Day 8: Correlation Analysis: Theory to Practice

Understanding Statistical Relationships Through Real-World Applications

January 7, 2025

1. Pearson Correlation Coefficient

Mathematical Foundation

r = Σ((x - μx)(y - μy)) / (σx σy)

Key Properties:

Range: -1 to +1
-1: Perfect negative correlation
0: No linear correlation
+1: Perfect positive correlation

Real-World Applications

1. Economic Indicators

GDP vs. Employment Rate

Strong positive correlation (+0.82)
As GDP increases, employment typically rises
Used in economic forecasting

2. Medical Research

Blood Pressure vs. Age

Moderate positive correlation (+0.65)
Blood pressure tends to increase with age
Helps in preventive healthcare

3. Environmental Studies

Temperature vs. Ice Cream Sales

Strong positive correlation (+0.95)
Higher temperatures lead to increased sales
Used in inventory management

2. Spearman Rank Correlation

Mathematical Foundation

ρ = 1 - (6Σd²) / (n(n² - 1))

Advantages:

Works with non-normal distributions
Handles non-linear relationships
Resistant to outliers
Suitable for ordinal data

Real-World Applications

1. Education

Study Time vs. Test Rankings

Strong positive correlation (+0.78)
More study time generally leads to better rankings
Used in academic counseling

2. Sports Analytics

Player Ranking vs. Salary

Moderate positive correlation (+0.72)
Higher-ranked players tend to earn more
Used in contract negotiations

3. Customer Satisfaction

Service Quality vs. Customer Loyalty

Strong positive correlation (+0.85)
Better service leads to increased loyalty
Used in business strategy

When to Use Which?

Scenario	Best Choice
Linear relationship expected	Pearson
Ranked data	Spearman
Outliers present	Spearman
Non-normal distribution	Spearman

Code Implementation

Sample implementation:

import numpy as np
from scipy import stats
import pandas as pd

# Sample data
x = np.array([1, 2, 3, 4, 5])
y = np.array([2, 4, 5, 4, 5])

# Pearson correlation
pearson_corr, _ = stats.pearsonr(x, y)
print(f"Pearson correlation: {pearson_corr:.2f}")

# Spearman correlation
spearman_corr, _ = stats.spearmanr(x, y)
print(f"Spearman correlation: {spearman_corr:.2f}")

Day 8: Correlation Analysis: Theory to Practice

1. Pearson Correlation Coefficient

Mathematical Foundation

Key Properties:

Real-World Applications

1. Economic Indicators

2. Medical Research

3. Environmental Studies

2. Spearman Rank Correlation

Mathematical Foundation

Advantages:

Real-World Applications

1. Education

2. Sports Analytics

3. Customer Satisfaction

When to Use Which?

Code Implementation

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