Exploring Two-Variable Data

Two-variable data let us ask whether two characteristics are associated 关联 – whether knowing one tells you something about the other. An explanatory variable 解释变量 (the "input") may help predict a response variable 响应变量 (the "output"). Association is not the same as causation.

A two-way table 双向表 (contingency table) counts individuals by two categorical variables at once. A marginal distribution 边缘分布 is a row or column total written as a fraction of the grand total (the totals themselves are just counts). Comparing the inside cells shows whether the variables are related.

A conditional distribution 条件分布 is the distribution of one variable within a fixed category of the other (found by dividing each cell by its row or column total). If the conditional distributions differ across groups, the two variables are associated; if they are the same, there is no association. Segmented bar charts 分段条形图 or mosaic plots display them.

A scatterplot 散点图 plots each individual as a point, explanatory variable on the $x$-axis and response on the $y$-axis. Describe it with DUFS: Direction (positive/negative), Unusual features (outliers, clusters), Form (linear or curved), and Strength (how tightly the points follow the pattern) – always in context.

The correlation coefficient 相关系数 $r$ measures the strength and direction of a linear relationship. It runs from $-1$ to $1$: near $\pm 1$ is strong linear, near $0$ is weak linear. $r$ has no units and does not change if you swap the variables. Warnings: $r$ only measures linear strength, it is not resistant to outliers, and a strong $r$ does not prove causation.

The least-squares regression line 最小二乘回归线 predicts the response: $\hat{y}=a+bx$, where $\hat{y}$ is the predicted response. The slope 斜率 $b$ is the predicted change in $y$ per one-unit increase in $x$; the $y$-intercept 截距 $a$ is the predicted $y$ when $x=0$. Interpret both in context and with units – a graded skill. Avoid extrapolation 外推 (predicting far outside the data).

Worked example. A study of hours studied ($x$) and test score ($y$) gives $\hat{y}=20+3x$. The slope means each extra hour of study is associated with a predicted $3$-point increase. A student who studies $5$ hours is predicted to score $\hat{y}=20+3(5)=35$.

A residual 残差 is actual minus predicted, $y-\hat{y}$: how far a point sits above (+) or below (−) the line. A residual plot 残差图 graphs residuals against $x$. If it shows no pattern (random scatter), a linear model is appropriate; a curved or fanning pattern means the linear model is a poor fit.

Worked example. Continuing the study above, a student who studied $5$ hours actually scored $40$. The residual is $y-\hat{y}=40-35=+5$: the line under-predicted by $5$ points, so this point sits above the line.

The line minimizes the sum of squared residuals. Its fit is measured by:

• $s$, the standard deviation of the residuals – the typical prediction error, in the response's units.
• $r^2$, the coefficient of determination 决定系数 – the proportion of the variation in $y$ that the linear model explains (a value between $0$ and $1$; multiply by $100$ to state it as a percent). Report it in context: "$r^2 = 0.81$ means 81% of the variation in $y$ is explained by the linear relationship with $x$."

Some points strongly affect the line. A high-leverage 高杠杆 point has an extreme $x$-value; an influential 有影响的 point noticeably changes the slope or $r$ when removed; an outlier here is a point with a large residual. When the pattern is curved, transform a variable (e.g. take a log) to straighten it, then fit a line to the transformed data.

• On a scatterplot describe direction, form, strength, and outliers; $r$ ranges $-1$ to $1$.
• Correlation is not causation — a lurking variable can drive both.
• Interpret the slope of the least-squares line in context ("per one unit of $x$, predicted $y$ changes by $b$").
• Check a residual plot: no pattern means a line fits; a curve means it does not. Avoid extrapolation.
• $r^2$ is the fraction of variation in $y$ explained by the model.