📚 Uncertainty | 不确定度
Every physical measurement has an uncertainty, whether it comes from the resolution of an instrument, the judgement of the experimenter, or the random variation between repeat readings. Understanding uncertainty allows a physicist to state how reliable a result is, to compare results with accepted values, and to decide where experimental improvements are most needed. In AQA A-level Physics, uncertainty questions appear in both theory papers and practical assessments.
每一次物理测量都有不确定度，它可能来自仪器的分辨率、实验者的判断，或重复读数之间的随机变化。理解不确定度使物理学家能够说明结果的可靠程度，将结果与公认值进行比较，并确定最需要改进实验的地方。在 AQA A-level 物理中，不确定度问题会出现在理论试卷和实验评估中。
1. Accuracy and Precision | 准确度与精密度
Accuracy describes how close a measured value is to the true or accepted value. A measurement is accurate if systematic errors are small. Precision describes how close repeated measurements are to each other, reflecting the size of random errors. An experiment can be precise but inaccurate, for example when a digital balance is poorly zeroed but gives very repeatable readings.
准确度描述测量值接近真实值或公认值的程度。如果系统误差小，测量就是准确的。精密度描述重复测量值彼此接近的程度，反映随机误差的大小。一个实验可以精密但不准确，例如一台数字天平零点未校准，却给出非常重复的读数。
Accuracy and precision are often confused in exam questions. A metre rule may have a precision of 1 mm but if its end is worn, every length may be too large by a constant amount, so accuracy is poor. Repeating the measurement does not correct this systematic error.
考试题中经常混淆准确度与精密度。一把米尺可能具有 1 mm 的精密度，但如果尺端磨损，每次长度都可能恒定偏大，因此准确度差。重复测量无法修正这一系统误差。
2. Types of Error: Systematic and Random | 误差类型：系统误差与随机误差
Systematic errors affect every reading in the same way and are not reduced by repeating measurements. Common sources include zero errors, wrongly calibrated instruments and parallax when reading a scale from a fixed angle. Random errors cause readings to scatter unpredictably and can be reduced by taking repeats and averaging.
系统误差以相同方式影响每次读数，不能通过重复测量来减小。常见来源包括零点误差、仪器校准不当以及从固定角度读数时的视差。随机误差使读数不可预测地分散，可以通过多次重复并取平均值来减小。
Zero error occurs when an instrument reads non-zero when it should read zero. For example, a newton meter reads 0.2 N with no load. This zero reading must be subtracted from every measurement, or the instrument should be adjusted before use.
零点误差指仪器在应当读数为零时却显示非零。例如，牛顿计在无负载时读数为 0.2 N。必须从每次测量中减去该零点读数，或者在使用前调整仪器。
3. Absolute, Fractional and Percentage Uncertainty | 绝对、相对与百分比不确定度
Fractional uncertainty is absolute uncertainty divided by the measured value, and percentage uncertainty is fractional uncertainty multiplied by 100% centage uncertainty is particularly useful when comparing measurements of different sizes.
绝对不确定度是以与测量本身相同单位表示的不确定度，例如 ±0.1 mm。相对不确定度是绝对不确定度除以测量值，百分比不确定度是相对不确定度乘以 100%。在比较不同大小的测量值时，百分比不确定度特别有用。
%Δx = (Δx / x) × 100%
ement of 25 mm with absolute uncertainty ±0.5 mm has percentage uncertainty (0.5/25) × 100% = 2% e uncertainty applies to a smaller reading of 5 mm, the percentage uncertainty rises to 10%. This shows why measuring small quantities usually gives larger percentage uncertainty.
测量值 25 mm，绝对不确定度为 ±0.5 mm，百分比不确定度为 (0.5/25) × 100% = 2%。如果同样的绝对不确定度用于较小的 5 mm 读数，百分比不确定度将上升至 10%。这说明为什么测量小量通常会产生更大的百分比不确定度。
4. Uncertainty in a Single Measurement | 单次测量的不确定度
the resolution stated by the manufacturer. If the measurement is difficult to judge because of fluctuating readings, a larger uncertainty may be justified.
对于模拟刻度，读数不确定度通常取最小分度值的一半。对于数字仪器，不确定度至少为最后一位显示数字的 ±1，或制造商标明的分辨率。如果由于读数波动难以判断，可以合理采用更大的不确定度。
V. If the reading fluctuates between 1.52 V and 1.56 V, the half-range is 0.02 V, which is larger than the resolution and should be used instead.
如果仪器数字显示为 0.01 V，不确定度为 ±0.01 V。如果读数在 1.52 V 和 1.56 V 之间波动，极差的一半为 0.02 V，大于分辨率，应改用该值。
5. Combining Uncertainties: Addition and Subtraction | 组合不确定度：加减法
or Q = A − B, the rule is ΔQ = ΔA + ΔB. Even when subtracting, uncertainties never cancel.
当两个被测量 A 和 B 相加或相减得到结果 Q 时，绝对不确定度相加。这给出了最大可能的不确定度。对于 Q = A + B 或 Q = A − B，规则为 ΔQ = ΔA + ΔB。即使做减法，不确定度也绝不会相互抵消。
ΔQ = ΔA + ΔB
that the measured value changes but the absolute uncertainty remains the sum of the individual absolute uncertainties.
计算示例：A = 10.0 ± 0.2 cm，B = 5.0 ± 0.1 cm。和 A + B = 15.0 ± 0.3 cm。差 A − B = 5.0 ± 0.3 cm。注意测量值改变，但绝对不确定度仍然是各自绝对不确定度之和。
6. Combining Uncertainties: Multiplication, Division and Powers | 组合不确定度：乘除与幂
ision, the fractional or percentage uncertainties are added. For Q = AB or Q = A/B, we use %ΔQ = %ΔA + % power, the percentage uncertainty is multiplied by the power. For Q = Aⁿ, the rule becomes %ΔQ = n × %ΔA.
对于乘法或除法，相对或百分比不确定度相加。对于 Q = AB 或 Q = A/B，使用 %ΔQ = %ΔA + %ΔB。当一个量被升到幂次时，百分比不确定度要乘以该幂次。对于 Q = Aⁿ，规则变为 %ΔQ = n × %ΔA。
%ΔQ = %ΔA + %ΔB and %ΔQ = n × %ΔA
For example, if a cube' side is measured as 2.0 ± 0.1 cm, its volume percentage uncertainty is 3 × (0.1/2.0 × 100%) = 15% ertainty, so the absolute uncertainty doubles but the percentage uncertainty is unchanged: %ΔQ = %ΔA.
例如，若测得立方体边长为 2.0 ± 0.1 cm，则体积的百分比不确定度为 3 × (0.1/2.0 × 100%) = 15%。当公式包含常数时，例如 Q = 2A，常数 2 没有不确定度，因此绝对不确定度翻倍，但百分比不确定度不变：%ΔQ = %ΔA。
7. Uncertainty from Repeated Measurements | 重复测量的不确定度
d quote the result as mean ± half-range, unless the question specifies standard deviation.
当获得一组重复读数时，最佳估计值是平均值。A-level 中绝对不确定度的一个简单估计是极差的一半：(最大值 − 最小值)/2。这覆盖了大部分散布范围，且易于计算。除非题目指定使用标准差，否则应把结果表示为平均值 ± 极差的一半。
mm and the range is 0.04 mm, so the uncertainty is ±0.02 mm. The result is 0.53 ± 0.02 mm.
示例：某金属丝直径五次读数（单位 mm）为 0.52、0.54、0.53、0.55、0.51。平均值是 0.53 mm，极差是 0.04 mm，因此不确定度为 ±0.02 mm。结果为 0.53 ± 0.02 mm。
8. Error Bars and Graphs | 误差棒与图表
Error bars show the absolute uncertainty in each plotted point. The vertical bar extends from y − Δy to y + Δy. If the horizontal axis has significant uncertainty, horizontal error bars are also drawn. Error bars allow a visual judgement of whether a straight line can reasonably fit all the data.
误差棒表示每个数据点对应的
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