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HSSLIVE Plus One Economics Chapter 17: Correlation Notes

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Correlation analysis examines the relationship between two variables, determining the direction and strength of their association without implying causation. The scatter diagram provides a visual representation of this relationship, while Karl Pearson’s coefficient of correlation quantifies it numerically (ranging from -1 to +1). A positive correlation indicates that variables move in the same direction, while a negative correlation shows an inverse relationship. Spearman’s rank correlation offers an alternative method for ordinal data or when dealing with non-linear relationships. Correlation analysis serves as a fundamental tool in economics for studying relationships between factors like income and consumption, price and demand, or education and wages, providing insights for economic theories and policy formulation.

Chapter 17: Correlation

Introduction

Correlation measures the strength and direction of the relationship between two variables. It tells us how changes in one variable are associated with changes in another variable.

Types of Correlation

1. Positive Correlation

When both variables increase or decrease together, the correlation is positive. For example, study time and exam scores typically have a positive correlation.

2. Negative Correlation

When one variable increases as the other decreases, the correlation is negative. For example, time spent on social media and productivity might have a negative correlation.

3. No Correlation

When there is no apparent relationship between the variables, there is no correlation. For example, shoe size and intelligence would typically show no correlation.

Measuring Correlation

Karl Pearson’s Coefficient of Correlation

Also known as the Pearson correlation coefficient or r, this is the most commonly used measure of correlation.

Formula: r = ∑(x – x̄)(y – ȳ) / √[∑(x – x̄)² × ∑(y – ȳ)²]

or the simplified computational formula: r = [n∑xy – (∑x)(∑y)] / √[n∑x² – (∑x)²] × [n∑y² – (∑y)²]

Properties:

  • The value of r lies between -1 and +1
  • r = +1 indicates perfect positive correlation
  • r = -1 indicates perfect negative correlation
  • r = 0 indicates no correlation
  • The closer r is to +1 or -1, the stronger the correlation

Example: For x values: 5, 10, 15, 20, 25 And y values: 10, 12, 14, 16, 18

Step 1: Calculate necessary values ∑x = 75, ∑y = 70, ∑xy = 1130, ∑x² = 1375, ∑y² = 1000, n = 5

Step 2: Apply the formula r = [5(1130) – (75)(70)] / √[5(1375) – (75)²] × [5(1000) – (70)²] = [5650 – 5250] / √[6875 – 5625] × [5000 – 4900] = 400 / √[1250 × 100] = 400 / √125000 = 400 / 353.55 = 0.9814

This indicates a very strong positive correlation.

Spearman’s Rank Correlation Coefficient

Used when data is ordinal or when the relationship between variables isn’t linear.

Formula: rs = 1 – [6∑d² / n(n² – 1)]

Where:

  • d is the difference in ranks
  • n is the number of pairs

Correlation vs. Causation

An important principle in statistics is that “correlation does not imply causation.” Just because two variables are correlated doesn’t mean one causes the other. There could be:

  • A third variable causing both
  • Coincidental correlation
  • Reverse causation

Applications in Computer Science

In computer applications, correlation is used for:

  • Data mining and pattern recognition
  • Feature selection in machine learning
  • Anomaly detection
  • Predictive analytics
  • Recommendation systems
  • Performance optimization
  • Network analysis
  • Image and signal processing

Complete Chapter-wise Hsslive Plus One Economics Notes

Our HSSLive Plus One Economics Notes cover all chapters with key focus areas to help you organize your study effectively:

Economics: Indian Economic Development

Economics: Statistics for Economics

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