Class 11 Statistics for Economics Chapter 6: Correlation

Introduction

In daily life, many variables are related to each other.

Examples

• Higher temperature → More ice cream sales
• Higher income → Higher consumption
• Higher price → Lower demand
• More rainfall → Higher agricultural production

The statistical method used to study such relationships is called Correlation.

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Meaning of Correlation

Definition

Correlation is a statistical technique used to measure the relationship between two variables.

It shows:

• Whether variables are related.
• Direction of relationship.
• Strength of relationship.

Important Point

Correlation measures association, not causation.

Just because two variables move together does not mean one causes the other.

Example

Ice cream sales and drowning deaths may rise together in summer.

Reason:

• Hot weather increases both.
• Ice cream does not cause drowning.

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Types of Correlation

1. Positive Correlation

When two variables move in the same direction.

Examples

• Income and Consumption
• Rainfall and Crop Production
• Temperature and Ice Cream Sales

Characteristics

X Increases	Y Increases
X Decreases	Y Decreases

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2. Negative Correlation

When two variables move in opposite directions.

Examples

• Price and Demand
• Study Time and Failure Rate

Characteristics

X Increases	Y Decreases
X Decreases	Y Increases

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3. Zero Correlation

When no relationship exists between two variables.

Examples

• Shoe Size and Bank Balance
• Number of Birds and Birth Rate

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Methods of Studying Correlation

There are three important methods:

1. Scatter Diagram
2. Karl Pearson’s Coefficient of Correlation
3. Spearman’s Rank Correlation

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1. Scatter Diagram

Meaning

A scatter diagram is a graphical method used to study the relationship between two variables.

Procedure

• Plot values of X and Y on graph paper.
• Each pair becomes one point.
• Observe the pattern formed.

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Types of Scatter Diagrams

A. Positive Correlation

Points move upward from left to right.

     *
   *
 *
*

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B. Negative Correlation

Points move downward from left to right.

*
  *
    *
      *

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C. No Correlation

Points are scattered randomly.

*     *
   *
      *
 *

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D. Perfect Positive Correlation

All points lie exactly on an upward line.

Correlation = +1

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E. Perfect Negative Correlation

All points lie exactly on a downward line.

Correlation = -1

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Advantages of Scatter Diagram

✔ Simple

✔ Easy to understand

✔ Shows nature of relationship

✔ Can identify non-linear relationships

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2. Karl Pearson’s Coefficient of Correlation

Meaning

Karl Pearson developed a numerical measure of correlation.

It measures:

• Direction
• Degree (strength)

of linear relationship between two variables.

Symbol

$$r$$r

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Formula

Direct Formula

$$r= \frac{N\Sigma XY-(\Sigma X)(\Sigma Y)} {\sqrt{[N\Sigma X^2-(\Sigma X)^2][N\Sigma Y^2-(\Sigma Y)^2]}}$$r=[NΣX2−(ΣX)2][NΣY2−(ΣY)2]​NΣXY−(ΣX)(ΣY)​

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Interpretation of r

Value of r	Meaning
+1	Perfect Positive Correlation
-1	Perfect Negative Correlation
0	No Linear Correlation
+0.8 to +1	Strong Positive Correlation
+0.2 to +0.8	Moderate Positive Correlation
Near 0	Weak Correlation

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Properties of Karl Pearson’s Coefficient

1. No Unit

r is a pure number.

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2. Range

$$-1 \le r \le +1$$−1≤r≤+1

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3. Positive Value

Indicates positive correlation.$$r > 0$$r>0

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4. Negative Value

Indicates negative correlation.$$r < 0$$r<0

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5. Zero Value

No linear relationship.$$r = 0$$r=0

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6. Perfect Correlation

$$r = +1$$r=+1

or$$r = -1$$r=−1

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7. Independent of Origin and Scale

Changing units does not affect r.

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Advantages of Karl Pearson’s Method

✔ Most accurate method

✔ Gives numerical result

✔ Measures strength and direction

✔ Widely used

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Limitations

✘ Measures only linear relationship

✘ Affected by extreme values

✘ Difficult calculations

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Step Deviation Method

Used when numerical values are large.

Transformation

$$U=\frac{X-A}{h}$$U=hX−A​ $$V=\frac{Y-B}{k}$$V=kY−B​

Where:

• A and B = Assumed Means
• h and k = Common Factors

Property

$$r_{UV}=r_{XY}$$rUV​=rXY​

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3. Spearman’s Rank Correlation

Meaning

Developed by:

Charles Edward Spearman

Used when:

• Exact measurements are unavailable.
• Data is in ranks.
• Qualitative characteristics are studied.

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Examples

• Beauty
• Honesty
• Intelligence
• Leadership

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Formula

$$r_s= 1-\frac{6\Sigma D^2} {n(n^2-1)}$$rs​=1−n(n2−1)6ΣD2​

Where:

• D = Difference between ranks
• n = Number of observations

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Steps to Calculate Rank Correlation

Step 1

Assign ranks.

Step 2

Find differences between ranks.$$D = R_x – R_y$$D=Rx​−Ry​

Step 3

Calculate$$D^2$$D2

Step 4

Apply formula.

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Properties of Rank Correlation

Range

$$-1 \le r_s \le +1$$−1≤rs​≤+1

Interpretation

Same as Karl Pearson’s coefficient.

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Advantages of Rank Correlation

✔ Easy to calculate

✔ Suitable for qualitative data

✔ Less affected by extreme values

✔ Useful when actual measurements are unavailable

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Limitations

✘ Less accurate than Karl Pearson’s method

✘ Uses ranks instead of actual values

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Repeated Ranks (Tied Ranks)

When two or more observations have the same value, they are assigned the average rank.

Example

Ranks 4 and 5 are tied.

Assigned Rank:$$\frac{4+5}{2}=4.5$$24+5​=4.5

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Difference Between Karl Pearson and Spearman Rank Correlation

Basis	Karl Pearson	Spearman
Data Type	Actual Values	Ranks
Accuracy	More Accurate	Less Accurate
Variables	Quantitative	Qualitative/Ranked
Extreme Values	Affected	Less Affected
Complexity	More	Less

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Correlation vs Causation

Correlation

Shows relationship between variables.

Causation

Shows cause-and-effect relationship.

Important

Correlation does not imply causation.

Example:

• Ice cream sales ↑
• Drowning deaths ↑

Both are caused by higher temperature.

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Key Formulae for Examination

Karl Pearson’s Coefficient

$$r= \frac{N\Sigma XY-(\Sigma X)(\Sigma Y)} {\sqrt{[N\Sigma X^2-(\Sigma X)^2][N\Sigma Y^2-(\Sigma Y)^2]}}$$r=[NΣX2−(ΣX)2][NΣY2−(ΣY)2]​NΣXY−(ΣX)(ΣY)​

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Spearman Rank Correlation

$$r_s= 1-\frac{6\Sigma D^2} {n(n^2-1)}$$rs​=1−n(n2−1)6ΣD2​

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Step Deviation

$$U=\frac{X-A}{h}$$U=hX−A​ $$V=\frac{Y-B}{k}$$V=kY−B​

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One-Page Revision

Correlation

Relationship between two variables.

Types

• Positive
• Negative
• Zero

Methods

1. Scatter Diagram
2. Karl Pearson Correlation
3. Spearman Rank Correlation

Range

$$-1 \le r \le +1$$−1≤r≤+1

Perfect Positive

$$r = +1$$r=+1

Perfect Negative

$$r = -1$$r=−1

No Correlation

$$r = 0$$r=0

Spearman Formula

$$r_s= 1-\frac{6\Sigma D^2} {n(n^2-1)}$$rs​=1−n(n2−1)6ΣD2​

All Possible MCQs, Fill in the Blanks, True/False, Assertion-Reason, Very Short, Short and Long Questions

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A. Multiple Choice Questions (MCQs)

1. Correlation studies the relationship between:

a) One variable
b) Two variables
c) Three variables
d) Four variables

Ans: b) Two variables

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2. Correlation measures:

a) Causation
b) Covariation
c) Production
d) Classification

Ans: b) Covariation

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3. Correlation does not imply:

a) Relationship
b) Association
c) Causation
d) Direction

Ans: c) Causation

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4. When two variables move in the same direction, it is called:

a) Negative correlation
b) Positive correlation
c) Zero correlation
d) Perfect correlation

Ans: b) Positive correlation

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5. When one variable increases and the other decreases:

a) Positive correlation
b) Perfect correlation
c) Negative correlation
d) Zero correlation

Ans: c) Negative correlation

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6. Which method provides a visual presentation of correlation?

a) Mean
b) Median
c) Scatter Diagram
d) Mode

Ans: c) Scatter Diagram

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7. If all points lie exactly on an upward sloping line:

a) Perfect Negative Correlation
b) Perfect Positive Correlation
c) No Correlation
d) Zero Correlation

Ans: b) Perfect Positive Correlation

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8. If all points lie exactly on a downward sloping line:

a) Perfect Positive Correlation
b) Zero Correlation
c) Perfect Negative Correlation
d) Weak Correlation

Ans: c) Perfect Negative Correlation

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9. Karl Pearson’s coefficient is represented by:

a) M
b) D
c) r
d) R

Ans: c) r

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10. Karl Pearson’s coefficient measures:

a) Non-linear relationship
b) Linear relationship
c) Qualitative relationship
d) Random relationship

Ans: b) Linear relationship

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11. The value of r always lies between:

a) 0 and 100
b) –1 and +1
c) –10 and +10
d) 0 and 1

Ans: b) –1 and +1

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12. If r = +1:

a) Perfect Positive Correlation
b) Perfect Negative Correlation
c) No Correlation
d) Weak Correlation

Ans: a) Perfect Positive Correlation

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13. If r = –1:

a) Positive Correlation
b) Perfect Positive Correlation
c) Perfect Negative Correlation
d) No Correlation

Ans: c) Perfect Negative Correlation

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14. If r = 0:

a) Perfect Correlation
b) No Linear Correlation
c) Positive Correlation
d) Negative Correlation

Ans: b) No Linear Correlation

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15. Karl Pearson’s coefficient is:

a) A pure number
b) A percentage
c) A ratio
d) A frequency

Ans: a) A pure number

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16. Spearman’s Rank Correlation was developed by:

a) Karl Pearson
b) Bowley
c) C.E. Spearman
d) Fisher

Ans: c) C.E. Spearman

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17. Spearman’s method is based on:

a) Frequencies
b) Means
c) Ranks
d) Percentages

Ans: c) Ranks

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18. Rank correlation is useful for:

a) Quantitative variables only
b) Qualitative variables
c) Frequency distributions
d) Time series

Ans: b) Qualitative variables

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19. Which correlation method is less affected by extreme values?

a) Pearson’s Correlation
b) Scatter Diagram
c) Spearman’s Correlation
d) Covariance

Ans: c) Spearman’s Correlation

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20. Which method is more accurate when exact measurements are available?

a) Spearman Rank Correlation
b) Pearson Correlation
c) Scatter Diagram
d) None

Ans: b) Pearson Correlation

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B. Fill in the Blanks

1. Correlation measures the ______ between two variables.
   Ans: relationship
2. Correlation measures ______ and not causation.
   Ans: covariation
3. Positive correlation means variables move in the ______ direction.
   Ans: same
4. Negative correlation means variables move in ______ directions.
   Ans: opposite
5. Scatter Diagram is a ______ method of studying correlation.
   Ans: graphical
6. Karl Pearson’s coefficient is denoted by ______.
   Ans: r
7. The value of r lies between ______ and ______.
   Ans: –1, +1
8. Perfect positive correlation is represented by ______.
   Ans: +1
9. Perfect negative correlation is represented by ______.
   Ans: –1
10. Spearman’s correlation is based on ______.
    Ans: ranks
11. Rank correlation is useful when data cannot be ______ measured.
    Ans: precisely
12. Karl Pearson’s coefficient measures only ______ relationships.
    Ans: linear
13. A high value of r indicates a ______ relationship.
    Ans: strong
14. A low value of r indicates a ______ relationship.
    Ans: weak
15. Correlation coefficient has no ______.
    Ans: unit

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C. True or False

1. Correlation implies causation.
   Ans: False
2. Correlation measures relationship between variables.
   Ans: True
3. Positive correlation means variables move together.
   Ans: True
4. Negative correlation means variables move in opposite directions.
   Ans: True
5. Scatter Diagram gives a visual representation.
   Ans: True
6. Karl Pearson’s coefficient measures linear relationships.
   Ans: True
7. The value of r can exceed +1.
   Ans: False
8. The value of r can be less than –1.
   Ans: False
9. Spearman’s method uses ranks.
   Ans: True
10. Rank correlation is unaffected by extreme values.
    Ans: True
11. Pearson’s method is more accurate than rank correlation.
    Ans: True
12. If r = 0, there is no linear relationship.
    Ans: True
13. Perfect correlation occurs when r = ±1.
    Ans: True
14. Correlation coefficient has units.
    Ans: False
15. Scatter Diagram can show non-linear relationships.
    Ans: True

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D. Match the Following

Column A	Column B
Positive Correlation	Variables move together
Negative Correlation	Variables move oppositely
r = +1	Perfect Positive Correlation
r = –1	Perfect Negative Correlation
r = 0	No Linear Correlation
Karl Pearson	Linear Correlation
Spearman	Rank Correlation
Scatter Diagram	Graphical Method
Covariation	Correlation
C.E. Spearman	Rank Correlation

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E. Assertion and Reason Questions

1.

Assertion (A): Correlation measures relationship between variables.

Reason (R): Correlation always implies causation.

Ans: A is true, R is false.

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2.

Assertion (A): Pearson’s coefficient measures linear relationship.

Reason (R): It should be used only for linear data.

Ans: Both A and R are true and R is the correct explanation.

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3.

Assertion (A): Spearman’s correlation is useful for qualitative data.

Reason (R): Qualitative characteristics can be ranked.

Ans: Both A and R are true and R is the correct explanation.

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4.

Assertion (A): The value of r lies between –1 and +1.

Reason (R): r is a pure number.

Ans: Both A and R are true but R is not the correct explanation.

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F. One Word Answer Questions

1. Relationship between variables?
   Ans: Correlation
2. Symbol of correlation coefficient?
   Ans: r
3. Correlation when variables move together?
   Ans: Positive
4. Correlation when variables move oppositely?
   Ans: Negative
5. Visual method of correlation?
   Ans: Scatter Diagram
6. Correlation based on ranks?
   Ans: Spearman
7. Perfect positive correlation value?
   Ans: +1
8. Perfect negative correlation value?
   Ans: –1
9. No linear correlation value?
   Ans: 0
10. Developer of Rank Correlation?
    Ans: Spearman

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G. Very Short Answer Questions (1 Mark)

1. Define correlation.
2. What is positive correlation?
3. What is negative correlation?
4. What is scatter diagram?
5. What is Karl Pearson’s coefficient?
6. What is Spearman’s rank correlation?
7. State the range of r.
8. What does r = 0 indicate?
9. What does r = +1 indicate?
10. What does r = –1 indicate?

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H. Short Answer Questions (3 Marks)

1. Explain correlation.
2. Differentiate between positive and negative correlation.
3. Explain Scatter Diagram.
4. State advantages of Scatter Diagram.
5. Explain Karl Pearson’s coefficient.
6. State properties of correlation coefficient.
7. Explain Spearman’s rank correlation.
8. Why is rank correlation used?
9. Differentiate between Pearson and Spearman methods.
10. Explain the concept of covariation.

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I. Long Answer Questions (5 Marks)

1. Explain the meaning and importance of correlation.
2. Describe different types of correlation.
3. Explain Scatter Diagram method with merits.
4. Explain Karl Pearson’s coefficient of correlation.
5. Discuss properties of correlation coefficient.
6. Explain Spearman’s rank correlation and its uses.
7. Differentiate between Karl Pearson and Spearman correlation.
8. Explain correlation and causation with examples.
9. Discuss merits and limitations of correlation analysis.
10. Explain situations where rank correlation is preferred.

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Important Formula-Based Questions

Karl Pearson’s Coefficient

$$r= \frac{N\Sigma XY-(\Sigma X)(\Sigma Y)} {\sqrt{[N\Sigma X^2-(\Sigma X)^2][N\Sigma Y^2-(\Sigma Y)^2]}}$$r=[NΣX2−(ΣX)2][NΣY2−(ΣY)2]​NΣXY−(ΣX)(ΣY)​

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Spearman’s Rank Correlation

$$r_s= 1-\frac{6\Sigma D^2} {n(n^2-1)}$$rs​=1−n(n2−1)6ΣD2​

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Step Deviation

$$U=\frac{X-A}{h}$$U=hX−A​ $$V=\frac{Y-B}{k}$$V=kY−B​

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Important HOTS / Competency-Based Questions

1. Why does correlation not imply causation?
2. Can two variables have correlation without any real relationship?
3. Why should a scatter diagram be drawn before calculating Pearson’s coefficient?
4. Why is Spearman’s method suitable for intelligence and beauty?
5. Can two variables have zero Pearson correlation but still be related?
6. Why is Pearson’s coefficient called a pure number?
7. Explain the importance of correlation in economics.
8. Why are repeated ranks corrected in Spearman’s method?
9. Which method would you prefer for qualitative data and why?
10. Explain the significance of the value of r in decision-making.