Bivariate Statistics | 双变量统计分析方法

📚 Bivariate Statistics | 双变量统计分析方法

Bivariate statistics concerns the analysis of two variables measured on the same set of subjects, with the aim of understanding whether and how they co-vary. This is a core topic in the IB Mathematics curriculum, appearing in both Analysis and Approaches and Applications and Interpretation.

双变量统计研究的是在同一组对象上测量的两个变量，目的是理解它们之间是否以及如何共同变化。这是 IB 数学课程的核心主题，同时出现在分析与方法（AA）和应用与解释（AI）中。

1. Scatter Plots and Visual Trends | 散点图与视觉趋势

A scatter plot is the fundamental graphical tool in bivariate analysis. Each point on the plot corresponds to a paired observation \((x_i, y_i)\), where the horizontal axis represents the explanatory variable and the vertical axis represents the response variable.

散点图是双变量分析中最基本的图形工具。图中每个点对应一对观测值 \((x_i, y_i)\)，横轴表示解释变量，纵轴表示响应变量。

When examining a scatter plot, three primary features deserve attention:

在审视散点图时，有三个主要特征值得关注：

Direction: a positive trend slopes upward, a negative trend slopes downward, and no direction indicates no apparent linear association.
方向：正向趋势向上倾斜，负向趋势向下倾斜，无方向则表明无明显线性关联。
Form: the pattern may be linear, curved, or clustered; the form guides the choice of the most suitable statistical model.
形式：模式可能是线性的、弯曲的或分群的；形式决定了最合适统计模型的选择。
Strength: the degree to which points cluster around a central pattern determines how strongly the variables are associated.
强度：点围绕中心模式聚集的程度决定了变量之间关联的强度。

2. Pearson Correlation Coefficient r | 皮尔逊相关系数 r

The Pearson product-moment correlation coefficient, denoted by \(r\), quantifies the strength and direction of a linear relationship between two quantitative variables. Its value always lies between \(-1\) and \(+1\), inclusive.

皮尔逊积矩相关系数（记作 \(r\)）量化了两个定量变量之间线性关系的强度与方向。其值始终介于 \(-1\) 和 \(+1\) 之间（含端点）。

The formula for \(r\) is commonly expressed as follows:

\(r\) 的公式通常表述如下：

r = Σ((xᵢ − x̄)(yᵢ − ȳ)) ÷ √(Σ(xᵢ − x̄)² × Σ(yᵢ − ȳ)²)

Here, \(x̄\) and \(ȳ\) denote the sample means of the two variables. The numerator is the sum of the products of deviations from the respective means, while the denominator normalises this sum to the product of the standard deviations.

其中 \(x̄\) 和 \(ȳ\) 分别表示两个变量的样本均值。分子是各变量偏离其均值之积的总和，分母则将该总和标准化为标准差之积。

Interpretation guidance for IB examinations: \(r = 1\) indicates a perfect positive linear relationship, \(r = −1\) a perfect negative linear relationship, and \(r = 0\) no linear relationship. Values such as \(|r| ≥ 0.8\) are typically considered strong, while \(|r| ≤ 0.3\) is considered weak.

IB 考试中的解释指引：\(r = 1\) 表示完全正线性关系，\(r = −1\) 表示完全负线性关系，\(r = 0\) 表示无线性关系。通常 \(|r| ≥ 0.8\) 视为强相关，而 \(|r| ≤ 0.3\) 视为弱相关。

3. Covariance and Its Role | 协方差及其作用

Covariance measures the joint variability of two variables. For a sample, it is defined as the average of the products of deviations from the means:

协方差度量两个变量的联合变异性。对于样本而言，其定义为各变量偏离均值之乘积的平均值：

sₓᵧ = Σ((xᵢ − x̄)(yᵢ − ȳ)) ÷ (n − 1)

A positive covariance indicates that the variables tend to move in the same direction, whereas a negative covariance indicates that they move in opposite directions. However, covariance is scale-dependent and difficult to interpret directly; the correlation coefficient is the standardised version of covariance.

正协方差表明变量倾向于同向变动，而负协方差表明它们反向变动。然而，协方差受尺度影响，难以直接解读；相关系数正是协方差的标准化形式。

In fact, the relationship between \(r\) and the sample covariance is \(r = sₓᵧ / (sₓ × sᵧ)\), where \(sₓ\) and \(sᵧ\) are the sample standard deviations of \(x\) and \(y\).

事实上，\(r\) 与样本协方差的关系为 \(r = sₓᵧ / (sₓ × sᵧ)\)，其中 \(sₓ\) 和 \(sᵧ\) 分别是 \(x\) 和 \(y\) 的样本标准差。

4. The Least-Squares Regression Line | 最小二乘回归线

When a linear relationship exists between two variables, we may fit a regression line of the form \(y = a + bx\), where \(b\) is the slope and \(a\) is the \(y\)-intercept. The least-squares principle chooses the line that minimises the sum of squared vertical distances between observed points and the fitted line.

当两个变量之间存在线性关系时，我们可以拟合形如 \(y = a + bx\) 的回归线，其中 \(b\) 是斜率，\(a\) 是 \(y\) 截距。最小二乘原理选择使观测点到拟合线的垂直距离平方和最小的那条直线。

The slope and intercept are calculated as follows:

斜率和截距的计算方法如下：

b = r × (sᵧ ÷ sₓ) , a = ȳ − b × x̄

Equivalently, \(b = sₓᵧ / sₓ²\). The regression line always passes through the point \((x̄, ȳ)\), the centroid of the data set.

等价地，\(b = sₓᵧ / sₓ²\)。回归线始终经过数据集的质心 \((x̄, ȳ)\)。

On a graphing calculator, the regression line is obtained via the statistical computation menu. IB students must know how to interpret the values of \(a\) and \(b\) in context, including their units.

在图形计算器上，通过统计计算菜单可获得回归线。IB 学生必须能够结合语境解读 \(a\) 和 \(b\) 的值，包括它们的单位。

5. Residuals and Residual Plots | 残差与残差图

A residual is the difference between an observed value and the value predicted by the regression line at the same \(x\):

残差是观测值与回归线在相同 \(x\) 处预测值之间的差：

eᵢ = yᵢ − ŷᵢ

The sum of all residuals in a least-squares regression is always zero (apart from rounding error). Residual plots are crucial for assessing model adequacy: a suitable linear model should yield residuals scattered randomly around zero with no systematic pattern.

最小二乘回归中所有残差的总和始终为零（舍入误差除外）。残差图对于评估模型适用性至关重要：合适的线性模型应使残差随机散布在零附近，无系统模式。

Common residual patterns to recognise:

需要识别的常见残差模式：

Random scatter: the linear model is appropriate.
随机散布：线性模型是合适的。
Curved pattern: a nonlinear model may fit better.
弯曲模式：非线性模型可能拟合更好。
Increasing spread (fan shape): indicates heteroscedasticity, violating the assumption of constant variance.
展宽加大（扇形）：表明方差不齐，违反方差恒定的假设。

6. Coefficient of Determination R² | 决定系数 R²

The coefficient of determination, \(R²\), represents the proportion of the total variation in the response variable that is explained by the regression model. For simple linear regression, \(R² = r²\).

决定系数 \(R²\) 表示回归模型所解释的响应变量总变异所占的比例。在简单线性回归中，\(R² = r²\)。

Total variation can be decomposed into the sum of squares of regression and the sum of squares of residuals:

总变异可分解为回归平方和与残差平方和：

SS_total = SS_regression + SS_residual , R² = SS_regression ÷ SS_total

For example, if \(R² = 0.84\), then 84 percent of the variability in the response variable is accounted for by the explanatory variable, and the remaining 16 percent is due to other factors or random error. In IB examinations, \(R²\) is commonly used to compare the suitability of competing models.

例如，若 \(R² = 0.84\)，则响应变量 84% 的变异性可由解释变量解释，其余 16% 归因于其他因素或随机误差。在 IB 考试中，\(R²\) 常用于比较备选模型的适用性。

7. Interpretation of Slope and Intercept | 斜率与截距的解读

In the regression equation \(\hat{y} = a + bx\), the slope \(b\) represents the predicted change in the response variable \(y\) for a one-unit increase in the explanatory variable \(x\). The intercept \(a\) predicts the value of \(y\) when \(x = 0\).

在回归方程 \(\hat{y} = a + bx\) 中，斜率 \(b\) 表示解释变量 \(x\) 每增加一个单位时响应变量 \(y\) 的预测变化量。截距 \(a\) 预测当 \(x = 0\) 时 \(y\) 的值。

Students must exercise caution when interpreting the intercept: if the range of the data does not include \(x = 0\), then the intercept has no practical interpretation and should not be extrapolated beyond the observed domain.

学生在解读截距时必须谨慎：若数据范围不包含 \(x = 0\)，则截距没有实际意义，不应外推至观测域之外。

Consider a study examining the relationship between hours studied and examination score, where the regression line is \(\hat{y} = 32.5 + 4.8x\). The slope of 4.8 means that each additional hour of study is associated with a predicted increase of 4.8 marks in the examination score.

考虑一项研究，考察学习小时数与考试分数之间的关系，假设回归线为 \(\hat{y} = 32.5 + 4.8x\)。斜率 4.8 意味着每额外学习一小时，考试分数预测增加 4.8 分。

8. Interpolation and Extrapolation | 内插与外推

Interpolation refers to making predictions within the observed range of the explanatory variable; this is generally safe and statistically valid. Extrapolation refers to making predictions outside this range; this is risky because the linear relationship may not persist beyond the observed data.

内插是指在解释变量的观测范围内进行预测；这通常是安全且统计上有效的。外推是指在观测范围之外进行预测；这存在风险，因为线性关系在观测数据之外可能不再持续。

For example, suppose height and shoe size data are collected from a sample of students aged 14 to 18. Predicting shoe size for a 16-year-old student is interpolation, while predicting shoe size for a 40-year-old adult is extrapolation and likely inaccurate.

例如，假设从 14 至 18 岁的学生样本中收集身高和鞋码数据。预测 16 岁学生的鞋码属于内插，而预测 40 岁成年人的鞋码则属于外推，很可能不准确。

Key points for examination:

考试要点：

State clearly whether a prediction is interpolation or extrapolation.
明确说明预测属于内插还是外推。
Mention the limitations and risks of extrapolation when applying the regression model outside the data range.
在将回归模型应用于数据范围之外时，提及外推的局限性和风险。

9. Outliers and Influential Points | 异常值与强影响点

An outlier is an observation whose value deviates markedly from the overall pattern of the data. In regression analysis, an influential point is an observation whose removal would substantially change the fitted regression line.

异常值是其取值显著偏离整体数据模式的观测。在回归分析中，强影响点是指其移除会显著改变拟合回归线的观测。

Outliers in bivariate data can be detected using the following methods:

双变量数据中的异常值可以通过以下方法检测：

Visually, by inspecting the scatter plot for points far from the main cluster.
数值上，通过检查残差图中的大绝对值残差来识别。
Numerically, by examining residuals with large absolute values in the residual plot.
直观上，通过检查散点图中远离主体集群的点来识别。

It is essential to distinguish between an outlier in the \(x\)-direction and an outlier in the \(y\)-direction: a point with an extreme \(x\)-value but lying close to the regression line may still exert disproportionate influence on the slope.

区分 \(x\) 方向异常值和 \(y\) 方向异常值至关重要：\(x\) 值极端但靠近回归线的点仍可能对斜率产生不成比例的影响。

10. Correlation vs. Causation | 相关关系与因果关系

Perhaps the most frequently tested conceptual point in IB examinations is that correlation does not imply causation. A strong correlation between two variables may arise from several mechanisms, only one of which is a causal relationship.

IB 考试中最常考查的概念点或许就是：相关并不意味着因果。两个变量之间的强相关可能源于多种机制，其中只有一种是因果关系。

Possible explanations for an observed association:

观察到的关联可能的原因解释：

Direct causation: changes in one variable cause changes in the other.
直接因果：一个变量的变化引起另一个变量的变化。
Reverse causation: the response variable actually affects the explanatory variable.
反向因果：响应变量实际上影响着解释变量。
Confounding: a third (lurking) variable influences both variables, creating a spurious association.
混杂因素：第三个（潜藏的）变量同时影响两个变量，产生虚假关联。
Coincidence: the association occurs purely by chance, especially in small samples.
巧合：关联纯属偶然发生，尤其在小样本中。

In examination answers, students should explicitly identify whether the study is observational or experimental. Causal conclusions can only be justified by a well-designed randomised experiment, never by an observational study alone.

在考试作答中，学生应明确指出研究是观察性研究还是实验性研究。因果结论只能由设计良好的随机实验来证明，仅凭观察性研究是不够的。

11. Transformations and Nonlinear Models | 变换与非线性模型

When the scatter plot exhibits a curved pattern, the linear model is inappropriate. In such cases, transforming one or both variables may linearise the relationship, enabling the use of linear regression techniques.

当散点图呈现弯曲模式时，线性模型不再适用。在这种情况下，对其中一个或两个变量进行变换可能使关系线性化，从而能够使用线性回归技术。

Common transformations in the IB curriculum:

IB 课程中常见的变换：

Exponential growth: if \(y\) grows exponentially with \(x\), then \(\ln y\) plotted against \(x\) will be approximately linear.
指数增长：若 \(y\) 随 \(x\) 指数增长，则 \(\ln y\) 对 \(x\) 作图将近似为直线。
Power model: if \(y = kx^n\), then \(\ln y\) against \(\ln x\) yields a linear pattern with slope \(n\).
幂模型：若 \(y = kx^n\)，则 \(\ln y\) 对 \(\ln x\) 作图将产生斜率为 \(n\) 的线性模式。
Quadratic trend: plotting \(y\) against \(x²\) may linearise a parabolic relationship.
二次趋势：将 \(y\) 对 \(x²\) 作图可能会使抛物线关系线性化。

When a logarithmic transformation is applied, the regression equation becomes \(\ln \hat{y} = a + bx\), which is equivalent to \(\hat{y} = e^{a} × e^{bx}\). Students must be able to convert between logarithmic and exponential forms.

当应用对数变换时，回归方程变为 \(\ln \hat{y} = a + bx\)，等价于 \(\hat{y} = e^{a} × e^{bx}\)。学生必须能够在对数形式和指数形式之间进行转换。

12. Worked Example and Exam Strategies | 例题与应试策略

Consider the following table showing the number of hours studied \(x\) and the corresponding examination scores \(y\) for five students:

考虑下表，展示五名学生的学习小时数 \(x\) 与相应的考试分数 \(y\)：

x (hours)	1	2	3	4	5
y (score)	40	55	58	70	82

First, we compute the means: \(x̄ = 3\) and \(ȳ = 61\). The covariance is \(sₓᵧ = 22\), the variance of \(x\) is \(sₓ² = 2.5\), and therefore the slope is \(b = 22 ÷ 2.5 = 8.8\). The intercept is \(a = 61 − 8.8 × 3 = 34.6\). The regression line is \(\hat{y} = 34.6 + 8.8x\).

首先，计算均值：\(x̄ = 3\)，\(ȳ = 61\)。协方差 \(sₓᵧ = 22\)，\(x\) 的方差 \(sₓ² = 2.5\)，因此斜率 \(b = 22 ÷ 2.5 = 8.8\)。截距 \(a = 61 − 8.8 × 3 = 34.6\)。回归线为 \(\hat{y} = 34.6 + 8.8x\)。

To find the correlation coefficient, we need \(sᵧ² = 82\), so \(r = 22 ÷ √(2.5 × 82) ≈ 0.974\). Thus \(R² ≈ 0.949\), meaning approximately 94.9 percent of the variation in scores is explained by study hours.

为求相关系数，我们需要 \(sᵧ² = 82\)，所以 \(r = 22 ÷ √(2.5 × 82) ≈ 0.974\)。因而 \(R² ≈ 0.949\)，意味着大约 94.9% 的分数变异可由学习小时数解释。

Exam strategies to remember:

需要牢记的应试策略：

Always denote variables clearly and state the regression equation with symbols.
始终清晰标明变量并写出带有符号的回归方程。
Use your GDC efficiently; practice locating the statistics menus before the examination.
高效使用图形计算器；在考前练习定位统计菜单。
Justify the appropriateness of a linear model by referencing the scatter plot and residual plot.
通过引用散点图和残差图来论证线性模型的适用性。
Check the limits of interpolation and avoid unjustified extrapolation in your conclusions.
注意内插的范围限制，并在结论中避免无充分理由的外推。

By mastering the concepts of correlation, regression, residuals, and model evaluation, students can confidently approach any bivariate statistics question in the IB examination.

通过掌握相关、回归、残差和模型评估等概念，学生可以自信地应对 IB 考试中的任何双变量统计问题。

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Bivariate Statistics | 双变量统计分析方法