Correlation & Regression: Coefficients, Equations & Interpretation

TAKEAWAYS:

• Correlation ≠ Causation: A strong correlation (e.g., 0.84) means variables move together, but doesn’t prove one causes the other (e.g., ice cream sales and drowning deaths both rise in summer).
• Regression Lines Predict: The equation Ŷ = a + bX lets you forecast Y (e.g., sales) from X (e.g., ad spend), with b showing the impact of a 1-unit change in X.
• Coefficient of Determination (r²): Converts r (e.g., 0.84) to explained variance (70.56%), answering “How much of Y’s change is due to X?”
• Two Regression Lines: Y on X and X on Y have different slopes unless r = ±1; their intersection gives the means (X̄, Ȳ).
• Real-World Hooks: Used by eSewa (predicting transaction delays from server load), Daraz (forecasting order volumes from promo spend), and Ncell (estimating data usage from call duration).

1. Correlation: Measuring Relationships

Definition & Key Formula

Correlation measures how two variables move together. The Pearson’s coefficient (r) ranges from -1 to +1:

• r = +1: Perfect positive (e.g., study hours vs. exam scores).
• r = -1: Perfect negative (e.g., temperature vs. heating costs).
• r = 0: No linear relationship.

Formula: $$r=\frac{\text{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}=\frac{\sum (X_i-\bar{X})(Y_i-\bar{Y})}{(n-1){\sigma }_X{\sigma }_Y}$$ Where:

• Cov(X,Y): Covariance (how X and Y vary together).
• σ_X, σ_Y: Standard deviations of X and Y.

Worked Example 1: Calculating r from Covariance

Given:

• Cov(X,Y) = 18
• Var(X) = 16 → σ_X = 4
• Var(Y) = 81 → σ_Y = 9

Solution: $$r=\frac{18}{4\times 9}=\frac{18}{36}=0.5$$ Interpretation: Weak positive correlation (25% of Y’s variance is explained by X).

Coefficient of Determination (r²)

Converts r to percentage of variance explained: $$r^2=(0.5{)}^2=0.25(25\%)$$ Exam Tip: If r = 0.84, then r² = 0.7056 → 70.56% of Y’s changes are due to X.

2. Regression Analysis: Predicting Outcomes

Why Regression?

Correlation shows strength; regression shows direction and prediction.

• Simple Linear Regression: Models Y as a linear function of X: $$\hat{Y}=a+bX$$
  • Ŷ: Predicted Y (e.g., sales).
  • a: Y-intercept (value of Y when X=0).
  • b: Slope (change in Y per 1-unit change in X).

Calculating Regression Coefficients

Given:

• X̄ = 15, Ȳ = 20
• σ_X = 4, σ_Y = 3
• r = 0.7

Slope (b): $$b=r\cdot \frac{{\sigma }_Y}{{\sigma }_X}=0.7\times \frac{3}{4}=0.525$$

Intercept (a): $$a=\bar{Y}-b\bar{X}=20-0.525\times 15=12.125$$

Regression Equation: $$\hat{Y}=12.125+0.525X$$

Worked Example 2: Regression from Raw Data

Data: Promotional expenses (X) vs. sales (Y) for a product.

Steps:

1. Calculate means: X̄ = 10, Ȳ = 20.57.
2. Compute b and a (as above).
3. Predict sales for X=8: $$\hat{Y}=12.125+0.525\times 8=16.325(\text{Rs }1,632,500)$$

3. Two Regression Lines

For any bivariate data, there are two regression lines:

1. Y on X: Ŷ = a + bX (predicts Y from X).
2. X on Y: X̂ = c + dY (predicts X from Y).

Key Property:

• The two lines intersect at (X̄, Ȳ).
• Their slopes are reciprocals: b₁ = r(σ_Y/σ_X), b₂ = r(σ_X/σ_Y).

Worked Example 3: Finding Means from Regression Equations

Given:

• Regression 1: 4X – 5Y + 33 = 0 → Y = 0.8X + 6.6
• Regression 2: 20X – 9Y – 107 = 0 → Y = (20/9)X – 11.89

Solution: At intersection (X̄, Ȳ): $$0.8\bar{X}+6.6=\frac{20}{9}\bar{X}-11.89$$ Solve for X̄ = 9, then Ȳ = 13.8.

4. Real-World Applications

In the Real World

1. eSewa Transaction Delays

   • Idea Used: Regression predicts delay time (Y) from server load (X).
   • How: Engineers use Ŷ = 2 + 0.5X to estimate wait times during festivals (e.g., X=1000 users → Ŷ=502 seconds).

2. Daraz Sales Forecasting

   • Idea Used: Correlation (r=0.85) between promo spend (X) and sales (Y).
   • How: Marketing teams use r²=0.72 to justify budgets: “72% of sales growth comes from ads.”

3. Ncell Data Usage

   • Idea Used: Regression models data consumption (Y) from call duration (X).
   • Equation: Ŷ = 50 + 2X (50MB base + 2MB per minute).

Worked Example 4: NEPSE Stock Returns

Scenario: An investor notes r=0.6 between a stock’s price (X) and market index (Y). If σ_X=10 and σ_Y=15, predict the stock’s return (Ŷ) when the index rises by 5 units (X=5).

Solution:

1. Calculate b: $$b=0.6\times \frac{15}{10}=0.9$$
2. Find a (assuming X̄=100, Ȳ=120): $$a=120-0.9\times 100=30$$
3. Predict Ŷ for X=5: $$\hat{Y}=30+0.9\times 5=34.5$$ Interpretation: The stock is expected to rise by 34.5 units when the index rises by 5.

5. Common Pitfalls & Exam Tips

Mistakes to Avoid

Exam Tip: Step-by-Step for Full Marks

1. For r questions:

   • Show the formula: r = Cov(X,Y) / (σ_X σ_Y).
   • Calculate covariance and standard deviations separately.
   • Interpret r and r² clearly (e.g., “64% of variance in Y is explained by X”).

2. For regression:

   • Write both equations (Y on X and X on Y).
   • Show calculations for b and a using r, σ, and means.
   • Plot the data and trendline (even if not required, it helps visualization).

3. Interpretation:

   • Correlation: “There is a [weak/moderate/strong] [positive/negative] linear relationship.”
   • Regression: “For every 1 unit increase in X, Y increases/decreases by b units.”

6. Comparison Table: Correlation vs. Regression

7. Practice Questions (Exam-Style)

1. Given:

   • Cov(X,Y) = 30, Var(X) = 25, Var(Y) = 64.
   • Find: r and interpret r².
   • Answer: $$r=\frac{30}{\sqrt{25}\times \sqrt{64}}=\frac{30}{5\times 8}=0.75$$ r² = 0.5625 → 56.25% of Y’s variance is explained by X.

2. Given:

   • Regression equations: Y = 2X + 10 and X = 0.4Y + 5.
   • Find: X̄ and Ȳ.
   • Answer: Solve the system at intersection → X̄=10, Ȳ=30.

8. Visual Summary

mindmap
  root((Correlation & Regression))
    Correlation
      Definition: Strength/direction of linear relationship
      Formula: r = Cov(X,Y) / (σ_X σ_Y)
      Range: -1 to +1
      Interpretation: "r = 0.84 → Strong positive"
    Regression
      Purpose: Prediction
      Equation: Ŷ = a + bX
      Coefficients: b = r(σ_Y/σ_X), a = Ȳ - bX̄
      Two Lines: Y on X and X on Y
    Applications
      Business: Sales forecasting, risk assessment
      Tech: eSewa delay prediction, Daraz promo planning
      Finance: Stock returns, loan default risk
    Exam Tips
      Show all steps
      Interpret r and r² clearly
      Plot data for regression