📚 Precision | 精密度
In physics, precision refers to the degree of agreement among repeated measurements of the same quantity. A precise measurement set has little scatter, even if the values may not be close to the true value.
在物理学中，精密度是指对同一物理量进行重复测量时各结果之间的一致程度。一组精密的测量数据散布很小，即使这些数值可能并不接近真实值。
1. Definition of Precision | 精密度的定义
Precision is a measure of how close repeated measurements are to one another. If you measure a length five times and obtain 10.2 cm, 10.3 cm, 10.2 cm, 10.3 cm and 10.2 cm, the results are highly precise because the spread is only 0.1 cm.
精密度是衡量重复测量结果之间相互接近程度的量。如果你将一个长度测量五次，得到 10.2 cm、10.3 cm、10.2 cm、10.3 cm 和 10.2 cm，这些结果具有很高的精密度，因为离散程度仅为 0.1 cm。
Precision is independent of whether the measured values are correct. A precise but inaccurate result may arise from a poorly calibrated instrument that nevertheless gives very repeatable readings.
精密度与测量值是否正确无关。一个精密度高但不准确的结果可能来自一台校准不良但读数高度可重复的仪器。
2. Precision vs Accuracy | 精密度与准确度
Accuracy describes how close a measured value is to the true or accepted value. Precision describes how close repeated measurements are to each other. These two concepts are often confused, but they are distinct.
准确度描述测量值与真实值或公认值之间的接近程度。精密度描述重复测量结果之间的接近程度。这两个概念常被混淆，但它们是不同的。
The classic target-diagram analogy is useful: four shots clustered tightly in one corner show high precision but low accuracy; shots scattered across the target show low precision; shots clustered near the bullseye show both high accuracy and high precision.
经典的靶图类比对此很有帮助：四支箭紧紧聚在靶心一角，代表精密度高但准确度低；箭矢散布在靶面上代表精密度低；箭矢聚在靶心附近则代表准确度和精密度都高。
| Accurate | Not accurate | |
| Precise | Tight cluster near true value | Tight cluster away from true value |
| Not precise | Wide scatter centred on true value | Wide scatter away from true value |
3. Measurement Uncertainty | 测量不确定度
mple, a reading of 5.2 ± 0.1 cm means the true length is likely between 5.1 cm and 5.3 cm.
每一次测量都存在不确定度。不确定度量化了真实值可能落入的范围。例如，读数 5.2 ± 0.1 cm 表示真实长度很可能在 5.1 cm 与 5.3 cm 之间。
Uncertainty can be estimated from the instrument scale, from repeated readings using the range, or from statistical analysis of a data set. Higher precision generally leads to a smaller uncertainty, assuming no systematic errors dominate.
不确定度可以通过仪器刻度、利用极差对重复读数进行估计，或通过对数据集进行统计分析来获得。在没有系统误差主导的情况下，更高的精密度通常对应更小的不确定度。
uncertainty = ± (range ÷ 2) for repeated measurements
4. Systematic and Random Errors | 系统误差与随机误差
Systematic errors shift all measurements in the same direction by a constant or proportional amount. They affect accuracy but not precision. Examples include a zero error on a balance or a stretched tape measure.
系统误差使所有测量结果朝相同方向偏移一个恒定或成比例的量。它们影响准确度，但不影响精密度。例如，天平未调零或卷尺被拉伸。
Random errors cause unpredictable scatter in measurements. They affect precision directly. Reading an analogue scale from slightly different angles, or small temperature fluctuations during an experiment, produce random errors.
随机误差导致测量结果产生不可预测的散布。它们直接影响精密度。从略微不同的角度读取模拟刻度，或实验期间微小的温度波动，都会产生随机误差。
5. Instrument Precision and Resolution | 仪器精密度与分辨率
The resolution of an instrument is the smallest change in the measured quantity that can be detected. A ruler marked in millimetres has a finer resolution than one marked in centimetres, so it can give more precise readings.
仪器的分辨率是能够被检测到的被测物理量的最小变化。以毫米为刻度的尺子比以厘米为刻度的尺子分辨率更高，因此能给出更精密的读数。
However, resolution is not the only factor. A digital multimeter with many digits may still have internal noise that limits its practical precision. The instrument's calibration certificate or manufacturer's specification often states its precision.
然而，分辨率并不是唯一因素。一台显示位数很多的数字万用表仍可能因内部噪声而限制其实际精密度。仪器的校准证书或制造商的说明书中通常会注明其精密度。
6. Significant Figures | 有效数字
The number of significant figures in a measurement reflects its precision. A value of 12.3 has three significant figures, implying uncertainty in the last digit. Writing 12.30 suggests four significant figures and a higher precision than 12.3, although both represent the same numeric amount.
测量结果的有效数字位数反映其精密度。数值 12.3 有三位有效数字，意味着最后一位存在不确定度。写成 12.30 暗示四位有效数字，比 12.3 的精密度更高，尽管两者数值大小相同。
When taking readings, you should record the value to the full resolution of the instrument. For an analogue scale, you may estimate one decimal place beyond the smallest division, which improves the precision of the reading.
读数时应记录到仪器的完整分辨率。对于模拟刻度，可以估读到最小分度下一位，从而提高读数的精密度。
7. Calculating Precision: Range and Standard Deviation | 计算精密度：极差与标准差
The range is the simplest measure of precision. It is the difference between the largest and smallest measured values. A small range indicates high precision. For a set of repeated measurements, the uncertainty is often taken as half the range.
极差是表示精密度最简单的量。它是最大测量值与最小测量值之差。极差小表示精密度高。对于一组重复测量，不确定度常取极差的一半。
Standard deviation is a more robust statistical measure. It uses every value in the data set. The smaller the standard deviation σ, the more precise the measurements are. For a sample of n measurements, the standard deviation is given by:
标准差是一种更稳健的统计量。它使用了数据集中的每一个数值。标准差 σ 越小，测量越精密。对于 n 次测量样本，标准差的计算公式为：
σ = √[ Σ(xᵢ - x̄)² ÷ (n - 1) ]
Here xᵢ is each individual measurement and x̄ is the mean. In AQA A-level Physics, you are expected to calculate the mean and use the range or standard deviation to estimate uncertainty.
其中 xᵢ 是每一次测量值，x̄ 是平均值。在 AQA A-Level 物理中，你应会计算平均值，并使用极差或标准差来估算不确定度。
8. Precision in Experimental Design | 实验设计中的精密度
Good experimental design improves precision by controlling variables and reducing random error. Repeating measurements and calculating averages smooths out some random fluctuations, giving a more precise final result.
良好的实验设计通过控制变量和减小随机误差来提高精密度。重复测量并计算平均值可以平滑部分随机波动，从而获得更精密的最终结果。
Using appropriate instruments is crucial. If a timer measures to the nearest 0.1 s but the reaction occurs within 20 s, the percentage uncertainty is 0.5%. A stopwatch with 0.01 s resolution reduces this to 0.05%, improving precision.
选用合适的仪器至关重要。如果计时器精确到 0.1 s，而反应在 20 s 内完成，则百分比不确定度为 0.5%。分辨率为 0.01 s 的秒表会将其降低到 0.05%，从而提高精密度。
9. Sources of Random Error | 随机误差的来源
Random errors arise from factors that cannot be perfectly controlled. Common sources include human reaction time when starting and stopping a timer, parallax error when reading an analogue scale, and background vibrations or electrical noise.
随机误差来自无法完全控制的因素。常见来源包括启动和停止计时器时的人为反应时间、读取模拟刻度时的视差，以及背景振动或电噪声。
Temperature changes, air currents, and fluctuations in power supply can also produce random scatter. Taking readings at different times of day or in different positions can introduce additional variability into the data.
温度变化、气流和电源波动也会产生随机散布。在一天中不同时间或不同位置读数会给数据引入额外的变异性。
- Human reaction time | 人的反应时间
- Parallax error | 视差
- Electrical noise | 电噪声
- Vibration | 振动
- Temperature fluctuation | 温度波动
10. Improving Precision | 提高精密度
To improve precision, you can repeat measurements several times and take the mean. You can also use instruments with higher resolution, such as a micrometer instead of a ruler, or a digital sensor instead of an analogue meter.
要提高精密度，你可以多次重复测量并取平均值。也可以使用分辨率更高的仪器，例如用千分尺代替尺子，或用数字传感器代替模拟表。
Automated data logging removes human reaction time and can record values at a constant rate, eliminating many random errors. Reducing the influence of external variables by shielding the experiment or maintaining constant temperature also helps.
自动数据采集可以消除人为反应时间，并以恒定速率记录数值，从而排除许多随机误差。通过屏蔽实验或保持恒温来减少外部变量的影响也有帮助。
N increased → σ of the mean = σ ÷ √N
11. Reporting Measurements with Uncertainty | 报告测量结果与不确定度
For example, a mass of 25.3 g with an uncertainty of ± 0.2 g is written as (25.3 ± 0.2) g.
在 AQA A-Level 物理中，测量结果应附上适当的不确定度。例如，质量为 25.3 g，不确定度为 ± 0.2 g，写作 (25.3 ± 0.2) g。
When combining measurements, uncertainties are added. Multiplying readings requires adding percentage uncertainties. These rules allow the final result to carry an uncertainty that reflects the precision of the original data.
组合测量时，不确定度需相加。对读数进行乘除时，需要将百分比不确定度相加。这些规则使最终结果的不确定度能够反映原始数据的精密度。
| Operation | Uncertainty rule |
| Addition / subtraction | Absolute uncertainties add |
| Multiplication / division | Percentage uncertainties add |
| Power | Multiply percentage uncertainty by the power |
12. Conclusion | 结论
Precision is essential for reliable physics experiments, but it is only one part of measurement quality. A precise measurement without accuracy is misleading. AQA A-level physics rewards students who understand the distinction, calculate uncertainties correctly, and justify their choice of instruments.
精密度对于可靠的物理实验至关重要，但它只是测量质量的一部分。没有准确度的精密测量会误导人。AQA A-Level 物理奖励那些理解两者区别、正确计算不确定度并能论证仪器选择的学生。
By repeating readings, using suitable instruments, and reporting uncertainties properly, you can maximise precision and produce results that are both meaningful and defensible.
通过重复读数、选用合适的仪器并正确报告不确定度，你可以最大化精密度，并得出既有意义又经得起推敲的结果。
Published by TutorHao | Physics Revision Series | aleveler.com
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