Analysing Results | 结果分析

📚 Analysing Results | 结果分析

In A-Level Physics, the ability to analyse experimental results is just as important as collecting data. Raw measurements carry little meaning until they are processed, combined with uncertainties, and displayed in a clear graphical form. For CIE learners, particularly those preparing for Papers 3 and 5, mastering result analysis means confidently identifying trends, quantifying errors, and using mathematical techniques such as linearisation to test physical relationships. This article provides a structured guide to the core skills needed to turn laboratory numbers into reliable scientific conclusions.

在A-Level物理中，分析实验结果的能力与收集数据同样重要。原始测量值只有在经过处理、与不确定度结合并以清晰的图形形式展示后，才具有意义。对于CIE学生，尤其是准备试卷3和试卷5的同学，掌握结果分析意味着能够自信地识别趋势、量化误差，并运用线性化等数学手段来检验物理关系。本文提供了一个结构化的指南，涵盖将实验室数字转化为可靠科学结论所需的核心技能。

1. Significant Figures and Scientific Notation | 有效数字与科学计数法

Every measured value should reflect the precision of the instrument used. The number of significant figures indicates the certainty of the measurement; for example, a ruler reading of 15.3 cm has three significant figures, while a micrometer might give 15.30 mm (four). When recording data in a table, all values in the same column must be stated to the same number of decimal places, matching the resolution of the measuring device. Scientific notation is useful for very large or small quantities, and it must be used consistently in calculations and on graph axes.

每个测量值都应反映出所使用仪器的精度。有效数字的位数表明了测量的确定性；例如，用直尺测量的读数15.3 cm有三位有效数字，而千分尺可能给出15.30 mm（四位）。在表格中记录数据时，同一列的所有值必须保留相同的小数位数，并与测量设备的分辨率相匹配。科学计数法对于非常大或小的量很有用，并且在计算和坐标轴上必须一致使用。

• Always match the decimal places to the instrument’s precision.
• 小数位数始终与仪器精度匹配。
• Convert to scientific notation when numbers are beyond 1000 or below 0.01.
• 当数值超过1000或小于0.01时，转换为科学计数法。
• In calculations, carry extra significant figures and round only at the final answer.
• 计算时保留额外的有效数字，仅在最终答案进行四舍五入。

---

2. Absolute, Relative and Percentage Uncertainties | 绝对、相对和百分比不确定度

a thermometer marked in 0.5 °C divisions has an absolute uncertainty of ±0.25 °C. The relative uncertainty is the absolute uncertainty divided by the measured value, and the percentage uncertainty multiplies that ratio by 100. Expressing uncertainty as a percentage makes it easier to compare the precision of different measurements.

单次读数的绝对不确定度通常是最小刻度的一半（或数字仪器的分辨率）。例如，一支分度值为0.5 °C的温度计，其绝对不确定度为±0.25 °C。相对不确定度是绝对不确定度除以测量值，而百分比不确定度将该比值乘以100。将不确定度表示为百分比，可以更容易地比较不同测量值的精度。

Type / 类型	Formula / 公式	Example / 例子
Absolute / 绝对	±Δx	Length = 25.0 ± 0.5 cm
Relative / 相对	Δx / x	0.5 / 25.0 = 0.02
Percentage / 百分比	(Δx / x) × 100%	0.02 × 100% = 2%

---

3. Recording Repeat Measurements | 记录重复测量值

When the same quantity is measured several times, the spread of values reveals random error. The best estimate is the arithmetic mean. The absolute uncertainty can be taken as half the range (for small numbers of repeats), or the standard deviation if enough data are collected. A table of repeat readings should always include the mean and the calculated uncertainty. This practice is essential when the experiment demands a measure of reliability, such as finding the period of a pendulum.

当同一物理量被多次测量时，数值的分散程度反映了随机误差。最佳估计值是算术平均值。绝对不确定度可以取极差的一半（对于少量重复测量），或者在有足够数据时取标准偏差。重复读数的表格应始终包含平均值和计算出的不确定度。这一做法在实验需要可靠性度量时至关重要，例如测定单摆的周期。

• For three readings, uncertainty ≈ (max - min)/2.
• 对于三次读数，不确定度 ≈ (最大值 - 最小值)/2。
• Mean value should quote the same number of decimal places as individual measurements.
• 平均值的小数位数应与单个测量值相同。
• Record absolute uncertainty to 1 or 2 significant figures.
• 绝对不确定度保留1或2位有效数字。

---

4. Combining Uncertainties: Addition and Subtraction | 合并不确定度：加法与减法

If two quantities are added or subtracted, the absolute uncertainties simply add. For example, when measuring a temperature rise θ₂ - θ₁, the absolute uncertainty in the rise is Δθ₁ + Δθ₂. This is because the uncertainties can both act in the same direction, maximising the error. Even when subtracting, uncertainties never cancel—they always combine to give a larger total absolute uncertainty.

如果两个量相加或相减，绝对不确定度直接相加。例如，测量温升θ₂ - θ₁时，温升的绝对不确定度为Δθ₁ + Δθ₂。这是因为不确定度可能作用于同一方向，从而最大化误差。即使在做减法时，不确定度也永不相消——它们总是合并成一个更大的总绝对不确定度。

If Q = A ± B, then ΔQ = ΔA + ΔB

若 Q = A ± B，则 ΔQ = ΔA + ΔB

---

5. Combining Uncertainties: Multiplication, Division and Powers | 合并不确定度：乘除与幂次

sum of the percentage uncertainties in A and B. When a quantity is raised to a power, its percentage uncertainty is multiplied by that power. For instance, if kinetic energy depends on v², the percentage uncertainty in KE from velocity alone is 2 × (% v). These rules are vital for propagating experimental errors through derived quantities.

对于乘法和除法，百分比（或相对）不确定度相加。如果Q = A × B 或 Q = A / B，则Q的百分比不确定度是A和B的百分比不确定度之和。当一个量被乘方时，其百分比不确定度乘以该幂次。例如，若动能依赖于v²，则仅速度带来的动能百分比不确定度为2 × (v的百分比不确定度)。这些规则对于通过导出量传播实验误差至关重要。

Operation / 运算	Rule for uncertainty / 不确定度规则
Q = A × B	%ΔQ = %ΔA + %ΔB
Q = A / B	%ΔQ = %ΔA + %ΔB
Q = An	%ΔQ = n × %ΔA

---

6. Plotting Graphs: Axes, Scales and Points | 绘制图表：坐标轴、标度和描点

Graphs reveal relationships hidden in columns of numbers. Both axes must be labelled with the quantity and unit, and the scale should be chosen so that points occupy more than half the grid in each direction. Avoid awkward divisions (e.g., 3, 7, 11 per cm). Data points should be plotted as fine crosses or dots surrounded by a small circle, and their coordinates must be readable without needing to calculate from the scale. A descriptive title helps the reader understand the graph’s purpose.

图表能够揭示隐藏在数字列中的关系。两个坐标轴都必须标注物理量和单位，标度应选择使得数据点在每个方向上占据超过一半的网格。避免不方便划分的标度（如每厘米3、7、11个单位）。数据点应绘制为细交叉或带小圆圈的圆点，且不通过标度计算就能直接读出坐标。一个描述性的标题有助于读者理解图表的目的。

• X-axis: independent variable; Y-axis: dependent variable.
• X轴：自变量；Y轴：因变量。
• Use units like m, s, kg, N; scale in 1s, 2s, 5s or multiples of 10.
• 使用m、s、kg、N等单位；标度采用1、2、5或10的倍数。
• Plot each point with a sharp pencil dot and enclose in a small circle.
• 用削尖的铅笔点画点，并用小圆圈圈住。

---

广告 / Sponsored

7. Error Bars and Uncertainty Rectangles | 误差棒和不确定度矩形

izontal bar by ±Δx. If both variables have significant uncertainty, an uncertainty rectangle (or cross‑ st fit passes within the error margins and to estimate the gradient uncertainty with worst‑fit lines.

误差棒直观地表示测量中的绝对不确定度。垂直误差棒在数据点上下各延伸±Δy；水平误差棒延伸±Δx。如果两个变量都有显著的不确定度，则应绘制不确定度矩形（或十字形棒）。这些图形表示能让你判断最佳拟合线是否穿过误差界限，并利用最差拟合线估算斜率的不确定度。

• Draw error bars as solid lines; total length = 2 × absolute uncertainty.
• 用实线绘制误差棒；总长度 = 2 × 绝对不确定度。
• If only one variable has uncertainty, vertical bars alone are acceptable.
• 若只有一个变量有不确定度，仅绘制垂直误差棒也可接受。
• Error bars must be clearly visible and not confused with grid lines.
• 误差棒必须清晰可见，不与网格线混淆。

---

8. Lines of Best Fit and Worst Fit | 最佳拟合线与最差拟合线

The line of best fit is drawn to pass through or as close to as many error bars as possible, with roughly equal numbers of points above and below the line. It does not have to go through the origin unless there is a valid physical reason. To find the uncertainty in the gradient, two worst‑fit lines are often drawn: the steepest and shallowest lines that still pass through the error bars of most points. The difference between their slopes gives an estimate of the gradient uncertainty.

最佳拟合线的绘制应尽可能穿过或接近更多的误差棒，且线上下的点数大致相等。除非有合理的物理理由，否则它不必通过原点。要计算斜率的不确定度，通常绘制两条最差拟合线：一条最陡线和一条最浅线，它们仍能穿过大多数点的误差棒。它们斜率之间的差值给出了斜率不确定度的估计。

Line / 线	How to draw / 如何绘制
Best fit / 最佳拟合	Balanced distribution of points; minimise vertical deviations.
Steepest / 最陡	Passes through bottom of first error bar and top of last, or vice versa.
Shallowest / 最浅	Opposite extreme to steepest line.

Δgradient ≈ (gradient_steep - gradient_shallow) / 2

Δ斜率 ≈ (最陡斜率 - 最浅斜率) / 2

---

9. Determining Gradient and Intercept Uncertainties | 确定斜率和截距的不确定度

Once the best‑ ference between extreme gradients. The intercept uncertainty is found by reading the intercepts of the steepest and shallowest lines and halving the difference. Always quote the final result as value ± absolute uncertainty, and ensure the number of decimal places matches the uncertainty. For a straight line through the origin, the gradient uncertainty alone may be sufficient.

一旦画出了最佳拟合线、最陡线和最浅线，从每条线的大三角形计算斜率。斜率的不确定度是极端斜率差值的一半。截距的不确定度通过读取最陡和最浅线的截距并取差值的一半得到。始终以数值 ± 绝对不确定度的形式给出最终结果，并确保小数位数与不确定度匹配。对于通过原点的直线，仅斜率不确定度可能就足够了。

• Use points on the line (not data points) for triangle.
• 使用线上的点（而非数据点）构建三角形。
• Gradient = (y₂ - y₁) / (x₂ - x₁); select widely separated coordinates.
• 斜率 = (y₂ - y₁) / (x₂ - x₁)；选择相距较远的坐标。
• Express gradient with units, e.g., m/s².
• 标明斜率的单位，如 m/s²。

---

10. Linearising Power‑Law Relationships | 幂律关系的线性化

Many experimental relationships are not straight lines in their raw form. For example, the period T of a simple pendulum relates to length L by T ∝ √L, giving a curve when T is plotted against L. By squaring the period, T² ∝ og x yields a straight line with gradient n. Transforming variables to obtain a linear graph is a key skill in CIE Paper 5 analysis.

许多实验关系在原始形式下并不是直线。例如，单摆的周期T与摆长L的关系为T ∝ √L，若绘制T对L的图会得到一条曲线。将周期平方后，T² ∝ L成为一条过原点的直线，斜率为4π²/g。类似地，若y = k xⁿ，绘制log y对log x的图会得到一条斜率为n的直线。通过变量变换得到线性图是CIE试卷5分析中的一项关键技能。

Relationship / 关系	Linearised form / 线性化形式	Gradient / 斜率
T = 2π√L/g	T² vs L	4π²/g
s = ut + ½at²	s/t vs t	½a
y = k xⁿ	log y vs log x	n

---

11. Using Logarithms to Test Exponential and Power Laws | 使用对数检验指数和幂律关系

When the relationship is exponential, y = A e^kx y = kx + ln A. A graph of ln y against x is linear with gradient k. For a power law, base‑ nd intercept. Students must include error bars on the logged points by converting the original absolute uncertainties into logarithmic form: Δ(ln y) ≈ Δy / y.

当关系为指数形式y = A e^kx时，取自然对数得到ln y = kx + ln A。此时ln y对x的图为一条直线，斜率为k。对于幂律关系，可以使用以10为底或自然对数：log y = n log x + log k。在这两种情况下，绘制取对数后的量可以将常数从斜率和截距中提取出来。学生必须通过将原始绝对不确定度转换为对数形式来在对数点上添加误差棒：Δ(ln y) ≈ Δy / y。

• Use ln for exponential decay (capacitor discharge, radioactive decay).
• 指数衰减使用ln（电容放电、放射性衰变）。
• Use log (any base) for power‑law tests; gradient gives the exponent.
• 使用log（任意底数）进行幂律检验；斜率给出指数。
• Quote uncertainties in ln to 1 or 2 decimal places.
• ln的不确定度保留1或2位小数。

---

12. Evaluating Experiment Quality and Suggesting Improvements | 评估实验质量并提出改进

the experimental value with an accepted or theoretical value using percentage difference: |(experimental - accepted)| / accepted × 100% hen it should be linear), identify systematic errors such as zero errors, incorrect assumptions, or uncontrolled variables. Propose specific improvements like using a fiducial marker, digital sensors, or better thermal insulation.

分析完成后，你必须反思结果的可靠性。使用百分比差异将实验值与公认值或理论值进行比较：|(实验值 - 公认值)| / 公认值 × 100%。较小的百分比差异表明准确度高。若图的最佳拟合线不支持预期的关系（例如，本应是线性关系却出现了弯曲），则需识别系统误差，如零位误差、不正确的假设或未控制的变量。提出具体的改进措施，例如使用基准标记、数字传感器或更好的隔热措施。

• Quote percentage difference to show accuracy, not just precision.
• 指出百分比差异以表明准确度，而不仅仅是精密度。
• Identify if intercept is significantly non‑zero; this often reveals a systematic error.
• 判断截距是否显著非零；这通常揭示了一个系统误差。
• Suggest improvements linked to the main source of error, e.g., “use a longer pendulum to reduce timing uncertainty”.
• 提出的改进应与主要误差来源相关，例如“使用更长的单摆以减少计时不确定度”。

---

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）