IB Mathematics: Correlation and Regression Analysis | IB数学：相关性与回归分析

📚 IB Mathematics: Correlation and Regression Analysis | IB数学：相关性与回归分析

Correlation and regression are fundamental tools in IB Mathematics for exploring relationships between two variables. Correlation measures the strength and direction of a linear association, while regression provides an equation that summarises the trend and allows predictions. This article revises the essential concepts, formulas, and common pitfalls you need for your exams.

相关性与回归分析是IB数学中探索两个变量之间关系的基本工具。相关性用于衡量线性关联的强度与方向，而回归则给出一个概括趋势并允许预测的方程。本文将复习考试所需的核心概念、公式以及常见误区。

1. Scatter Diagrams | 散点图

A scatter diagram is a graph in which each data point is plotted on a coordinate plane, with one variable on the horizontal axis and the other on the vertical axis. It visually reveals the overall pattern of the relationship.

散点图是将每个数据点绘制在坐标平面上的图形，一个变量在横轴，另一个变量在纵轴。它直观地揭示关系的总体模式。

Direction: A positive relationship appears as points rising from left to right; a negative relationship appears as points falling from left to right.
方向：正相关表现为点从左到右上升；负相关表现为点从左到右下降。
Form: We look for linear, quadratic, exponential, or other curved patterns.
形式：寻找线性、二次、指数或其他曲线模式。
Strength: Tight clustering around a line indicates a strong relationship; wide spread indicates a weak one.
强度：点紧密围绕一条线表明关系强；点分散表明关系弱。
Outliers: Points far from the general pattern can strongly influence both correlation and regression.
外点：远离总体模式的点可能对相关性和回归产生强烈影响。

When drawing a scatter diagram, always label axes clearly and use a suitable scale. In the IB exam, you may be asked to describe the relationship in words before calculating any numerical measure.

绘制散点图时，务必清楚标记坐标轴并使用合适的比例。在IB考试中，你可能需要在计算数值之前先用语言描述关系。

2. Covariance and Correlation | 协方差与相关

Covariance measures how two variables change together. For paired data \((x_i, y_i)\), the sample covariance is defined as

协方差衡量两个变量如何一起变化。对于成对数据 \((x_i, y_i)\)，样本协方差定义为

sₓᵧ = Σ(xᵢ − x̄)(yᵢ − ȳ) / (n − 1)

where \(x̄\) and \(ȳ\) are the sample means and \(n\) is the number of data pairs.

其中 \(x̄\) 和 \(ȳ\) 是样本均值，\(n\) 是数据对数量。

However, covariance depends on the units of measurement, making it difficult to compare across different datasets. Correlation standardises covariance so that it always lies between −1 and 1.

然而，协方差依赖测量单位，难以在不同数据集之间比较。相关性对协方差进行标准化，使其始终落在 −1 和 1 之间。

r = sₓᵧ / (sₓ sᵧ)

Here \(sₓ\) and \(sᵧ\) are the sample standard deviations of \(x\) and \(y\).

其中 \(sₓ\) 和 \(sᵧ\) 分别是 \(x\) 和 \(y\) 的样本标准差。

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

Pearson’s correlation coefficient \(r\) measures the strength and direction of a linear relationship between two quantitative variables. Its formula is

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与方向。其公式为

r = Σ(xᵢ − x̄)(yᵢ − ȳ) / √[Σ(xᵢ − x̄)² Σ(yᵢ − ȳ)²]

Equivalently, using the covariance and standard deviations: \(r = sₓᵧ / (sₓ sᵧ)\).

等价地，用协方差和标准差表示：\(r = sₓᵧ / (sₓ sᵧ)\)。

\(r = +1\): perfect positive linear correlation.
\(r = +1\)：完全正线性相关。
\(r = −1\): perfect negative linear correlation.
\(r = −1\)：完全负线性相关。
\(r = 0\): no linear correlation.
\(r = 0\)：无线性相关。

Values of \(r\) near +1 or −1 indicate a strong linear relationship; values near 0 indicate a weak one. However, \(r\) only describes linear association — a curved relationship may yield \(r\) close to 0 even if the variables are strongly related.

\(r\) 接近 +1 或 −1 表示强线性关系；接近 0 表示弱关系。然而，\(r\) 只描述线性关联——即使变量关系很强，如果是曲线关系，\(r\) 也可能接近 0。

In IB exams, you are usually expected to compute \(r\) using a GDC rather than by hand, but you must understand the formula and its properties.

在IB考试中，通常要求使用图形计算器计算 \(r\)，而非手算，但你必须理解公式及其性质。

4. Spearman’s Rank Correlation Coefficient (ρ) | 斯皮尔曼等级相关系数

Spearman’s rank correlation coefficient, denoted \(\rho\) (rho), measures the monotonic relationship between two variables. It is calculated using the ranks of the data rather than the raw values.

斯皮尔曼等级相关系数用 \(\rho\) 表示，衡量两个变量之间的单调关系。它使用数据的秩而不是原始值进行计算。

The formula is

公式为

ρ = 1 − 6 Σdᵢ² / [n(n² − 1)]

where \(dᵢ\) is the difference between the ranks of each pair and \(n\) is the number of data pairs.

其中 \(dᵢ\) 是每一对数据秩的差值，\(n\) 是数据对数量。

Rank each variable separately from 1 to \(n\), giving the smallest value rank 1.
分别对每个变量从1到\(n\)排序，最小值为第1名。
If there are tied ranks, assign the average of the ranks they would have received.
如果存在并列名次，则赋予它们原有名次的平均值。
Compute the difference \(dᵢ\) for each pair, square it, and sum.
计算每一对数据的秩差 \(dᵢ\)，取平方并求和。

Spearman’s \(\rho\) is suitable when data are ordinal, or when the relationship is monotonic but not necessarily linear. It is less sensitive to outliers than Pearson’s \(r\).

斯皮尔曼 \(\rho\) 适用于有序数据，或关系为单调但不一定线性的情况。它对异常值不如皮尔逊 \(r\) 敏感。

In the IB formula booklet, the formula is given without ties. When ties exist, an alternative definition based on the Pearson correlation of the ranks is used; your GDC can handle this automatically.

在IB公式手册中，给出的公式不涉及并列。当存在并列时，可使用基于秩的皮尔逊相关替代定义；图形计算器会自动处理。

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

Regression differs from correlation: regression describes how the dependent variable \(y\) changes, on average, with the independent variable \(x\). The least-squares regression line minimises the sum of squared vertical distances from the data points to the line.

回归不同于相关：回归描述因变量 \(y\) 如何随自变量 \(x\) 平均变化。最小二乘回归线使数据点到直线的竖直距离平方和最小。

The regression line has the form

回归线具有形式

y = a + bx

where \(b\) is the slope and \(a\) is the y-intercept. The slope and intercept are given by

其中 \(b\) 是斜率，\(a\) 是 y 轴截距。斜率和截距由下式给出

b = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)², a = ȳ − b x̄

The regression line always passes through the mean point \((x̄, ȳ)\). In IB, you are expected to find \(a\) and \(b\) using your GDC, but you must know how to interpret them.

回归线总是经过均值点 \((x̄, ȳ)\)。在IB中，要求使用图形计算器求出 \(a\) 和 \(b\)，但你必须懂得如何解释它们。

It is important to distinguish between \(y\)-on-\(x\) and \(x\)-on-\(y\) regression lines. In \(y\)-on-\(x\), we minimise vertical distances; in \(x\)-on-\(y\), we minimise horizontal distances. The two lines are not the same unless \(r = ±1\).

重要的是区分 \(y\) 关于 \(x\) 和 \(x\) 关于 \(y\) 的回归线。在 \(y\) 关于 \(x\) 时，最小化竖直距离；在 \(x\) 关于 \(y\) 时，最小化水平距离。除非 \(r = ±1\)，否则两条线不同。

6. Coefficient of Determination (R²) | 决定系数

The coefficient of determination, \(R²\), is the square of the Pearson correlation coefficient \(r\). It represents the proportion of the total variation in the dependent variable \(y\) that is explained by the regression model using \(x\).

决定系数 \(R²\) 是皮尔逊相关系数 \(r\) 的平方。它表示因变量 \(y\) 的总变差中，由回归模型通过 \(x\) 解释的比例。

R² = r²

For example, if \(r = 0.8\), then \(R² = 0.64\), meaning 64% of the variation in \(y\) is explained by its linear relationship with \(x\). The remaining 36% is due to other factors or random error.

例如，若 \(r = 0.8\)，则 \(R² = 0.64\)，意味着 \(y\) 中64%的变差可由它与 \(x\) 的线性关系解释。其余36%来自其他因素或随机误差。

\(R²\) always lies between 0 and 1. A high \(R²\) indicates a good fit of the regression line to the data, while a low \(R²\) indicates that the linear model explains little of the variability.

\(R²\) 始终在 0 与 1 之间。较高的 \(R²\) 表明回归线对数据拟合较好，较低的 \(R²\) 表明线性模型解释的变差很少。

Note that \(R²\) is only meaningful for linear regression with a single explanatory variable. In IB, you may be asked to state the value of \(R²\) and interpret it in context.

注意 \(R²\) 仅对单解释变量的线性回归有意义。在IB中，你可能会被要求写出 \(R²\) 的值并结合情境解读。

7. Residuals and Model Fit | 残差与模型拟合

A residual is the vertical difference between an observed value \(yᵢ\) and the corresponding predicted value \(\hat{y}ᵢ\) from the regression line:

残差是观测值 \(yᵢ\) 与回归线对应预测值 \(\hat{y}ᵢ\) 之间的竖直差：

eᵢ = yᵢ − ŷᵢ

Residuals show how far each point lies from the line. The sum of the residuals is always zero for a least-squares regression line.

残差显示每个点偏离直线多远。对于最小二乘回归线，残差之和始终为零。

A residual plot is a scatter diagram of residuals against the independent variable \(x\). It helps assess whether a linear model is appropriate.

残差图是以残差对自变量 \(x\) 绘制的散点图。它有助于评估线性模型是否恰当。

If residuals are randomly scattered around zero with no systematic pattern, the linear model is appropriate.
如果残差在零附近随机分布，没有系统性模式，线性模型是恰当的。
If residuals show a curved pattern or increasing spread, a linear model may not be suitable.
如果残差呈现曲线模式或离散度增大，线性模型可能不合适。

In IB questions, you may be shown a residual plot and asked to comment on the validity of the regression model. Always refer to the plot’s randomness, not just the size of the residuals.

在IB题目中，你可能会看到残差图并被要求评论回归模型的有效性。始终要提及图的随机性，而不仅仅是残差的大小。

8. Interpolation vs Extrapolation | 内插与外推

Once the regression line is found, it can be used to predict values of \(y\) for given values of \(x\). Two types of prediction exist:

一旦找到回归线，就可以用它预测给定 \(x\) 时的 \(y\) 值。有两种预测类型：

Interpolation: predicting \(y\) for a value of \(x\) that lies within the range of the observed data.
内插：对观测数据范围内 \(x\) 的取值进行 \(y\) 的预测。
Extrapolation: predicting \(y\) for a value of \(x\) that lies outside the range of the observed data.
外推：对观测数据范围之外 \(x\) 的取值进行 \(y\) 的预测。

Interpolation is generally reliable because the model is supported by actual data. Extrapolation is risky because the linear relationship may not continue beyond the observed range.

内插通常可靠，因为模型有实际数据支持。外推有风险，因为线性关系在观测范围之外可能不再延续。

For example, if data are collected for ages 10 to 18, predicting a student’s test score at age 15 is interpolation; predicting at age 25 is extrapolation and should be treated with caution.

例如，如果收集了10至18岁的数据，预测15岁学生的考试成绩属于内插；预测25岁则属于外推，应谨慎对待。

In exam questions, you should always state whether a prediction is an interpolation or extrapolation and comment on its reliability.

在考试题目中，你应始终说明预测是内插还是外推，并评论其可靠性。

9. Using a GDC for Correlation and Regression | 使用图形计算器进行相关与回归分析

The IB Mathematics syllabus allows the use of a graphing display calculator (GDC) for computation. You need to know how to enter paired data and obtain the required statistics.

IB数学教学大纲允许使用图形显示计算器进行计算。你需要知道如何输入成对数据并获得所需统计量。

Typical steps on a GDC (e.g., TI-Nspire or Casio) are:

在图形计算器上（如TI-Nspire或Casio）的典型步骤为：

Enter the \(x\) and \(y\) lists into the statistics editor.
将 \(x\) 和 \(y\) 列表输入统计编辑器。
Plot a scatter diagram to visualise the data.
绘制散点图以观察数据。
Run the linear regression (y = a + bx) command; read off \(a\), \(b\), \(r\), and \(R²\).
运行线性回归（y = a + bx）命令；读出 \(a\)、\(b\)、\(r\) 和 \(R²\)。
Use the regression equation to predict values via the ‘calc’ or ‘value’ feature.
使用回归方程通过”计算”或”求值”功能预测数值。

For Spearman’s \(\rho\), you must either rank the data manually and then use the correlation command on the ranks, or use a GDC function that directly computes rank correlation if available.

对于斯皮尔曼 \(\rho\)，你必须手动将数据排序，然后对秩使用相关命令；如果计算器有直接计算秩相关的功能，也可以使用。

Always state the GDC output to the required degree of accuracy and link it to the context of the question.

始终按照题目要求的精度写出计算器输出，并将其与题目情境相联系。

10. Common Pitfalls and Examination Tips | 常见误区与考试提示

Many students lose marks in correlation and regression questions due to small but avoidable errors. Here are the most common ones:

许多学生在相关性与回归题目中因为一些细小但可避免的错误失分。以下是最常见的误区：

Confusing correlation with causation: A high correlation does not mean one variable causes the other. Refer to “association” rather than “causation” unless there is a clear experimental basis.
混淆相关与因果：高相关并不意味着一个变量导致另一个变量。除非有明确的实验依据，否则应说”关联”而非”因果”。
Using the wrong regression line: If you predict \(y\) from \(x\), use \(y\)-on-\(x\); if you predict \(x\) from \(y\), use \(x\)-on-\(y\).
使用错误的回归线：如果由 \(x\) 预测 \(y\)，使用 \(y\) 关于 \(x\) 的回归；如果由 \(y\) 预测 \(x\)，使用 \(x\) 关于 \(y\) 的回归。
Forgetting the units: The slope \(b\) has units of \(y\) per unit of \(x\); the intercept \(a\) has the same units as \(y\).
忘记单位：斜率 \(b\) 的单位是 \(y\) 的单位除以 \(x\) 的单位；截距 \(a\) 的单位与 \(y\) 相同。
Ignoring outliers: A single outlier can dramatically change \(r\) and the regression line. Always check the scatter diagram first.
忽略外点：一个外点可以显著改变 \(r\) 和回归线。务必先检查散点图。
Misusing \(R²\): \(R²\) is the proportion of variance explained, not the correlation itself.
误用 \(R²\)： \(R²\) 是被解释的方差比例，而不是相关性本身。
Rounding too early: Keep full precision in intermediate calculations, and round only at the final answer.
过早四舍五入：在中间计算中保留完整精度，只在最终答案处四舍五入。

Finally, always read the wording of the question: “use your regression equation to estimate…” means you must substitute the given \(x\) value, while “interpret the slope” asks for a contextual meaning.

最后，务必仔细阅读题目措辞：”使用你的回归方程估计……”意味着你需要代入给定的 \(x\) 值，而”解释斜率”要求给出情境中的含义。

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IB Mathematics: Correlation and Regression Analysis | IB数学：相关性与回归分析