Correlation & RegressionJEE Main

How tightly two variables move together — interactive Mathematics simulation for IIT-JEE.

Concept

Pearson's r measures LINEAR association between two variables: r = cov(x,y)/(σₓσᵧ), always in [−1, 1]. The regression line of y on x minimises vertical squared errors and passes through (x̄, ȳ) with slope b = r·σᵧ/σₓ. There are TWO regression lines (y-on-x and x-on-y); they coincide only when |r| = 1, and r² is the geometric mean of their slopes.

Key formula

r = \frac{\sum(x-\bar x)(y-\bar y)}{\sqrt{\sum(x-\bar x)^2 \sum(y-\bar y)^2}}; \quad b_{yx} = r\frac{\sigma_y}{\sigma_x}; \quad b_{yx}\,b_{xy} = r^2

Derivation

Least squares: minimise Σ(yᵢ − a − bxᵢ)². ∂/∂a = 0 forces the line through (x̄, ȳ); ∂/∂b = 0 gives b = Sxy/Sxx.

Rewrite b = r·σᵧ/σₓ using r's definition. Noise inflates σᵧ without adding covariance, dragging r toward 0 — watch it live in the scatter.

Scenarios to explore

Correlation & Regression — Pearson r, two regression lines and the noise slider.

Real-world applications

Economics: price–demand relationships.
Medicine: dose–response strength.
ML: r² as the baseline goodness-of-fit metric.

JEE exam tips

Both regression coefficients share r's sign; their product is r² ≤ 1 — instant consistency check.
r is unchanged by shifting/scaling either variable (positive scale).
Angle between the two regression lines: tanθ = (1−r²)/r · σₓσᵧ/(σₓ²+σᵧ²).

Common mistakes

Correlation ⟹ causation — it does NOT.
Using r for curved relationships (r measures LINEAR association only).
Confusing the two regression lines — 'y on x' predicts y.

Exam traps to avoid

b_yx·b_xy must be ≤ 1: if given coefficients multiply above 1, the data is impossible.
r = 0 means no LINEAR relation — a perfect parabola can still lurk.