J Finding the correlation coefficient and the equation of the regression line | 求相关系数与回归线方程

📚 J Finding the correlation coefficient and the equation of the regression line | 求相关系数与回归线方程

In IB Mathematics, understanding how two variables relate to each other is a fundamental statistical skill. This article will guide you through finding the Pearson correlation coefficient and the equation of the regression line, both essential for analysing bivariate data.

在IB数学中，理解两个变量之间的相互关系是一项基本的统计技能。本文将指导你求Pearson相关系数和回归线方程的方法，这两者对于分析二元数据至关重要。

1. Introduction to Bivariate Data | 二元数据简介

Bivariate data consists of pairs of observations (x, y) from two variables, such as height and weight, or study hours and exam scores. We are often interested in describing the nature and strength of the linear relationship between them.

二元数据由两个变量的成对观测值 (x, y) 组成，例如身高和体重，或学习时间与考试分数。我们通常关心的是描述它们之间线性关系的性质和强度。

Correlation measures the degree to which two variables move together, while regression allows us to model this relationship and make predictions.

相关性衡量两个变量共同变化的程度，而回归则使我们能够建模这种关系并做出预测。

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2. Visualising Relationships: Scatter Plots | 可视化关系：散点图

Always start by plotting the data as a scatter diagram. The pattern of points can reveal whether a linear model is appropriate and if there are any outliers.

始终首先将数据绘制成散点图。点的分布模式可以揭示线性模型是否合适，以及是否存在异常值。

A scatter plot showing an upward trend suggests positive correlation; a downward trend suggests negative correlation. If no pattern is visible, the variables may be uncorrelated.

呈上升趋势的散点图表明正相关；呈下降趋势则表明负相关。若看不到任何模式，变量可能不相关。

Identifying a roughly elliptical shape of points supports the use of Pearson’s r. Suspicious points far from the main cluster should be investigated as outliers before proceeding.

识别出大致呈椭圆形的点云支持使用 Pearson r。远离主簇的可疑点应在继续分析前作为异常值进行调查。

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3. The Pearson Correlation Coefficient r | Pearson相关系数 r

The Pearson correlation coefficient, denoted by r, is a numerical measure of the strength and direction of a linear relationship. Its value always lies between -1 and 1 inclusive.

Pearson相关系数，记作 r，是衡量线性关系强度和方向的数值量度。其值总在 -1 到 1 之间（含端点）。

If r = 1, there is a perfect positive linear correlation. If r = -1, there is a perfect negative linear correlation. r = 0 indicates no linear correlation.

若 r = 1，则存在完全正线性相关；若 r = -1，则存在完全负线性相关；r = 0 表示无线性相关。

The formula provided in the IB formula booklet is:

r = Sxy / √(Sxx × Syy)

where Sxy, Sxx and Syy are sums of squares and cross-products computed from the data.

其中 Sxy、Sxx 和 Syy 是由数据计算得到的平方和与交叉乘积和。

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4. Manual Calculation of r | 手动计算 r

To compute r without technology, first find the summary statistics: n, Σx, Σy, Σx², Σy², Σxy. Then calculate:

要在不使用技术的情况下计算 r，首先求出汇总统计量：n、Σx、Σy、Σx²、Σy²、Σxy。然后计算：

Sxy = Σxy − (Σx × Σy) / n

Sxx = Σx² − (Σx)² / n

Syy = Σy² − (Σy)² / n

Finally, substitute into r = Sxy / √(Sxx × Syy).

最后代入 r = Sxy / √(Sxx × Syy) 即可。

Consider a small data set: (1,2), (3,5), (4,4).
n = 3, Σx = 8, Σy = 11, Σx² = 26, Σy² = 45, Σxy = 33.
Sxy = 33 − (8×11)/3 = 3.667; Sxx = 26 − 64/3 = 4.667; Syy = 45 − 121/3 = 4.667.
r = 3.667 / √(4.667 × 4.667) ≈ 0.785.

考虑一个小数据集：(1,2), (3,5), (4,4)。
n = 3, Σx = 8, Σy = 11, Σx² = 26, Σy² = 45, Σxy = 33。
Sxy = 33 − (8×11)/3 = 3.667; Sxx = 26 − 64/3 = 4.667; Syy = 45 − 121/3 = 4.667。
r = 3.667 / √(4.667 × 4.667) ≈ 0.785。

This process is tedious by hand, so using a GDC is recommended in exams.

手动计算过程繁琐，因此在考试中建议使用图形计算器（GDC）。

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5. Using GDC for Correlation | 使用图形计算器求相关系数

IB exams expect you to find r efficiently using a graphing calculator. Enter the x-values in one list and y-values in another, then run a linear regression statistical test.

IB考试期望你熟练使用图形计算器求 r。将 x 值输入一个列表，y 值输入另一个列表，然后运行线性回归统计检验。

For TI-Nspire: Menu → Statistics → Stat Calculations → Linear Regression (a+bx). Ensure the output includes the correlation coefficient r.

对于 TI-Nspire：菜单 → 统计 → 统计计算 → 线性回归 (a+bx)。确保输出中包含相关系数 r。

For Casio fx-CG50: enter data in List 1 and List 2, press CALC (F2), select SET, choose Linear Reg and confirm to display r, a and b.

对于 Casio fx-CG50：在 List 1 和 List 2 中输入数据，按 CALC (F2)，选择 SET，选取 Linear Reg 并确认以显示 r、a 和 b。

Always write down the value of r rounded to three significant figures, unless asked otherwise, and note that the calculator gives r² as well.

除非另有要求，通常将 r 值四舍五入到三位有效数字并写下，同时注意计算器也会给出 r²。

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6. Understanding and Interpreting r | 理解与解释 r

An absolute value of r close to 1 indicates a strong linear relationship. A rule of thumb: 0.0 ≤ |r| < 0.5 weak, 0.5 ≤ |r| < 0.8 moderate, 0.8 ≤ |r| ≤ 1 strong.

r 的绝对值接近 1 表示强线性关系。经验法则：0.0 ≤ |r| < 0.5 弱相关，0.5 ≤ |r| < 0.8 中等相关，0.8 ≤ |r| ≤ 1 强相关。

However, a high correlation does not imply causation. It only measures linear association.

然而，高相关性并不意味着因果关系。它仅衡量线性关联。

Also, r is sensitive to outliers; one extreme point can drastically change the value. Always examine the scatter plot alongside r.

此外，r 对异常值敏感；一个极端点可能会大大改变其值。务必结合散点图一同审视 r。

When reporting r, state both the direction (positive or negative) and the strength, e.g. ‘There is a strong positive linear correlation (r

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