If r = −1, the correlation is:

Perfect negative. All points lie on a falling straight line.

Price and demand of a commodity usually show:

Negative correlation. As price rises, demand falls.

Height and weight of students usually show:

Positive correlation. Taller students tend to weigh more.

If Σxy = 200, n = 10, x̄ = 4 and ȳ = 3, then cov(x, y) is:

8. 200/10 − 4 × 3 = 20 − 12 = 8.

Maharashtra State Board · HSC Commerce · Class 11 (Std XI)

Class 11 Maths and Statistics Part 2 Chapter 5: Correlation

Short notes, key concepts, quick revision and practice MCQs with answers

Correlation studies whether two variables move together. The chapter explains positive, negative and zero correlation, draws scatter diagrams, defines covariance and computes Karl Pearson's coefficient of correlation, with its properties and interpretation.

Key concepts: Correlation

Correlation: The mutual relationship between two variables. If a change in one is accompanied by a change in the other, the variables are correlated.
Types of correlation: Positive (both move in the same direction, e.g. height and weight), negative (opposite directions, e.g. price and demand) and zero (no linear relationship).
Scatter diagram: Plot the pairs (x, y) as points. Points rising from left to right show positive correlation, falling show negative, scattered without pattern show no correlation; points on a line show perfect correlation.
Covariance: cov(x, y) = (1/n) Σ(xᵢ − x̄)(yᵢ − ȳ) = (1/n) Σxᵢyᵢ − x̄ȳ. cov(x, x) = var(x).
Karl Pearson's coefficient: r = cov(x, y)/(σₓ σᵧ). Also called the product moment correlation coefficient.
Properties of r: −1 ≤ r ≤ 1; r has no units; its numerical value is unaffected by change of origin and scale; r and cov(x, y) have the same sign.
Interpretation: r = 1: perfect positive; r = −1: perfect negative; r = 0: no linear correlation; values near ±1 indicate strong and near 0 weak correlation.

Quick revision: important points

cov(x, y) = r σₓ σᵧ.
If u = x − a and v = y − b, then r_uv = r_xy.
If cov(x, y) = 0, then r = 0.
Correlation measures only linear relationship; r = 0 does not rule out a non-linear relation.
Price and demand, temperature and woollen sales: negative. Income and expenditure: positive.

Practice MCQs with answers

Karl Pearson's coefficient of correlation lies between:
1. 0 and 1
2. −1 and 1
3. −∞ and ∞
4. 0 and ∞
Show answer
Answer: B. −1 and 1
−1 ≤ r ≤ 1.
If r = −1, the correlation is:
1. Perfect positive
2. Perfect negative
3. Zero
4. Weak negative
Show answer
Answer: B. Perfect negative
All points lie on a falling straight line.
Price and demand of a commodity usually show:
1. Positive correlation
2. Negative correlation
3. No correlation
4. Perfect positive correlation
Show answer
Answer: B. Negative correlation
As price rises, demand falls.
Height and weight of students usually show:
1. Positive correlation
2. Negative correlation
3. No correlation
4. Perfect negative correlation
Show answer
Answer: A. Positive correlation
Taller students tend to weigh more.
If Σxy = 200, n = 10, x̄ = 4 and ȳ = 3, then cov(x, y) is:
1. 8
2. 12
3. 20
4. 32
Show answer
Answer: A. 8
200/10 − 4 × 3 = 20 − 12 = 8.

All Class 11 Maths and Statistics chapters

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