them using the equation:
在AQA A-Level物理中，你需要熟悉绝对不确定度和百分比不确定度，并能用以下公式进行换算：
Percentage uncertainty = (absolute uncertainty / measured value) × 100%
百分比不确定度 = (绝对不确定度 / 测量值) × 100%
7. Random Errors in Graphs and Data Analysis | 图表和数据分析中的随机误差
When plotted on a graph, random errors are seen as a scatter of data points that do not all lie exactly on the line or curve of best fit. In physics experiments, you are often asked to draw a best-fit line and to comment on whether the scatter is consistent with the estimated uncertainties.
当数据绘制在图表上时，随机误差表现为数据点并未全部精确落在最佳拟合直线或曲线上。在物理实验中，常要求你画出最佳拟合线，并判断数据点的散布是否与估计的不确定度一致。
If a plotted point has error bars, the bar should be drawn from the maximum value to the minimum value, centred on the measured point. A well-drawn line of best fit should pass through all error bars if the uncertainties are realistic and no systematic error is present. If a point lies far from the line even allowing for its error bar, it may be an anomalous result that should be investigated.
如果绘制的点带有误差棒，误差棒应从最大值延伸到最小值，以测量点为中心。如果不确定度合理且不存在系统误差，则一条恰当的拟合线应穿过所有误差棒。如果某一点即使考虑其误差棒后仍远离直线，则该点可能为异常结果，需要调查原因。
Random errors can also be detected from the residuals – the vertical differences between the data points and the best-fit line. If these residuals appear random and roughly constant in size, the scatter is consistent with random uncertainty. If they show a curved pattern or drift, a systematic effect or a non-linear relationship may be present.
随机误差也可以通过残差检测——即数据点与拟合线之间的垂直差异。如果残差呈现随机分布且大小大致恒定，则散布与随机不确定度一致。如果残差呈弯曲模式或漂移，则可能存在系统效应或非线性关系。
8. Propagation of Random Errors | 随机误差的传播
When a result is calculated from several measured quantities, the uncertainties from those quantities must be combined. This is called error propagation. For random errors, the following rules apply in A-Level physics:
当结果由多个测量量计算得出时，必须将这些量的不确定度组合起来，这称为误差传播。对于随机误差，A-Level物理中适用以下规则：
When adding or subtracting quantities, add the absolute uncertainties: Δ(a ± b) = Δa + Δb.
当物理量相加或相减时，绝对不确定度相加：Δ(a ± b) = Δa + Δb。
When multiplying or dividing quantities, add the percentage uncertainties: Δ(a × b)/|a × b| = Δa/|a| + Δb/|b|.
当物理量相乘或相除时，百分比不确定度相加：Δ(a × b)/|a × b| = Δa/|a| + Δb/|b|。
When a quantity is raised to a power n, multiply the percentage uncertainty by n: Δ(aⁿ)/|aⁿ| = n × Δa/|a|.
当物理量取n次幂时，百分比不确定度乘以n：Δ(aⁿ)/|aⁿ| = n × Δa/|a|。
) kg and V = (2.0 ± 0.1) × 10⁻⁴ m³. The percentage uncertainty in m is (0.005/0.500) × 100% = 1%, and the percentage uncertainty in V is (0.1/2.0) × 100% = 5%. Therefore, the percentage uncertainty in ρ is 1% + 5% = 6%. The density is 0.500 / 2.0 × 10⁻⁴ = 2500 kg m⁻³, so the absolute uncertainty is 6% of 2500 = 150 kg m⁻³. The result is reported as (2.5 ± 0.2) × 10³ kg m⁻³.
例如，圆柱体的密度ρ = m / V，其中m = (0.500 ± 0.005) kg，V = (2.0 ± 0.1) × 10⁻⁴ m³。m的百分比不确定度为(0.005/0.500) × 100% = 1%，V的百分比不确定度为(0.1/2.0) × 100% = 5%。因此，ρ的百分比不确定度为1% + 5% = 6%。密度为0.500 / 2.0 × 10⁻⁴ = 2500 kg m⁻³，所以绝对不确定度为2500的6%，即150 kg m⁻³。结果记录为(2.5 ± 0.2) × 10³ kg m⁻³。
9. Reporting Results with Random Errors | 报告带随机误差的结果
2.346 °C with an uncertainty of 0.12 °C should be reported as (22.35 ± 0.12) °C or, if the uncertainty is rounded to one significant figure, (22.35 ± 0.1) °C.
报告最终结果时，应写明平均值、相应不确定度和正确的单位。按照惯例，不确定度保留一位有效数字，测量值修约到与不确定度相同的小数位。例如，温度为22.346 °C，不确定度为0.12 °C，应记录为(22.35 ± 0.12) °C；若不确定度修约为一位有效数字，则为(22.35 ± 0.1) °C。
In a practical write-up, you should also compare the final uncertainty with the spread of the data. If the range of readings is much larger than the instrument resolution, random errors dominate the uncertainty. If the range is similar to the resolution, both the instrument limit and random variation contribute equally. Including this analysis demonstrates a strong understanding of experimental error.
在实验报告中，你还应将最终不确定度与数据散布进行比较。如果测量读数的极差远大于仪器分辨率，则随机误差主导不确定度。如果极差与分辨率相近，则仪器极限和随机变化贡献相当。加入这样的分析，能够展示你对实验误差的深入理解。
10. Exam Tips on Random Errors | 关于随机误差的考试技巧
In AQA A-Level Physics exams, questions on random errors often appear in the context of required practicals. Common marks are awarded for identifying that repeating readings reduces random errors, calculating a mean, estimating uncertainty from the range, and drawing error bars. You should be able to explain why taking more readings improves reliability but does not eliminate systematic error.
在AQA A-Level物理考试中，随机误差相关题目通常结合必修实验出现。常见得分点包括：指出重复测量可减少随机误差、计算平均值、从极差估计不确定度以及绘制误差棒。你应该能够解释为什么增加读数次数能提高可靠性，但不能消除系统误差。
When answering such questions, use the correct terminology: 'precision' refers to the spread of results; 'accuracy' refers to closeness to the true value; 'uncertainty' quantifies the range of possible values; and 'anomalous result' is a reading that does not fit the pattern. Avoid vague statements such as 'take more readings to make the experiment more accurate' – instead, say 'repeat readings to reduce the effect of random errors and obtain a more reliable mean.'
回答此类问题时，应使用正确的术语：“精密度”指结果的散布程度；“准确度”指与真实值的接近程度；“不确定度”量化可能值的范围；“异常结果”指不符合规律的一次读数。避免含糊说法，如“多测几次让实验更准确”，而应说“重复读数以减小随机误差的影响，获得更可靠的平均值”。
Finally, practice converting between absolute and percentage uncertainties, and know how to combine them in addition, subtraction, multiplication, division, and power calculations. These skills are frequently tested and are essential for achieving top marks in the practical-based examination questions.
最后，练习绝对不确定度与百分比不确定度之间的转换，并掌握它们在加减乘除和幂运算中组合的方法。这些技能经常被考查，也是在基于实验的考题中获得高分的关键。
Published by TutorHao | Physics Revision Series | aleveler.com
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