学ぶ Correlation and Covariance

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Understanding how variables relate to each other is essential in statistics. Correlation and covariance are two fundamental measures that help you quantify these relationships. Both measure the direction of a linear relationship between two variables, but they differ in scale and interpretation.

Covariance

Indicates whether two variables move together;
Positive covariance: both variables increase or decrease together;
Negative covariance: one variable increases while the other decreases;
Magnitude depends on the scale of the variables, making comparison across datasets difficult.

Correlation

Standardizes the relationship by dividing the covariance by the product of the variables' standard deviations;
Results in a value between -1 and 1;
1 means a perfect positive linear relationship;
-1 means a perfect negative linear relationship;
0 means no linear relationship;
Unitless, so it is easier to interpret and compare.

Calculating Correlation and Covariance in Python

You can use libraries like numpy and pandas to calculate both covariance and correlation:

The cov() function in pandas computes the covariance matrix;
The corr() function gives you the correlation matrix.

These matrices show the pairwise relationships between all variables in your dataset.

Interpreting Results

A high correlation does not imply causation;
Outliers can impact both measures;
Correlation measures only linear relationships and may miss nonlinear associations.


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import numpy as np
import pandas as pd

# Realistic dataset: Student statistics
data = {
    "hours_studied": [2, 3, 5, 7, 9, 1, 4, 8, 6, 10],
    "exam_score":    [55, 60, 75, 85, 95, 50, 70, 90, 80, 98],
    "social_media":  [6, 5, 4, 2, 1, 7, 5, 2, 3, 0],
    "hours_slept":   [7, 6, 8, 7, 6, 8, 7, 6, 8, 7]
}
df = pd.DataFrame(data)

# Compute covariance matrix
cov_matrix = df.cov()
print("Covariance matrix:")
print(cov_matrix)

# Compute correlation matrix
corr_matrix = df.corr()
print("\nCorrelation matrix:")
print(corr_matrix)

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