Correlation — NCERT Class 11 Economics

How to Use This Page

This page provides complete NCERT exercise solutions for CBSE Class 11 Economics Chapter 7 — Correlation. Review the Key Concepts and formulas first, then attempt each numerical with step-by-step working. For Karl Pearson's and Spearman's methods, all calculation steps are shown. Verify your answers against these solutions.

Key Concepts (Quick Recap)

• Correlation: the degree and direction of linear relationship between two variables
• Types of Correlation: Positive (both increase), Negative (one increases, other decreases), Zero (no linear relation)
• Scatter Diagram: plot of paired values on X and Y axes; pattern shows type and degree of correlation
• Karl Pearson's Coefficient (r): measures linear correlation for quantitative data

• Alternative: r = (NΣXY − ΣXΣY) / √[(NΣX² − (ΣX)²)(NΣY² − (ΣY)²)]
• Range: −1 ≤ r ≤ +1
• Spearman's Rank Correlation (rₖ): uses ranks instead of actual values

• With tied ranks: rₖ = 1 − [6ΣD² + 1/12(m₁³ − m₁) + 1/12(m₂³ − m₂) + ...] / n(n² − 1)
• Used for qualitative data, ranked data, or when extreme values exist
• Properties of r: range −1 to +1, unit-free, symmetric (rₓᵧ = rᵧₓ), affected by extreme values
• Correlation ≠ Causation: correlation shows association, not cause-effect

Intext Questions

Section 7.1 — Scatter Diagram

Section 7.2 — Karl Pearson's Method

Chapter-End Exercise

Question 1

Calculate Karl Pearson's coefficient of correlation from the following data:

Answer (Direct Method using deviations from mean):

X̄ = (10+20+30+40+50)/5 = 150/5 = 30 Ȳ = (20+25+35+40+50)/5 = 170/5 = 34

r = Σxy / √(Σx² × Σy²) = 750 / √(1000 × 570) = 750 / √570000 = 750 / 754.98 = 0.993

Interpretation: Very high positive correlation.

Question 2

Calculate Karl Pearson's coefficient using the alternative formula:

r = (NΣXY − ΣXΣY) / √[(NΣX² − (ΣX)²)(NΣY² − (ΣY)²)]

= (5×1750 − 75×100) / √[(5×1375 − 75²)(5×2250 − 100²)]

= (8750 − 7500) / √[(6875 − 5625)(11250 − 10000)]

= 1250 / √[1250 × 1250]

= 1250 / 1250 = 1.0

Interpretation: Perfect positive correlation.

Question 3

Calculate Spearman's rank correlation coefficient:

Step 1: Assign ranks (highest = rank 1)

ΣD² = 0

rₖ = 1 − 6ΣD²/n(n² − 1) = 1 − 6(0)/6(36−1) = 1 − 0 = 1.0

Interpretation: Perfect positive rank correlation.

Question 4

Calculate Spearman's rank correlation:

ΣD² = 8, n = 5

rₖ = 1 − 6ΣD²/n(n² − 1) = 1 − 6(8)/5(25−1) = 1 − 48/120 = 1 − 0.4 = 0.6

Interpretation: Moderate positive rank correlation.

Question 5

Calculate Karl Pearson's r from the following:

X̄ = (6+2+10+4+8)/5 = 30/5 = 6 Ȳ = (9+11+5+8+7)/5 = 40/5 = 8

Σxy = −26, Σx² = 40, Σy² = 20

r = Σxy / √(Σx² × Σy²) = −26 / √(40 × 20) = −26 / √800 = −26/28.28 = −0.92

Interpretation: High negative correlation.

Question 6

Calculate Spearman's rank correlation when there are tied ranks:

Step 1: Assign ranks with tied ranks getting average rank

Ranks for X: 20, 20 → tied at positions 4 and 5 → average rank = (4+5)/2 = 4.5 each; 30 → 3; 40 → 2; 50 → 1

Ranks for Y: 10→5; 15, 15 → tied at positions 3 and 4 → average = 3.5 each; 25→2; 30→1

ΣD² = 1.50, n = 5

rₖ = 1 − 6ΣD²/n(n² − 1) = 1 − 6(1.5)/5(24) = 1 − 9/120 = 1 − 0.075 = 0.925

Question 7

What is the difference between positive and negative correlation?

Question 8

State the properties of correlation coefficient (r).

1. Range: r lies between −1 and +1, i.e., −1 ≤ r ≤ +1. 2. Unit-free: r has no unit of measurement. 3. Symmetric: rₓᵧ = rᵧₓ (correlation of X with Y equals correlation of Y with X). 4. Independent of origin and scale: r is not affected by change of origin (adding a constant) or scale (multiplying by a positive constant). 5. Does not imply causation: A high r does not mean X causes Y. 6. Measures linear relationship: r = 0 does not mean no relationship; it means no linear relationship. 7. Affected by extreme values: Outliers can significantly change r.

Question 9

What are the advantages of the scatter diagram?

1. Simple to construct — just plot paired values. 2. Gives a visual indication of the type (positive/negative/zero) and degree of correlation. 3. Does not require any calculation. 4. Helps identify outliers that may affect the correlation. 5. Reveals whether the relationship is linear or non-linear. 6. Useful preliminary step before calculating r.

Question 10

What are the limitations of Karl Pearson's coefficient?

1. Assumes linear relationship — cannot measure non-linear correlation. 2. Affected by extreme values — outliers distort r. 3. Does not indicate causation. 4. Requires quantitative data on both variables. 5. Can be misleading if data is not bivariate normal. 6. r = 0 means no linear correlation, but a non-linear relationship may exist.

Question 11

When is Spearman's rank correlation preferred over Karl Pearson's?

1. Data is qualitative and can only be ranked (e.g., beauty, honesty). 2. Only ranks are available, not actual values. 3. Extreme values are present that would distort Karl Pearson's r. 4. The relationship is not strictly linear. 5. The sample size is small. 6. Quick estimation of correlation is needed.

Question 12

"Correlation does not imply causation." Explain with examples.

1. Common cause: Both X and Y are affected by a third variable Z. Example: Ice cream sales and drowning deaths are positively correlated, but the common cause is summer heat — not that ice cream causes drowning.

2. Coincidence: Two variables may move together by chance without any real connection.

3. Reverse causation: Y may cause X, not the other way around.

4. Intervening variable: X affects Z, which affects Y.

Therefore, statistical correlation alone cannot establish cause and effect. Additional evidence and logical reasoning are required.

Internal Links

• [Chapter 7 Guide — Correlation](/school/cbse/class/11/economics/cbse-correlation)
• [All Class 11 Economics Solutions](/school/cbse/class/11/economics/solutions)
• [Class 11 Economics Subject Guide](/school/cbse/class/11/economics)