IB Mathematics: Analysing Relationships Between Variables in Bivariate Statistics | IB数学：双变量统计中变量关系分析

📚 IB Mathematics: Analysing Relationships Between Variables in Bivariate Statistics | IB数学：双变量统计中变量关系分析

Bivariate statistics is the branch of statistics that deals with two variables measured on the same subject or observation. In IB Mathematics, understanding how to analyse the relationship between two variables is essential for both Analysis and Approaches (AA) and Applications and Interpretation (AI) courses.

双变量统计是统计学中研究同一对象或同一观测上两个变量的分支。在IB数学中，理解如何分析两个变量之间的关系，对于数学分析与方法（AA）以及数学应用与解释（AI）课程都至关重要。

1. What Is Bivariate Data? | 什么是双变量数据

Bivariate data consists of pairs of values (x, y) collected from each subject. The variable x is often called the explanatory or independent variable, while y is called the response or dependent variable.

双变量数据由从每个对象收集的成对值 (x, y) 组成。变量 x 通常称为解释变量或自变量，而 y 称为响应变量或因变量。

Explanatory variable: used to explain or predict changes in the response variable.
解释变量：用于解释或预测响应变量的变化。
Response variable: the outcome that may depend on the explanatory variable.
响应变量：可能依赖于解释变量的结果。

For example, a study might record the number of hours studied (x) and the exam score (y) for each student. Here, hours studied is the explanatory variable because we expect it to influence the exam score.

例如，一项研究可能记录每位学生的学习小时数 (x) 和考试成绩 (y)。在这里，学习小时数是解释变量，因为我们期望它影响考试成绩。

2. Scatter Diagrams | 散点图

A scatter diagram is a graphical representation of bivariate data. Each point on the graph corresponds to one observation (x, y). It provides a quick visual impression of whether a relationship exists, and if so, what form it takes.

散点图是双变量数据的图形表示。图上的每个点对应一个观测值 (x, y)。它提供了一种快速直观的印象，帮助我们判断关系是否存在，以及如果存在，它呈现什么形式。

Positive correlation: y tends to increase as x increases.
正相关：随着 x 增加，y 趋于增加。
Negative correlation: y tends to decrease as x increases.
负相关：随着 x 增加，y 趋于减少。
No correlation: no obvious linear relationship exists.
无相关：不存在明显的线性关系。

When drawing a scatter diagram, always label the axes clearly and choose a scale that allows all points to be plotted accurately.

绘制散点图时，务必清晰标注坐标轴，并选择能够准确绘制所有点的刻度。

3. Pearson’s Product-Moment Correlation Coefficient r | 皮尔逊积矩相关系数 r

Pearson’s correlation coefficient r measures the strength and direction of a linear relationship between two quantitative variables.

皮尔逊相关系数 r 衡量两个定量变量之间线性关系的强度和方向。

r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² Σ(y − ȳ)²]

Here, x̄ and ȳ are the means of x and y respectively. The value of r always lies between −1 and 1 inclusive.

其中，x̄ 和 ȳ 分别是 x 和 y 的均值。r 的值总是在 −1 到 1 之间（含端点）。

r = 1: perfect positive linear relationship.
r = 1：完全正线性关系。
r = −1: perfect negative linear relationship.
r = −1：完全负线性关系。
r = 0: no linear relationship.
r = 0：不存在线性关系。

On the IB formula booklet, r is also written in terms of z-scores: r = Σ(zₓ zᵧ) / n, where zₓ and zᵧ are standardised values. This version is often quicker to use with a GDC.

在IB公式手册中，r 也可以用 z 分数表示：r = Σ(zₓ zᵧ) / n，其中 zₓ 和 zᵧ 是标准化值。这个版本在使用图形计算器（GDC）时往往更快。

4. Interpreting the Value of r | 解释 r 的值

The magnitude of r indicates the strength of the linear association, while the sign indicates its direction. However, correlation does not imply causation.

r 的大小表示线性关联的强度，而符号表示方向。然而，相关并不意味着因果。

Value of r	Interpretation
0.8 to 1.0 / −0.8 to −1.0	Very strong positive / negative linear relationship
0.6 to 0.8 / −0.6 to −0.8	Strong positive / negative linear relationship
0.4 to 0.6 / −0.4 to −0.6	Moderate positive / negative linear relationship
0.2 to 0.4 / −0.2 to −0.4	Weak positive / negative linear relationship
−0.2 to 0.2	Very weak or no linear relationship

These thresholds are approximate. In IB exam questions, you are expected to comment on r in context, not just quote a number.

这些界限是近似值。在IB考试题中，你需要在情境中评价 r，而不仅仅是引用一个数字。

5. The Coefficient of Determination r² | 决定系数 r²

The coefficient of determination r² represents the proportion of the variation in y that can be explained by the linear relationship with x.

决定系数 r² 表示 y 的变异中能够由与 x 的线性关系解释的比例。

r² = (correlation coefficient)²

For example, if r = 0.8, then r² = 0.64, meaning 64% of the variation in y is explained by the linear model. The remaining 36% is due to other factors or random variation.

例如，如果 r = 0.8，那么 r² = 0.64，意味着 y 中64%的变异可以由线性模型解释，其余36%由其他因素或随机变异造成。

r² is always between 0 and 1.
r² 总是在0和1之间。
r² is more useful than r for comparing models because it does not depend on the sign.
在比较模型时，r² 比 r 更有用，因为它不依赖符号。

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

The least squares regression line (LSRL) is the line that minimises the sum of the squared vertical distances from the data points to the line. Its equation is written as:

最小二乘回归线（LSRL）是使数据点到直线的垂直距离平方和最小的直线。其方程写为：

y = a + bx

where a is the y-intercept and b is the slope. On the GDC, some calculators use y = ax + b; always check which form your calculator uses.

其中 a 是 y 轴截距，b 是斜率。在使用图形计算器时，有些计算器使用 y = ax + b 的形式；务必确认你的计算器使用哪种形式。

The slope b can be computed as:

斜率 b 可计算为：

b = r × (sᵧ / sₓ)

where sₓ and sᵧ are the sample standard deviations of x and y. The intercept is then:

其中 sₓ 和 sᵧ 分别是 x 和 y 的样本标准差。截距则为：

a = ȳ − b x̄

This shows that the regression line always passes through the point (x̄, ȳ), the mean point.

这说明回归线总是通过均值点 (x̄, ȳ)。

7. Using the Regression Line for Prediction | 使用回归线进行预测

Once the regression line is found, it can be used to predict the value of y for a given x. This is called interpolation when the x value lies within the range of the observed data.

一旦找到回归线，就可以用它来预测给定 x 时的 y 值。当 x 值位于观测数据范围内时，这称为内插。

If the x value lies outside the observed range, the prediction is called extrapolation. Extrapolation is much less reliable because the linear trend may not continue beyond the data.

如果 x 值位于观测范围之外，则预测称为外推。外推的可靠性要低得多，因为线性趋势可能在数据范围之外不再延续。

Interpolation: generally reliable if the linear model fits well.
内插：如果线性模型拟合良好，通常可靠。
Extrapolation: risky; the relationship may change outside the observed range.
外推：有风险；关系在观测范围之外可能发生变化。

Always state clearly whether a prediction is interpolation or extrapolation in exam answers.

在考试答案中，务必明确说明预测是内插还是外推。

8. Residuals and the Fit of the Model | 残差与模型拟合

A residual is the difference between an observed y value and the value predicted by the regression line:

残差是观测到的 y 值与回归线预测值之间的差：

residual = y_observed − y_predicted

If the residuals are small and randomly scattered around zero when plotted against x, the linear model is appropriate. If the residual plot shows a curved pattern, a linear model may not be suitable.

如果残差较小，并且在以 x 为横轴的残差图中围绕零随机散布，则线性模型是合适的。如果残差图呈现弯曲模式，线性模型可能不合适。

Positive residual: observed y is above the regression line.
正残差：观测到的 y 高于回归线。
Negative residual: observed y is below the regression line.
负残差：观测到的 y 低于回归线。

9. Outliers and Influential Points | 离群值与影响点

An outlier is a data point that lies far away from the general trend. An influential point is a point whose removal would substantially change the regression line or correlation coefficient.

离群值是指远离总体趋势的数据点。影响点是指其删除会显著改变回归线或相关系数的点。

Outliers can distort both r and the regression equation. In the IB exam, you may be asked to determine whether removing an outlier improves the correlation or changes the slope significantly.

离群值可能扭曲 r 和回归方程。在IB考试中，你可能会被要求判断删除离群值是否提高了相关性或显著改变了斜率。

A point with an extreme x value is particularly influential because it can pull the regression line towards itself.

具有极端 x 值的点尤其具有影响力，因为它可以将回归线拉向自身。

10. Correlation vs. Causation | 相关与因果

One of the most important ideas in bivariate statistics is that correlation does not imply causation. Even a very strong correlation between two variables does not prove that one variable causes the other to change.

双变量统计中最重要的思想之一是相关并不蕴含因果。即使两个变量之间存在非常强的相关性，也不能证明一个变量导致另一个变量变化。

Possible explanations for a strong correlation include:

强相关可能的解释包括：

Direct cause: x causes y.
直接因果：x 导致 y。
Reverse cause: y causes x.
反向因果：y 导致 x。
Confounding variable: a third variable causes both x and y.
混杂变量：第三个变量同时导致 x 和 y。
Coincidence: the correlation appears by chance.
巧合：相关性偶然出现。

In exam responses, use phrases such as “There is evidence of a positive linear correlation, but this does not necessarily mean that x causes y.”

在考试回答中，请使用诸如”存在正线性相关的证据，但这并不一定意味着 x 导致 y”的表述。

11. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数

Spearman’s rank correlation coefficient rₛ measures the strength and direction of a monotonic relationship between two variables. It is especially useful when the data are ordinal or when the relationship is not linear but still consistently increasing or decreasing.

斯皮尔曼等级相关系数 rₛ 衡量两个变量之间单调关系的强度和方向。当数据是定序数据，或当关系不是线性但仍然持续递增或递减时，它尤其有用。

rₛ = 1 − (6 Σ d²) / (n(n² − 1))

where d is the difference between the ranks of each pair and n is the number of pairs. This formula is valid when all ranks are distinct.

其中 d 是每对数据排名之差，n 是数据对的数量。当所有排名都不同时，该公式有效。

Spearman’s rₛ also lies between −1 and 1, and it is interpreted similarly to Pearson’s r, but for monotonic relationships rather than linear ones.

斯皮尔曼 rₛ 也在 −1 和 1 之间，其解释与皮尔逊 r 类似，但针对的是单调关系而非线性关系。

12. Exam Tips and Common Mistakes | 考试技巧与常见错误

IB exam questions on bivariate statistics often involve interpreting graphs, calculating r or regression equations using a GDC, and writing a conclusion in context.

IB考试中关于双变量统计的问题通常涉及解释图形、使用GDC计算 r 或回归方程，以及在情境中写出结论。

Always identify which variable is x and which is y before calculating.
在计算之前，务必确定哪个变量是 x，哪个是 y。
Round r to 3 significant figures or as instructed by the question.
将 r 四舍五入到3位有效数字，或按题目要求进行。
Do not confuse the coefficient of determination r² with r itself.
不要将决定系数 r² 与 r 本身混淆。
When commenting on correlation, mention both strength and direction.
在评价相关性时，务必同时提到强度和方向。
Remember: correlation is not causation.
记住：相关不是因果。

By mastering these concepts, you will be well prepared for IB exam questions on bivariate statistics and able to critically analyse real-world data.

掌握了这些概念，你将为IB考试中有关双变量统计的问题做好充分准备，并能够批判性地分析现实世界中的数据。

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IB Mathematics: Analysing Relationships Between Variables in Bivariate Statistics | IB数学：双变量统计中变量关系分析