Quick Revision Notes: Economics (Class 11)

Chapter: Correlation

Correlation (सह-संबंध): It is a statistical technique that measures and analyzes the degree and direction of relationship between two or more variables.
Bivariate Distribution: A distribution in which two variables are studied simultaneously.
Example: Relationship between Price and Demand, or Income and Consumption.

Correlation can be classified on the basis of Direction and Degree.

Positive (Direct) Correlation: When two variables move in the same direction. If one increases, the other also increases; if one decreases, the other decreases.
- Example: Income and Consumption, Height and Weight.
Negative (Inverse) Correlation: When two variables move in opposite directions. If one increases, the other decreases.
- Example: Price and Demand of a commodity, Unemployment and Inflation.

Perfect Correlation ($r = \pm 1$): When a proportional change in one variable results in a proportional change in the other.
High Degree Correlation ($r$ close to $+1$ or $-1$): When variables change closely together.
Moderate Degree Correlation ($r$ around $0.5$): When the relationship is noticeable but not very strong.
Low Degree Correlation ($r$ close to $0$): When variables have a weak relationship.
Zero Correlation ($r = 0$): When there is no linear relationship between the two variables.

In Class 11 Economics, correlation is measured using three primary methods:

Scatter Diagram Method (Graphic Method)
Karl Pearson’s Coefficient of Correlation (Mathematical Method)
Spearman’s Rank Correlation Coefficient (Statistical Method for qualitative data)

Karl Pearson's method is the most widely used mathematical method. It is denoted by 'r'.

Direct / Actual Mean Method: r = Σ(x * y) / sqrt(Σ(x^2) * Σ(y^2)) Where:
- x = X - X̄ (deviation of X from actual mean)
- y = Y - Ȳ (deviation of Y from actual mean)
Shortcut / Assumed Mean Method: r = [N(Σdxdy) - (Σdx)(Σdy)] / [sqrt(NΣdx^2 - (Σdx)^2) * sqrt(NΣdy^2 - (Σdy)^2)] Where:
- N = Number of observations / pairs of values
- dx = X - A (deviation of X from assumed mean A)
- dy = Y - B (deviation of Y from assumed mean B)

Used when data cannot be measured quantitatively (e.g., beauty, honesty, intelligence) but can be ranked in order.

Case 1: When ranks are NOT equal (No Tie): R = 1 - [ (6 * Σ(d^2)) / (N^3 - N) ] Where:
- d = R_1 - R_2 (Difference between the ranks of two variables)
- N = Number of observations
Case 2: When ranks ARE equal (Tie in Ranks): When two or more items get the same rank, a correction factor is added to the numerator for each tied value. R = 1 - [ (6 * [ Σ(d^2) + (m_1^3 - m_1)/12 + (m_2^3 - m_2)/12 + ... ]) / (N^3 - N) ] Where:
- m = Number of times an item is repeated.

Limits: The value of $r$ always lies between $-1$ and $+1$ ($-1 \le r \le +1$).
Unit-Free: $r$ is independent of the choice of origin and scale (does not depend on units of measurement like kg, meters, rupees, etc.).
Symmetric: The correlation between X and Y ($r_{xy}$) is the same as between Y and X ($r_{yx}$).

Numerical Reminder: Always write the formula clearly before solving numerical questions in Pearson or Rank correlation.
Interpretation: If a question asks to interpret $r = -0.85$, state that it represents a High degree of negative correlation.
Limitation: Correlation does not imply causation. Just because two variables move together does not mean one causes changes in the other.