📚 Data analysis and modelling physics | 数据分析与物理建模
Physics progresses through a constant dialogue between experimental measurement and theoretical description. Data analysis is the process of transforming raw observations into statistically sound conclusions, while modelling uses mathematical structures to represent physical relationships. The combination of these skills allows physicists to extract meaning from uncertainty, compare competing explanations, and make predictions about the natural world.
物理学通过实验测量与理论描述之间的持续对话而进步。数据分析是将原始观测转化为统计上可靠结论的过程，而建模则使用数学结构表示物理关系。这些技能的结合使物理学家能够从不确定度中提取意义、比较竞争性解释并对自然界做出预测。
1. The role of measurement in physics | 测量在物理学中的作用
Every physical quantity, from length to electric current, is defined through its measurement procedure. Measurements are the cornerstone of experimental physics because they provide the numerical evidence needed to test laws and theories. Without careful quantification, physical concepts remain qualitative and cannot be used to make precise predictions.
每一种物理量，从长度到电流，都通过其测量程序来定义。测量是实验物理学的基石，因为它们提供了检验定律和理论所需的数值证据。如果缺乏精细的量化，物理概念将停留在定性层面，无法用于做出精确预测。
Measurements are never perfect. Every instrument has a finite resolution, and every observer introduces some degree of judgment. The true value of a quantity is an idealization; what we record is a measured value accompanied by an estimate of its uncertainty. Understanding this distinction is the first step toward meaningful data analysis.
测量永远不可能完美。每种仪器都具有有限的分辨力，每位观测者都会引入一定程度的判断。一个物理量的真值是一种理想化概念；我们记录的则是伴随其不确定度估计的测量值。理解这一区别是进行有意义数据分析的第一步。
2. Types of errors and uncertainties | 误差与不确定度的类型
Random errors cause readings to scatter symmetrically about the mean. They arise from unpredictable fluctuations in the measurement process, such as small variations in reaction time or electronic noise. Repeating the measurement and calculating the standard deviation quantifies this scatter. The precision of a measurement improves with more repeats because the standard error of the mean decreases as 1/√n.
随机误差导致读数在平均值两侧对称分散。它们源于测量过程中不可预测的波动，例如反应时间的微小变化或电子噪声。通过重复测量并计算标准差可以量化这种分散程度。随着重复次数增加，测量的精密度会提高，因为均值的标准误差按 1/√n 减小。
Systematic errors, in contrast, shift all readings in the same direction. They may be caused by a zero offset on a meter, a parallax error when reading a scale, or an incorrect calibration. Unlike random errors, repeating the measurement does not reveal systematic errors; they must be identified by using a different technique or comparing with a known standard. Accuracy refers to how close a measurement is to the accepted value, and systematic errors reduce accuracy even when precision is high.
相比之下，系统误差会使所有读数朝同一方向偏移。它们可能由仪表的零点偏差、读数时的视差或校准错误引起。与随机误差不同，重复测量并不能揭示系统误差；必须通过使用不同技术或与已知标准进行比较来识别。准确度指测量值接近公认值的程度，即使精密度很高，系统误差也会降低准确度。
± 0.1) cm, or in relative form, where the fractional uncertainty is 0.1/12.5 ≈ 0.008 (0.8%). Percentage uncertainty is the relative uncertainty multiplied by 100%. Distinguishing between these forms becomes crucial when combining measurements.
不确定度可以用绝对形式表示，例如长度为 (12.5 ± 0.1) cm，或者用相对形式表示，其中比例不确定度为 0.1/12.5 ≈ 0.008 (0.8%)。百分不确定度是相对不确定度乘以 100%。当组合多个测量结果时，区分这些形式至关重要。
3. Propagation of uncertainties | 不确定度的传递
add: ΔZ = ΔA + ΔB. For multiplication or division, such as Z = A × B, fractional uncertainties add in quadrature if one assumes normal distributions: (ΔZ/Z) = √[(ΔA/A)² + (ΔB/B)²]. In IB internal assessment work, students often begin with the simpler worst-case addition of fractional uncertainties: ΔZ/Z = ΔA/A + ΔB/B, which gives a conservative overestimate.
当推导量依赖多个测量变量时，最终不确定度由各个不确定度组合而成。最简单的规则适用于测量值相互独立且误差为随机的情况。对于加减运算，例如 Z = A + B，绝对不确定度相加：ΔZ = ΔA + ΔB。对于乘除运算，例如 Z = A × B，若假设正态分布，比例不确定度以平方和开根号方式组合：(ΔZ/Z) = √[(ΔA/A)² + (ΔB/B)²]。在 IB 内部评估作业中，学生往往先采用更简单的比例不确定度直接相加的方法：ΔZ/Z = ΔA/A + ΔB/B，这会给出保守的过高估计。
ties and independent variables. Practical examples include the period of a pendulum T = 2π√L/g, where the uncertainty in g derived from measurements of L and T depends on both ΔL and ΔT.
对于一般函数 f(x, y, ...)，不确定度可利用偏导数估算：(Δf)² ≈ (∂f/∂x)² (Δx)² + (∂f/∂y)² (Δy)² + ... 。该公式假定不确定度较小且变量相互独立。实际例子包括单摆周期 T = 2π√L/g，此时由 L 和 T 的测量值导出的 g 的不确定度同时依赖于 ΔL 和 ΔT。
Below is a summary of common propagation rules:
以下是常见传递规则的总结：
| Operation | Worst-case rule | Quadrature rule (independent) |
|---|---|---|
| Z = A ± B | ΔZ = ΔA + ΔB | ΔZ = √[(ΔA)² + (ΔB)²] |
| Z = A × B or A ÷ B | ΔZ/Z = ΔA/A + ΔB/B | ΔZ/Z = √[(ΔA/A)² + (ΔB/B)²] |
| Z = Aⁿ | ΔZ/Z = |n|(ΔA/A) | ΔZ/Z = |n|(ΔA/A) |
Using these rules, an experimenter can assign a defensible uncertainty to any calculated result, which is essential for comparing the result with theoretical predictions.
运用这些规则，实验者可以为任何计算结果赋予合理的、经得住推敲的不确定度，这对于将结果与理论预测进行比较至关重要。
4. Graphical analysis and linearization | 图形分析与线性化
Plotting data on a graph reveals trends that raw tables often hide. In physics, the goal of many investigations is to determine whether two variables follow a linear, power-law, or exponential relationship. Instead of performing a complex non-linear fit, scientists frequently transform the data to produce a straight line, a process called linearization. The resulting graph is not only easier to interpret, but its slope and intercept then have direct physical meaning.
将数据绘制在图上可以揭示原始表格常常隐藏的趋势。在物理学中，许多探究的目标是确定两个变量是否遵循线性、幂律或指数关系。科学家们通常不去进行复杂的非线性拟合，而是对数据进行变换以产生一条直线，这一过程称为线性化。所得图形不仅更易于解读，而且其斜率和截距具有直接的物理意义。
lotting ln y against t produces a straight line whose slope is −λ. A common mistake is to forget that the logarithm is dimensionless, so one must ensure the argument is a pure number, for instance by dividing by a unit before taking the log.
对于幂律关系 y = k·xⁿ，取对数得到 lg y = n·lg x + lg k。以 lg y 对 lg x 作图可得到一条斜率为 n、纵轴截距为 lg k 的直线。对于指数衰减 y = A·exp(−λt)，取自然对数可使其线性化：ln y = ln A − λt。以 ln y 对 t 作图可得到一条斜率为 −λ 的直线。一个常见错误是忘记对数必须是无量纲的，因此在取对数前应确保自变量为纯数，例如除以一个单位。
When constructing a graph, error bars should be included to represent the uncertainty in each data point. The best-fit line should be drawn by eye or computed, and its uncertainty can be estimated by drawing steepest and shallowest reasonable lines through the error bars. The range of slopes obtained gives an uncertainty in the gradient, which propagates to the final physical quantity.
在构建图形时，应包含误差棒以表示每个数据点的不确定度。最佳拟合线可通过目测或计算得出，其不确定度可借助穿过误差棒的最陡和最缓合理直线来估算。所得斜率范围给出了梯度的不确定度，该不确定度将进一步传递至最终的物理量。
Linearised form: lg y = n·lg x + lg k
5. Curve fitting and regression | 曲线拟合与回归
When the relationship between variables is linear, the method of least squares provides an objective way to determine the slope m and intercept c. The aim is to minimise the sum of the squared residuals: Σ (yᵢ − m·xᵢ − c)². The resulting formulas are:
当变量间的关系为线性时，最小二乘法提供了确定斜率 m 与截距 c 的客观方法。其目标是使残差平方和 Σ (yᵢ − m·xᵢ − c)² 最小。所得公式如下：
m = [N·Σ(xᵢyᵢ) − Σxᵢ·Σyᵢ] / [N·Σ(xᵢ²) − (Σxᵢ)²]
c = (Σyᵢ − m·Σxᵢ) / N
Modern software, including spreadsheets and data-analysis packages, routinely applies least squares to any user-defined function. However, blindly trusting the output without checking the assumptions can lead to misinterpretation. The residuals should be randomly scattered around zero; any systematic pattern suggests an inappropriate model or unaccounted-for systematic error.
现代软件（包括电子表格和数据分析软件包）能够轻松地对任何用户定义的函数应用最小二乘法。然而，若不检查假设而盲目信任输出，将可能导致误解。残差应随机散布在零附近；任何系统性图样都表明模型不合适或存在未考虑的系统误差。
The correlation coefficient r measures the strength of a linear relationship, while the coefficient of determination r² indicates the proportion of the variance in the dependent variable that is predictable from the independent variable. A high r² is not synonymous with a causal link, nor does it guarantee that a linear model is physically correct.
相关系数 r 衡量线性关系的强度，而决定系数 r² 则表示因变量中可由自变量预测的方差比例。高 r² 值并不等同于因果关系，也不能保证线性模型在物理上是正确的。
6. The χ² test for goodness of fit | 卡方拟合优度检验
When a set of experimental data is compared with a theoretical model that predicts the expected frequencies Oᵢ, the χ² statistic quantifies the agreement:
当一组实验数据与预测期望频数 Eᵢ 的理论模型进行比较时，χ² 统计量可量化其符合程度：
χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ
A small χ² indicates close agreement; a large χ² suggests that the model may be inadequate. The critical value depends on the number of degrees of freedom ν (number of categories minus the number of fitted parameters). In physics investigations, a χ² per degree of freedom (χ²/ν) close to 1 often signals a good fit, while a value much larger than 1 indicates either underestimation of experimental uncertainties or a genuinely poor model.
小的 χ² 值表示高度吻合；大的 χ² 值则暗示模型可能不适用。临界值取决于自由度 ν（类别数减去拟合参数个数）。在物理探究中，每个自由度的 χ² 值 (χ²/ν) 若接近 1，通常标志着良好的拟合，而远大于 1 则表明要么低估了实验不确定度，要么模型本身确实不佳。
For example, in a radioactivity experiment where a Geiger counter records counts in fixed time intervals, the expected distribution can be compared with the Poisson model using a χ² test. This provides a quantitative measure of whether the observed fluctuations are consistent with random decay. Such tests add rigour to conclusions that might otherwise be based on visual inspection of a graph.
例如，在放射性实验中，若盖革计数器在固定时间间隔内记录计数，可通过 χ² 检验将观察分布与泊松模型进行比较。这为判断观测波动是否与随机衰变一致提供了定量量度。此类检验可为原本仅凭肉眼观察图形得出的结论增添严谨性。
7. Model building and validation | 模型构建与验证
Building a physical model begins with simplifying assumptions — for instance, ignoring air resistance or treating a spring as massless. The model is then expressed mathematically, often as a differential equation or a functional relationship. Validation involves comparing the model's predictions with experimental data, not only in the region where it was calibrated but also under new conditions. A model that fails to extrapolate correctly must be revised or discarded.
构建物理模型始于简化假设——例如，忽略空气阻力或将弹簧视为无质量。随后，模型通过数学形式表达，通常是一个微分方程或函数关系。验证过程不仅要与模型校准区域内的实验数据进行比较，还要在新条件下进行检验。若模型无法正确外推，则必须修正或舍弃。
Good validation practice includes analyzing residuals, performing sensitivity tests, and checking the physical plausibility of fitted parameters. For example, if a thermal expansion model yields a negative value for the expansion coefficient of a metal, the model or the data analysis procedure is likely flawed. Parameter uncertainties obtained from covariance matrices or bootstrap methods reveal how tightly the data constrain the model.
良好的验证实践包括分析残差、执行灵敏度测试以及检查拟合参数的物理合理性。例如，若一个热膨胀模型给出的金属膨胀系数为负值，则该模型或数据分析过程很可能存在缺陷。通过协方差矩阵或自助法获得的参数不确定度，可揭示数据对模型的约束程度。
Every model has a domain of validity. Newtonian mechanics works perfectly for everyday speeds but fails at velocities approaching the speed of light. Recognizing these boundaries is as important as fitting the model itself.
每一个模型都有其有效域。牛顿力学在日常生活速度下表现完美，但在接近光速时将失效。认识到这些边界与拟合模型本身同等重要。
8. Comparing models: AIC and BIC | 模型比较：AIC与BIC
Often several competing models can explain the same dataset. The Akaike information criterion (AIC) and the Bayesian information criterion (BIC) offer quantitative methods to select the most parsimonious model — one that fits well but does not contain unnecessary parameters.
通常多个竞争模型都能解释同一数据集。赤池信息准则 (AIC) 与贝叶斯信息准则 (BIC) 提供了定量方法，以选择最简洁的模型——即拟合良好但未包含不必要参数的模型。
AIC = 2k − 2 ln(L)
BIC = k·ln(N) − 2 ln(L)
Here k is the number of estimated parameters, L is the maximised likelihood, and N is the number of data points. Lower values of AIC or BIC indicate a better trade-off between goodness of fit and complexity. BIC imposes a heavier penalty for extra parameters, making it more conservative in model selection. In IB extended essays or internal assessments where students compare, for example, a linear damping model with a quadratic damping model, reporting AIC
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