探索双变量数据

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.

Worked example. A school records how $200$ students took a course and whether they passed.

• Marginal distribution of result: $\tfrac{135}{200}=0.675$ passed and $\tfrac{65}{200}=0.325$ failed.
• Conditional distributions: online students passed at $\tfrac{60}{100}=0.60$; in-person students at $\tfrac{75}{100}=0.75$.

The conditional pass rates differ ($0.60$ vs $0.75$), so the format and the result are associated in these data. This alone does not show that the format causes the difference: students chose their format.

Divide by the correct total: “the proportion of online students who passed” uses the online total ($100$), while “the proportion of passes that were online” uses the pass total: $\tfrac{60}{135}\approx 0.44$.

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).

From summary statistics, the line comes from two formulas: the slope is $b=r\,\dfrac{s_y}{s_x}$, and the line always passes through $(\bar x,\bar y)$, so $a=\bar y-b\bar x$.

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 $\hat{y}=20+3(5)=35$.

Worked example: line from summary statistics. For another class, hours studied have $\bar x=6$ and $s_x=2$; scores have $\bar y=70$ and $s_y=10$; and $r=0.8$.

So $\hat y=46+4x$: each extra hour is associated with a predicted $4$-point increase. Also $r^2=0.64$: $64\%$ of the variation in scores is explained by the linear relationship with hours studied.

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.

Common mistakes. Computing predicted minus actual ($35-40=-5$): a residual is always actual minus predicted. Stating the slope as a certainty (“the score goes up by $3$”) instead of a predicted change, or without units and context. Interpreting the intercept when $x=0$ lies far outside the data.

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 $x$.”

Common mistakes. Saying $r=0.8$ means “$80\%$ of the variation is explained”: that statement belongs to $r^2=0.64$. Calling $r\approx0$ “no relationship”: a strong curve can have $r$ near $0$. Saying a high $r$ or $r^2$ shows a linear model is right: only the residual plot shows that.

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.

Answer each before reading the answers below.

1. Of 120 students, 50 play a sport, and 30 of those 50 are also in a music group. What proportion of sport players are in a music group?
2. A scatterplot of hours slept against reaction time gives $r=-0.85$. Describe the relationship.
3. A regression line is $\hat y=12+2.5x$. Predict $y$ when $x=8$, and find the residual for a point $(8,\ 29)$.
4. The data in question 3 have $x$ values from 0 to 10. Why is predicting $y$ at $x=40$ risky?
5. A student says: “The correlation is $0.9$, so $x$ causes $y$.” Correct the student.

1. $30/50=0.60$. This is a conditional proportion: it uses only the sport players as the whole.
2. A strong, negative, linear relationship: students who sleep more tend to have shorter reaction times.
3. $\hat y=12+2.5(8)=32$. Residual $=y-\hat y=29-32=-3$; the point lies 3 units below the line.
4. That is extrapolation: the pattern is known only for $x$ from 0 to 10, and nothing guarantees it continues to 40.
5. Correlation measures the strength of a linear association, not causation. A lurking variable could drive both, or the link could run the other way. Only a randomized experiment supports a cause-and-effect claim.

• 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.