📚 Random Errors in A-Level Physics | A-Level 物理中的随机误差
In A-level Physics practical work, no measurement is perfectly exact. Random errors are unpredictable fluctuations that cause readings to scatter around the true value. Understanding random errors is essential for judging the reliability of experimental data and for calculating uncertainties in the AQA specification.
在 A-level 物理实验中，任何测量都不是绝对精确的。随机误差是导致读数围绕真值波动的不可预测变化。理解随机误差对于判断实验数据的可靠性以及计算 AQA 考纲中的不确定度至关重要。
1. What Is a Random Error? | 什么是随机误差？
A random error is an unpredictable variation in a measurement that occurs without a consistent pattern. If you repeat the same measurement under the same conditions, random errors cause some readings to be slightly higher and some slightly lower than the true value.
随机误差是测量中出现的无规律、不可预测的变化。如果在相同条件下重复同一测量，随机误差会使一些读数略高于真值，另一些读数略低于真值。
Random errors are caused by many small, fluctuating factors such as reaction time, electrical noise or tiny changes in the environment. They cannot be eliminated completely, but they can be reduced by taking repeat readings and averaging.
随机误差由许多微小且波动的因素引起，例如反应时间、电噪声或环境的微小变化。随机误差无法完全消除，但可以通过重复测量并取平均值来减小。
2. Random Error vs Systematic Error | 随机误差与系统误差
A random error affects precision by making readings spread out unpredictably. A systematic error affects accuracy by shifting all readings in the same direction, often due to a faulty instrument or poor experimental technique.
随机误差通过使读数不可预测地分散而影响精密度。系统误差通常由于仪器故障或实验方法不当，使所有读数向同一方向偏移，从而影响准确度。
| Feature | Random error | 随机误差 | Systematic error | 系统误差 |
|---|---|---|
| Pattern | No consistent direction | 无固定方向 | Consistent bias | 固定偏差 |
| Effect | Reduces precision | 降低精密度 | Reduces accuracy | 降低准确度 |
| Can be reduced by | Repeating and averaging | 重复取平均 | Better calibration or technique | 校准或改进方法 |
In an AQA practical examination, you may be asked to identify whether a described source is random or systematic. A good rule is to ask: does this source make every reading too high or too low? If yes, it is systematic; if the sign changes unpredictably, it is random.
在 AQA 实验考试中，你可能需要判断所描述的来源是随机误差还是系统误差。一个简单的判断方法是：这个来源是否使每个读数都偏高或偏低？如果是，则属于系统误差；如果偏差方向不可预测地改变，则属于随机误差。
3. Common Sources of Random Error | 随机误差的常见来源
Random errors can arise from human reaction time when starting and stopping a stopwatch. Your response is never exactly the same from one trial to the next, so the measured time fluctuates.
随机误差可能来自启动和停止秒表时的人体反应时间。每次试验中你的反应不可能完全相同，因此测得的时间会波动。
Other sources include reading an analogue scale by eye, fluctuations in room temperature, air currents, signal noise in electronic circuits, and small variations in the release or launch of a projectile.
其他来源包括用眼睛读取模拟刻度、室温波动、空气流动、电子电路中的信号噪声，以及抛体释放或发射时的微小变化。
4. Why Repeating Measurements Helps | 为什么重复测量有助于减小随机误差
Random errors are equally likely to be positive or negative. When many independent readings are taken, positive and negative deviations tend to cancel out in the mean. Therefore the mean value is a more reliable estimate of the true value than a single reading.
随机误差出现正值和负值的概率大致相同。当进行多次独立测量时，正偏差和负偏差在平均值中往往会相互抵消。因此，平均值比单次读数更能可靠地估计真值。
Repeating also allows you to spot outliers that are clearly inconsistent with the rest of the data. In AQA required practicals, you should normally take at least three repeat measurements, but more repeats give a smaller random uncertainty.
重复测量还能帮助你发现与其他数据明显不一致的异常值。在 AQA 必修实验中，通常应至少重复测量三次，但更多次重复会给出更小的随机不确定度。
5. Mean, Range and Best Estimate | 平均值、极差与最佳估计
For a set of repeated readings x₁, x₂, x₃, ..., xₙ, the best estimate is the arithmetic mean. The range is the difference between the largest and smallest reading, and half the range is often used as the random uncertainty in A-level work.
对于一组重复读数 x₁、x₂、x₃、...、xₙ，最佳估计值就是算术平均值。极差是最大读数与最小读数之差，而极差的一半通常在 A-level 学习中用作随机不确定度。
x̄ = (x₁ + x₂ + ... + xₙ) / n
Δx = (xmax − xmin) / 2
.4 s, 12.7 s, 12.5 s and 12.8 s, the mean is 12.6 s and the random uncertainty is ± 0.2 s.
例如，如果单摆十次摆动的时间为 12.4 s、12.7 s、12.5 s 和 12.8 s，则平均值为 12.6 s，随机不确定度为 ± 0.2 s。
6. Precision and Accuracy | 精密度与准确度
Random errors limit the precision of an experiment. Precise results are close to each other, even if they are not close to the accepted value. Accurate results are close to the true or accepted value.
随机误差限制了实验的精密度。精密度高的结果彼此接近，即使它们并不接近公认值。准确度高的结果则接近真值或公认值。
A set of readings can be precise but inaccurate if a systematic error is present. For example, a voltmeter with a zero offset may give readings that are all close together but all too high by the same amount.
如果存在系统误差，一组读数可能精密度高但准确度低。例如，存在零点偏移的电压表可能给出彼此非常接近但都整体偏高的读数。
7. Expressing Random Uncertainty | 表示随机不确定度
When you state a result, you should include an absolute uncertainty with the unit. You can also express the random uncertainty as a percentage uncertainty to compare the quality of different measurements.
当你写出结果时，应注明带单位的绝对不确定度。你也可以将随机不确定度表示为百分不确定度，以便比较不同测量的质量。
% uncertainty = (Δx / x̄) × 100%
a length measurement is (1.25 ± 0.05) m, the percentage uncertainty is (0.05 / 1.25) × 100% = 4%. This is often used when combining uncertainties in calculations.
如果长度测量为 (1.25 ± 0.05) m，则百分不确定度为 (0.05 / 1.25) × 100% = 4%。在合并计算中的不确定度时，经常使用百分不确定度。
8. Error Bars on Graphs | 图表中的误差棒
Random uncertainty can be shown visually by adding error bars to a graph. A vertical error bar shows the range within which the true value of the plotted quantity is likely to lie.
随机不确定度可以通过在图表上添加误差棒来直观表示。竖直误差棒显示被标绘量的真值可能落在的范围。
When you plot a straight-line graph such as V against I, random scatter causes points to lie slightly away from the line of best fit. In AQA practical work, you may be expected to draw a worst-fit line or use error bars to estimate the uncertainty in the gradient.
当你绘制 V-I 等直线图时，随机散布会使数据点略微偏离最佳拟合线。在 AQA 实验工作中，你可能需要画出最坏拟合线，或使用误差棒来估计斜率的不确定度。
9. Propagation of Random Uncertainties | 随机不确定度的传播
When two measured quantities are added or subtracted, the absolute uncertainties add. This gives the overall random uncertainty in the result.
当两个测量量相加或相减时，绝对不确定度相加。这给出了结果中的总随机不确定度。
Δq = Δa + Δb
When quantities are multiplied or divided, you add percentage uncertainties instead. If a quantity is raised to a power, multiply the percentage
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