Correlation · 相关系数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| correlation/ˌkɒrɪˈleɪʃn/ | 相关性 | xiāng guān xìng |
One number for a linear trend
- Correlation 相关性 $r$ is a single number for the strength and direction of a linear relationship.
- Its sign matches the direction: $r>0$ positive, $r<0$ negative.
- Its size measures tightness: near $\pm 1$ = tight line, near $0$ = no linear trend.
- $r$ turns "looks fairly strong" into a precise, comparable value.
用一个数概括线性趋势
- 相关系数 $r$ 用一个数来表示线性关系的强度和方向。
- 它的符号与方向一致：$r>0$ 为正，$r<0$ 为负。
- 它的大小衡量紧密程度：接近 $\pm 1$ = 贴近直线，接近 $0$ = 没有线性趋势。
- $r$ 把“看起来相当强”变成一个精确、可比较的数值。
Always between −1 and 1
- Correlation is bounded: $-1 \le r \le 1$, always.
- $r=+1$: a perfect increasing line; $r=-1$: a perfect decreasing line.
- $r=0$: no linear association (the points may still have a non-linear pattern).
- $r$ has no units and doesn't change if you switch which variable is $x$.
永远在 −1 到 1 之间
- 相关系数是有界的：永远有 $-1 \le r \le 1$。
- $r=+1$：一条完美的上升直线；$r=-1$：一条完美的下降直线。
- $r=0$：没有线性关联（点仍可能有非线性的模式）。
- $r$ 没有单位，而且交换哪个是 $x$ 也不会改变它。
When r misleads
- $r$ measures linear strength only — a strong curve can give a middling $r$.
- A single outlier can pull $r$ up or down dramatically.
- So never report $r$ without first looking at the scatterplot.
- A high $r$ on a curved or outlier-driven plot is a false summary.
什么时候 r 会误导
- $r$ 只衡量线性强度——一条很强的曲线可能给出中等的 $r$。
- 单个离群值就能把 $r$ 大幅拉高或拉低。
- 所以报告 $r$ 之前，永远先看散点图。
- 在弯曲的或被离群值主导的图上，高 $r$ 是一个虚假的概括。
Correlation is not causation
- A large $r$ shows the variables move together, not that one causes the other.
- Ice-cream sales and drownings correlate — both rise in summer (heat is a lurking variable).
- Establishing cause needs an experiment, not just a high correlation.
- "Correlation does not imply causation" is the mantra of the whole unit.
相关不是因果
- 大的 $r$ 表明变量一起变动，而不是一个导致另一个。
- 冰淇淋销量和溺水事件相关——两者都在夏天上升（高温是潜伏变量）。
- 确立因果需要实验，而不只是高相关。
- “相关不蕴含因果”是整个单元的口头禅。
$r$ is a linear measure and is not resistant. A curved relationship can have a small $r$ even though the variables are tightly related, and one outlier can inflate or deflate $r$ a lot. Always look at the scatterplot before trusting $r$ — and remember a high $r$ never proves causation.
$r$ 是一个线性度量，而且不稳健。弯曲的关系即便变量紧密相关，$r$ 也可能很小；而一个离群值能把 $r$ 大幅抬高或压低。信任 $r$ 之前，永远先看散点图——并记住高 $r$ 永远不能证明因果。
Two datasets both have $r \approx 0.2$.
- Dataset A: a genuine weak, scattered linear trend — $r=0.2$ is honest.
- Dataset B: a strong U-shaped curve — the linear $r$ is near $0$, hiding a real relationship.
- Same $r$, totally different stories — which is why you must see the plot.
两个数据集的 $r$ 都约为 $0.2$。
- **数据集 A：**一个真实的、微弱而分散的线性趋势——$r=0.2$ 是诚实的。
- **数据集 B：**一条很强的 U 形曲线——线性 $r$ 接近 $0$，掩盖了真实的关系。
- 相同的 $r$，却是完全不同的故事——这正是你必须看图的原因。
Correlation $r$ measures the strength and direction of a linear relationship, with $-1 \le r \le 1$ (sign = direction, magnitude = tightness, no units). It is misleading for curved data or with outliers, so always check the scatterplot — and correlation never implies causation.
相关系数 $r$ 衡量线性关系的强度和方向，且 $-1 \le r \le 1$（符号 = 方向，大小 = 紧密程度，无单位）。对弯曲数据或有离群值时它会误导，所以总要检查散点图——而且相关永远不蕴含因果。
Correlation and the point cloud · 相关系数与点云
Raise the correlation and the points hug the line more tightly. · 提高相关系数，点会更紧地贴合直线。
What is the largest possible value of a correlation coefficient r? · 相关系数 r 可能取到的最大值是多少？
r ranges from −1 to 1, so the maximum is 1. · r 的取值范围是 −1 到 1，所以最大值是 1。
A correlation of r = 0 means there is definitely no relationship of any kind between the variables. · r = 0 意味着两个变量之间一定不存在任何形式的关系。
r = 0 means no LINEAR relationship; a strong curve can still exist. · r = 0 表示没有线性关系；仍可能存在很强的曲线关系。
Ice-cream sales and drownings both rise in summer, giving a high correlation. This is best explained by... · 冰淇淋销量和溺水事件都在夏天上升，产生高相关。最好的解释是……
Summer heat drives both — correlation without causation. · 夏天的高温同时驱动两者——有相关而无因果。
Which can make r a misleading summary? · 哪些情况会使 r 成为具有误导性的概括？
r is unitless; the real traps are curves, outliers, and skipping the plot. · r 无单位；真正的陷阱是曲线、离群值和跳过看图。
Correlation r measures the strength and direction of a ___ relationship (one word). · 相关系数 r 衡量一种 ___ 关系的强度和方向（一个词）。
r captures linear strength only. · r 只捕捉线性强度。