Pearson’s Linear Correlation | 皮尔逊线性相关

📚 Pearson’s Linear Correlation | 皮尔逊线性相关

In A-Level Biology, you often collect paired measurements, such as leaf length and leaf width, body mass and resting heart rate, or soil moisture and species diversity. Pearson’s linear correlation is a statistical method used to test whether two continuous variables are linearly related. It tells you how strongly and in which direction two variables move together.

在A-Level生物中，你经常收集成对测量数据，例如叶片长度和叶片宽度、体重和静息心率，或土壤湿度与物种多样性。皮尔逊线性相关是一种统计方法，用于检验两个连续变量是否呈线性关系。它告诉你两个变量一起变化的方向和强度。

1. What Is Pearson’s Linear Correlation? | 什么是皮尔逊线性相关？

Pearson’s correlation measures the strength and direction of a linear relationship between two continuous variables, usually denoted x and y. It answers the question: as x increases, does y tend to increase or decrease in a straight-line pattern? The method is named after Karl Pearson, who developed the mathematical basis of the coefficient in the late nineteenth century.

皮尔逊相关测量两个连续变量（通常记为 x 和 y）之间线性关系的强度和方向。它回答的问题是：随着 x 增加，y 是否倾向于沿直线模式增加或减少？该方法以卡尔·皮尔逊命名，他在十九世纪末发展了这个系数的数学基础。

It is a parametric test, meaning it assumes the data come from a population that meets certain conditions, such as approximate normality. In biology, it is useful when both variables are measured on interval or ratio scales, for example length in mm, mass in g, or enzyme activity in arbitrary units.

它是一种参数检验，这意味着它假设数据来自满足某些条件（如近似正态分布）的总体。在生物学中，当两个变量都用区间或比率尺度测量时，例如长度（mm）、质量（g）或酶活性（任意单位），该方法很有用。

2. The Correlation Coefficient, r | 相关系数 r

The symbol r is used for the Pearson correlation coefficient. It always lies between -1 and +1. If r = +1, there is a perfect positive linear correlation; if r = -1, there is a perfect negative linear correlation; if r = 0, there is no linear correlation.

符号 r 表示皮尔逊相关系数。它始终介于 -1 和 +1 之间。若 r = +1，则为完全正线性相关；若 r = -1，则为完全负线性相关；若 r = 0，则没有线性相关。

The formula for r is:

r 的计算公式为：

r = Σ((x – x̄)(y – ȳ)) ÷ √(Σ(x – x̄)² × Σ(y – ȳ)²)

In this formula, x̄ and ȳ are the means of the two variables. The numerator is called the covariance, and the denominator standardises the result so that r is not affected by the units of measurement. This allows you to compare correlation strengths between very different biological variables.

在该公式中，x̄ 和 ȳ 是两个变量的平均值。分子称为协方差，分母对结果进行标准化，使 r 不受测量单位的影响。这样你就能比较非常不同的生物变量之间的相关强度。

3. Interpreting r Values | 解读 r 值

The size of r indicates the strength of the linear association. Values close to +1 or -1 show a strong linear relationship, while values near 0 show a weak or absent linear relationship. The sign of r tells you the direction: positive means both variables tend to increase together, while negative means one increases as the other decreases.

r 的大小表示线性关联的强度。接近 +1 或 -1 的值显示强线性关系，而接近 0 的值显示弱线性关系或无线性关系。r 的符号表示方向：正号意味着两个变量往往一起增加，而负号意味着一个增加时另一个减少。

These cut-offs are not fixed rules; they depend on context. In biology, a correlation of r = 0.5 may be biologically important in field ecology, but very weak in a tightly controlled enzyme assay. Always interpret r alongside the scatter diagram and the sample size.

这些界限并不是固定规则，它们取决于具体情境。在生物学中，r = 0.5 在野外生态学中可能具有重要生物学意义，但在严格控制条件的酶实验中可能非常弱。解释 r 时始终要结合散点图和样本量。

4. Scatter Diagrams | 散点图

The first step in Pearson correlation is to plot the paired data as a scatter diagram. Each point represents one observation, with x on the horizontal axis and y on the vertical axis. The pattern of points gives a visual impression of whether a linear relationship exists.

皮尔逊相关的第一步是将成对数据绘制成散点图。每个点代表一个观测值，x 在横轴，y 在纵轴。点的分布模式可以直观显示是否存在线性关系。

A scatter diagram can reveal whether the relationship is linear, whether there are outliers, and whether the spread of points changes across the range. Pearson’s r should only be calculated if the pattern looks roughly linear. If the points form an obvious curve, r may be misleading even when the variables are strongly related.

散点图可以揭示关系是否呈线性、是否存在异常值，以及点的分布是否随范围变化。只有当图形看起来大致呈线性时，才应计算皮尔逊 r。如果点形成明显的曲线，即使变量关系很强，r 也可能产生误导。

5. Calculating Pearson’s r | 计算皮尔逊 r

To calculate r, use the following steps. First, find the means x̄ and ȳ. Second, subtract the mean from each value to obtain deviations. Third, multiply each pair of deviations and sum them. Fourth, square the deviations, sum them separately, and multiply the two sums. Finally,

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