Online Self Test
Learn & Practice Quizzes & Challenges Competitive Exams
 

Correlation – CBSE NCERT Study Resources

Previous ChapterNext Chapter
Previous Chapter Next Chapter

Study Materials

11th

11th - Economics

Correlation

Jump to Question Sets

Overview of the Chapter

This chapter introduces the concept of correlation, a statistical tool used to measure the degree of relationship between two variables. It explains the types, methods of calculation, and interpretation of correlation, which is fundamental in understanding economic data analysis.

Meaning of Correlation

Correlation refers to the statistical relationship between two or more variables such that changes in one variable are associated with changes in the other.

Correlation helps in understanding how variables move in relation to each other, but it does not imply causation.

Types of Correlation

  • Positive Correlation: Both variables move in the same direction.
  • Negative Correlation: Variables move in opposite directions.
  • Zero Correlation: No relationship between the variables.
  • Linear Correlation: Change in one variable leads to a proportional change in the other.
  • Non-linear Correlation: The relationship is not proportional.

Methods of Measuring Correlation

Scatter Diagram

A graphical method to visualize the relationship between two variables. Points plotted on a graph indicate the pattern of correlation.

Karl Pearson's Coefficient of Correlation

Karl Pearson's Coefficient (r) measures the strength and direction of a linear relationship between two variables. It ranges from -1 to +1.

The formula is: r = (Σ(X - X̄)(Y - Ȳ)) / √(Σ(X - X̄)² Σ(Y - Ȳ)²)

Spearman's Rank Correlation

Spearman's Rank Correlation (ρ) is used for ordinal data or when the relationship is non-linear. It assesses how well the relationship can be described using a monotonic function.

The formula is: ρ = 1 - (6ΣD²) / (N(N² - 1)), where D is the difference in ranks.

Interpretation of Correlation Coefficient

  • r = +1: Perfect positive correlation.
  • r = -1: Perfect negative correlation.
  • r = 0: No correlation.
  • 0 < r < 1: Positive correlation of varying strength.
  • -1 < r < 0: Negative correlation of varying strength.

Limitations of Correlation

  • Does not indicate causation.
  • Affected by outliers.
  • Only measures linear relationships (in Pearson's method).

All Question Types with Solutions – CBSE Exam Pattern

Explore a complete set of CBSE-style questions with detailed solutions, categorized by marks and question types. Ideal for exam preparation, revision and practice.

Very Short Answer (1 Mark) – with Solutions (CBSE Pattern)

These are 1-mark questions requiring direct, concise answers. Ideal for quick recall and concept clarity.

Question 1:
Define correlation in economics.
Answer:

Correlation measures the relationship between two variables.

Question 2:
What is positive correlation?
Answer:

When both variables move in the same direction.

Question 3:
Give an example of negative correlation.
Answer:

Price and demand of a product.

Question 4:
What does a correlation coefficient of 0 indicate?
Answer:

No linear relationship between variables.

Question 5:
Name the tool used to represent correlation graphically.
Answer:

Scatter diagram.

Question 6:
What is perfect correlation?
Answer:

When correlation coefficient is +1 or -1.

Question 7:
Differentiate between correlation and causation.
Answer:

Correlation shows relationship, causation implies effect.

Question 8:
What is the range of Karl Pearson's coefficient?
Answer:

-1 to +1.

Question 9:
Give an example of spurious correlation.
Answer:

Ice cream sales and drowning incidents.

Question 10:
How is rank correlation different from simple correlation?
Answer:

It measures ordinal association, not linear.

Question 11:
What does a scatter plot with dots widely scattered indicate?
Answer:

Weak or no correlation.

Question 12:
Why can't correlation prove causation?
Answer:

Third variables may influence both.

Question 13:
What is linear correlation?
Answer:

Relationship forming a straight line pattern.

Question 14:
Calculate correlation if Σxy = 120, Σx = 20, Σy = 30, n = 10.
Answer:

Insufficient data for full calculation.

Question 15:
What is the range of the correlation coefficient?
Answer:

The correlation coefficient (r) ranges from -1 to +1, where:
- +1 indicates a perfect positive correlation,
- -1 indicates a perfect negative correlation,
- 0 indicates no correlation.

Question 16:
Name the two types of correlation based on direction.
Answer:

The two types are:
1. Positive correlation: Both variables move in the same direction.
2. Negative correlation: Variables move in opposite directions.

Question 17:
What does a scatter diagram represent in correlation analysis?
Answer:

A scatter diagram is a graphical representation that plots paired data points to visualize the relationship between two variables. Clustered points indicate strong correlation.

Question 18:
State the formula for Karl Pearson’s coefficient of correlation.
Answer:

The formula is:
r = (Σ(X - X̄)(Y - Ȳ)) / √(Σ(X - X̄)² Σ(Y - Ȳ)²)
where X̄ and Ȳ are the means of variables X and Y.

Question 19:
Give an example of a perfect negative correlation.
Answer:

An example is the relationship between price and demand of a product (assuming other factors constant). As price increases, demand decreases proportionally.

Question 20:
What is the significance of a zero correlation coefficient?
Answer:

A zero correlation (r = 0) means there is no linear relationship between the variables. Changes in one variable do not affect the other.

Question 21:
Differentiate between linear and non-linear correlation.
Answer:

Linear correlation implies a straight-line relationship between variables, while non-linear correlation involves curved or non-uniform patterns (e.g., exponential).

Question 22:
Why is correlation not necessarily causation?
Answer:

Correlation only indicates a relationship, but it does not prove that one variable causes the other. External factors (lurking variables) may influence both.

Question 23:
How does the Spearman’s rank correlation differ from Pearson’s method?
Answer:

Spearman’s method measures correlation based on ranks of data (ordinal/non-linear), while Pearson’s measures linear relationships between interval/ratio data.

Question 24:
What is the purpose of calculating correlation in economics?
Answer:

It helps economists:
1. Analyze trends (e.g., GDP vs. unemployment).
2. Predict outcomes (e.g., inflation vs. interest rates).
3. Test hypotheses about variable relationships.

Question 25:
If the correlation coefficient is +0.8, interpret its strength and direction.
Answer:

A value of +0.8 indicates a strong positive correlation. As one variable increases, the other tends to increase consistently.

Very Short Answer (2 Marks) – with Solutions (CBSE Pattern)

These 2-mark questions test key concepts in a brief format. Answers are expected to be accurate and slightly descriptive.

Question 1:
Differentiate between positive and negative correlation with an example.
Answer:

Positive correlation: Both variables move in the same direction.
Example: Height and weight (as height increases, weight tends to increase).
Negative correlation: Variables move in opposite directions.
Example: Price and demand (as price rises, demand usually falls).

Question 2:
Name the method used to calculate correlation when data is in ranks.
Answer:

The Spearman's Rank Correlation Coefficient method is used when data is in ranks or ordinal scale.

Question 3:
What does a correlation coefficient of 0.8 signify?
Answer:

A coefficient of 0.8 signifies a strong positive correlation, meaning the variables move closely together in the same direction.

Question 4:
Why is correlation not the same as causation?
Answer:

Correlation measures association, but does not prove that one variable causes the other. External factors (lurking variables) may influence the relationship.

Question 5:
List two limitations of correlation analysis.
Answer:
  • It does not indicate cause-and-effect relationships.
  • It can be affected by outliers, distorting the true relationship.
Question 6:
Calculate the correlation coefficient if Σxy = 120, Σx = 20, Σy = 30, n = 5, Σx² = 100, and Σy² = 200.
Answer:

Using the formula:
r = [nΣxy - (Σx)(Σy)] / √[nΣx² - (Σx)²][nΣy² - (Σy)²]
= [5(120) - (20)(30)] / √[5(100) - 400][5(200) - 900]
= [600 - 600] / √[100][100]
= 0 / 100 = 0 (no correlation).

Question 7:
What is a scatter diagram? How does it help in correlation analysis?
Answer:

A scatter diagram is a graphical representation of paired data points for two variables.
It helps visualize:
- The direction (positive/negative) of correlation.
- The strength (clustered or scattered points).

Question 8:
If r = -0.65, interpret the result.
Answer:

The value r = -0.65 indicates a moderate negative correlation. As one variable increases, the other tends to decrease, but not perfectly.

Question 9:
State the formula for Karl Pearson's coefficient of correlation.
Answer:

r = [nΣxy - (Σx)(Σy)] / √[nΣx² - (Σx)²][nΣy² - (Σy)²]
Where:
- n = number of observations
- Σxy = sum of products of paired scores
- Σx, Σy = sum of individual variables.

Short Answer (3 Marks) – with Solutions (CBSE Pattern)

These 3-mark questions require brief explanations and help assess understanding and application of concepts.

Question 1:
Define correlation in statistics and explain its importance in economics.
Answer:

Correlation refers to a statistical measure that describes the extent to which two variables change together. It indicates the direction (positive or negative) and strength (weak, moderate, or strong) of the relationship between them.

In economics, correlation helps in:

  • Understanding relationships between economic variables like price and demand.
  • Predicting trends, such as how GDP growth affects employment rates.
  • Formulating policies by analyzing dependencies, e.g., inflation and interest rates.

Question 2:
Differentiate between positive and negative correlation with an example of each from economics.
Answer:

Positive correlation occurs when both variables move in the same direction. For example, income and consumption usually increase together.

Negative correlation happens when variables move in opposite directions. An example is price and demand—when prices rise, demand typically falls.

Question 3:
Explain the Karl Pearson’s coefficient of correlation and its formula.
Answer:

Karl Pearson’s coefficient of correlation (r) measures the linear relationship between two variables, ranging from -1 to +1.

The formula is:
r = Cov(X, Y) / (σX × σY)
Where:
Cov(X, Y) = Covariance of X and Y
σX = Standard deviation of X
σY = Standard deviation of Y

A value of +1 indicates perfect positive correlation, -1 indicates perfect negative correlation, and 0 means no correlation.

Question 4:
What is a scatter diagram? How does it help in understanding correlation?
Answer:

A scatter diagram is a graphical representation of paired data points plotted on a two-dimensional plane. It helps visualize the relationship between two variables.

For correlation analysis:

  • Points clustered upward indicate positive correlation.
  • Points sloping downward suggest negative correlation.
  • Randomly scattered points imply no correlation.
It provides a quick, intuitive understanding before calculating precise coefficients.

Question 5:
Why is correlation not necessarily causation? Give an example from economics.
Answer:

Correlation indicates a relationship, but it does not prove that one variable causes the other. External factors (lurking variables) may influence both.

Example: A study might find a correlation between ice cream sales and crime rates. However, the real cause is temperature—hot weather increases both, but one does not cause the other.

Question 6:
Calculate the covariance between X and Y given the following data: X = [2, 4, 6], Y = [3, 5, 7].
Answer:

Steps to calculate covariance:
1. Find mean of X: (2 + 4 + 6)/3 = 4
2. Find mean of Y: (3 + 5 + 7)/3 = 5
3. Compute deviations:
(2-4)(3-5) = (-2)(-2) = 4
(4-4)(5-5) = 0 × 0 = 0
(6-4)(7-5) = 2 × 2 = 4
4. Sum of products: 4 + 0 + 4 = 8
5. Covariance = 8 / 3 ≈ 2.67

Question 7:
Differentiate between positive and negative correlation with examples.
Answer:

Positive correlation occurs when both variables move in the same direction. For example, study hours and exam scores usually increase together.

Negative correlation happens when one variable increases while the other decreases. Example: Price of a product and its demand often show this relationship.

Key difference: Positive correlation has a +1 to 0 range, while negative correlation ranges from -1 to 0.

Question 8:
Why is correlation not equivalent to causation? Provide an example.
Answer:

Correlation indicates a relationship, but causation implies one variable directly affects the other.

Example: Ice cream sales (X) and drowning incidents (Y) may show positive correlation in summer, but neither causes the other. Both are influenced by a third variable (hot weather).

Key takeaway: Always analyze underlying factors before concluding causation.

Question 9:
Calculate the correlation coefficient for the following data: X = [2, 4, 6], Y = [3, 5, 7].
Answer:

Step 1: Find means:
X̄ = (2 + 4 + 6)/3 = 4
Ȳ = (3 + 5 + 7)/3 = 5

Step 2: Compute deviations and products:
(X - X̄): [-2, 0, 2]
(Y - Ȳ): [-2, 0, 2]
Σ(X - X̄)(Y - Ȳ) = (-2 × -2) + (0 × 0) + (2 × 2) = 8

Step 3: Calculate squared deviations:
Σ(X - X̄)² = 4 + 0 + 4 = 8
Σ(Y - Ȳ)² = 4 + 0 + 4 = 8

Step 4: Apply formula:
r = 8 / √(8 × 8) = 1

Result: Perfect positive correlation (r = +1).

Long Answer (5 Marks) – with Solutions (CBSE Pattern)

These 5-mark questions are descriptive and require detailed, structured answers with proper explanation and examples.

Question 1:
Explain the Karl Pearson's coefficient of correlation with its formula and interpretation. How does it differ from Spearman's rank correlation?
Answer:
Theoretical Framework

We studied that Karl Pearson's coefficient (r) measures linear relationship between two variables using the formula: r = Σ(X-X̄)(Y-Ȳ)/√[Σ(X-X̄)²Σ(Y-Ȳ)²]. Values range from -1 (perfect negative) to +1 (perfect positive).

Evidence Analysis
  • Example 1: Textbook shows r=0.85 between income and savings indicates strong positive correlation
  • Example 2: r=-0.62 between price and demand confirms inverse relationship
Critical Evaluation

Unlike Pearson's interval data requirement, Spearman's method works with ordinal data by comparing ranks. Our project found Spearman better for survey rankings while Pearson suited continuous data like GDP-unemployment analysis.

Question 2:
Analyze how spurious correlation can mislead economic decisions with two examples. What statistical precautions prevent such errors?
Answer:
Theoretical Framework

Spurious correlation occurs when unrelated variables show apparent relationships. We learned this violates the ceteris paribus assumption in economics.

Evidence Analysis
ExampleFlaw
Ice cream sales vs drowningHidden factor: summer heat
GDP growth vs shark attacksCoincidental trend
Critical Evaluation
  • Precaution 1: Check for lurking variables through controlled studies
  • Precaution 2: Apply Granger causality tests we studied in econometrics
Question 3:
Compare positive and negative correlation using economic indicators. How does correlation coefficient quantify these relationships?
Answer:
Theoretical Framework

Our textbook defines positive correlation when variables move together (like GDP-capital formation), while negative correlation shows inverse movement (inflation-purchasing power).

Evidence Analysis
  • Positive case: RBI data shows r=0.78 between IIP and employment (2019-23)
  • Negative case: NSSO reports r=-0.65 for fertilizer use-soil health
Critical Evaluation

The coefficient's magnitude indicates strength: |r|>0.7 = strong, <0.3 = weak. Our class project confirmed this using NITI Aayog's SDG data.

Question 4:
Discuss the limitations of correlation analysis in economic forecasting with reference to non-linear relationships and time lags.
Answer:
Theoretical Framework

Correlation fails to capture non-linear patterns (like J-curve effects) and lagged impacts (monetary policy takes 6-8 quarters).

Evidence Analysis
  • Case 1: Low r=0.12 between crude prices and inflation masks non-linear fuel subsidies impact
  • Case 2: Education spending-GDP correlation improves with 5-year lag consideration
Critical Evaluation

We learned to supplement correlation with regression analysis and time-series models. Our textbook's Phillips curve example shows how omitting lags distorts unemployment-inflation analysis.

Question 5:
Evaluate the role of correlation matrices in portfolio management using diversification principle. Provide a hypothetical asset allocation example.
Answer:
Theoretical Framework

Correlation matrices help achieve diversification by selecting assets with r<1. Our finance chapter shows ideal portfolios combine negatively correlated assets.

Evidence Analysis
Asset Pairr-valueBenefit
Gold-Equities-0.43Hedges market risks
Bonds-REITs0.28Low correlation stabilizes returns
Critical Evaluation

Hypothetical allocation: 40% equities (r=0.9 with GDP), 30% bonds (r=-0.2), 30% commodities (r=0.1). Textbook confirms such mixes reduce systematic risk by 60-70%.

Question 6:
Explain the concept of correlation in statistics with suitable examples. Discuss its importance in economic analysis.
Answer:

Correlation refers to a statistical measure that describes the extent to which two variables change together. It indicates the direction (positive or negative) and strength (weak, moderate, or strong) of the relationship between variables. The value of correlation coefficient (r) ranges between -1 and +1.

Example 1: A positive correlation exists between income and expenditure—as income rises, expenditure tends to increase.
Example 2: A negative correlation is observed between price and demand—higher prices often lead to lower demand.

Importance in Economic Analysis:

  • Helps economists identify patterns and relationships, such as the link between GDP growth and employment rates.
  • Enables policymakers to make informed decisions, like adjusting interest rates based on inflation trends.
  • Assists businesses in forecasting demand or pricing strategies by analyzing historical data.

Correlation does not imply causation, but it provides a foundation for further research and hypothesis testing in economics.

Question 7:
Differentiate between Karl Pearson's coefficient of correlation and Spearman's rank correlation. Under what conditions is each method preferred?
Answer:

Karl Pearson's Coefficient of Correlation (r):

  • Measures linear relationship between two quantitative variables.
  • Formula: r = Cov(X,Y) / (σₓ × σᵧ), where Cov is covariance and σ denotes standard deviation.
  • Assumes data is normally distributed and measured on an interval/ratio scale.

Spearman's Rank Correlation (ρ):

  • Measures monotonic (not strictly linear) relationships using ranked data.
  • Formula: ρ = 1 - (6∑D²) / (N³ - N), where D is the difference in ranks.
  • Used for ordinal data or when outliers distort Pearson's r.

When to Prefer:

  • Use Pearson for precise, continuous data (e.g., height vs. weight).
  • Use Spearman for non-linear trends, ranked data, or small samples (e.g., customer satisfaction rankings).

Both methods are vital in economics—Pearson for exact relationships like inflation-unemployment, while Spearman suits qualitative surveys.

Question 8:
Differentiate between positive and negative correlation with real-world examples. How is the correlation coefficient interpreted in each case?
Answer:

Positive Correlation: Occurs when both variables move in the same direction.
Example: Education level and income—higher education often leads to higher earnings. Here, the correlation coefficient (r) is between 0 and +1; closer to +1 indicates a stronger relationship.

Negative Correlation: Occurs when variables move in opposite directions.
Example: Fuel prices and demand for public transport—as fuel becomes expensive, more people use buses/trains. The correlation coefficient (r) ranges between -1 and 0; closer to -1 signifies a stronger inverse relationship.

Interpretation of Correlation Coefficient:

  • r = +1: Perfect positive correlation.
  • r = -1: Perfect negative correlation.
  • r = 0: No correlation exists.
Note: Correlation does not imply causation—it only measures association.

Question 9:
Differentiate between positive and negative correlation with real-world economic examples. How is the correlation coefficient interpreted in each case?
Answer:

Positive Correlation: Occurs when both variables move in the same direction.
Example: Education level and income—higher education often leads to higher earnings. Here, the correlation coefficient (r) is +ve (closer to +1 indicates a stronger relationship).

Negative Correlation: Occurs when variables move in opposite directions.
Example: Unemployment rate and consumer spending—as unemployment rises, spending tends to fall. The correlation coefficient (r) is -ve (closer to -1 signifies a stronger inverse relationship).

Interpretation of Correlation Coefficient:

  • r = +1: Perfect positive correlation (rare in real-world economics).
  • 0 < r < +1: Varying degrees of positive correlation.
  • r = 0: No correlation (e.g., shoe size and stock market performance).
  • -1 < r < 0: Varying degrees of negative correlation.
  • r = -1: Perfect negative correlation (rare).

Understanding these distinctions helps economists analyze trends, such as the impact of interest rates on investment or inflation on savings.

Question 10:
Explain the concept of correlation in statistics with suitable examples. Discuss its importance in economics and differentiate between positive and negative correlation with real-world instances.
Answer:

Correlation is a statistical measure that describes the extent to which two variables change together. It indicates the direction and strength of the relationship between variables, ranging from -1 to +1. A correlation of +1 signifies a perfect positive relationship, -1 indicates a perfect negative relationship, and 0 means no relationship.

Example: The relationship between income and expenditure is a classic example of positive correlation—as income increases, expenditure tends to rise. Conversely, the relationship between price and demand often shows negative correlation—when prices go up, demand usually falls.

Importance in Economics: Correlation helps economists analyze trends, make predictions, and formulate policies. For instance, understanding the correlation between education levels and income aids in designing better education policies to boost economic growth.

Difference Between Positive and Negative Correlation:

  • Positive Correlation: Both variables move in the same direction. Example: Higher rainfall leads to increased agricultural output.
  • Negative Correlation: Variables move in opposite directions. Example: Increased use of public transport reduces traffic congestion.

Understanding correlation is crucial for interpreting data accurately and making informed decisions in economics and beyond.

Question 11:
Explain the concept of correlation in statistics with suitable examples. Discuss its importance in economic analysis and distinguish between positive and negative correlation with real-world instances.
Answer:

Correlation is a statistical measure that describes the degree to which two variables move in relation to each other. It helps in understanding whether an increase or decrease in one variable corresponds to an increase or decrease in another. The value of correlation ranges between -1 and +1, where +1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 implies no correlation.


Example: The relationship between income and expenditure is a classic example of positive correlation. As income increases, expenditure tends to rise as well. On the other hand, the relationship between price and demand for a product often shows negative correlation—when prices go up, demand usually falls.


Importance in Economic Analysis: Correlation is crucial in economics as it helps policymakers and businesses make informed decisions. For instance, understanding the correlation between interest rates and investment can guide monetary policy. Similarly, businesses use correlation to analyze consumer behavior and market trends.


Positive vs. Negative Correlation:

  • Positive Correlation: Both variables move in the same direction. Example: Education level and income—higher education often leads to higher income.
  • Negative Correlation: Variables move in opposite directions. Example: Unemployment rate and GDP growth—when GDP grows, unemployment typically decreases.

Understanding these concepts helps in predicting trends and making data-driven decisions in economics.

Question 12:
Explain the concept of correlation in statistics with suitable examples. Discuss its importance in economics and distinguish between positive and negative correlation with real-world instances.
Answer:

Correlation refers to a statistical measure that describes the extent to which two variables change together. It indicates the direction and strength of the relationship between variables, ranging from -1 to +1. A value close to +1 signifies a strong positive correlation, while a value close to -1 indicates a strong negative correlation. A zero value means no correlation.

Example: In economics, correlation helps analyze relationships like:

  • Positive Correlation: As income increases, consumption expenditure also tends to rise (e.g., higher disposable income leads to more spending on goods).
  • Negative Correlation: As the price of a commodity increases, its demand usually decreases (e.g., expensive smartphones may see reduced sales).

Importance in Economics:

  • Helps in forecasting trends (e.g., predicting demand based on price changes).
  • Used in policy-making (e.g., analyzing the impact of interest rates on investments).
  • Identifies cause-and-effect relationships for better decision-making.

Key Differences:
1. Positive Correlation: Both variables move in the same direction (e.g., education level and income).
2. Negative Correlation: Variables move in opposite directions (e.g., unemployment and GDP growth).

Question 13:
Explain the concept of correlation in statistics with suitable examples. Discuss its importance in economics and distinguish between positive and negative correlation.
Answer:

Correlation is a statistical measure that describes the extent to which two variables change together. It indicates the direction and strength of the relationship between variables, ranging from -1 to +1. A correlation of +1 signifies a perfect positive correlation, while -1 indicates a perfect negative correlation. Zero means no correlation.

Example of Positive Correlation: As income increases, consumption expenditure also tends to increase. This is because higher income allows for greater spending.

Example of Negative Correlation: The relationship between price and demand for a product. As price rises, demand usually falls, assuming other factors remain constant.

Importance in Economics: Correlation helps economists analyze relationships between variables like GDP and unemployment, inflation and interest rates, or supply and demand. It aids in forecasting trends and formulating policies.

Difference Between Positive and Negative Correlation:

  • Positive Correlation: Both variables move in the same direction (e.g., education level and income).
  • Negative Correlation: Variables move in opposite directions (e.g., fuel prices and demand for electric vehicles).

Understanding correlation is crucial for making data-driven decisions in economics, such as predicting consumer behavior or assessing the impact of policy changes.

Question 14:
Explain the concept of correlation in statistics with the help of an example. Discuss its importance in economics and distinguish between positive and negative correlation with suitable illustrations.
Answer:

Correlation refers to a statistical measure that describes the extent to which two variables change together. It indicates the direction and strength of the relationship between variables but does not imply causation. The correlation coefficient (r) ranges from -1 to +1, where:

  • +1 indicates a perfect positive correlation.
  • -1 indicates a perfect negative correlation.
  • 0 means no correlation.

Example: In economics, the relationship between income and consumption is often positively correlated—as income rises, consumption tends to increase.

Importance in Economics: Correlation helps economists analyze trends, make predictions, and formulate policies. For instance, understanding the correlation between interest rates and investment aids in monetary policy decisions.

Positive vs. Negative Correlation:

  • Positive Correlation: Both variables move in the same direction. Example: Higher education levels often correlate with higher income.
  • Negative Correlation: Variables move in opposite directions. Example: The price of a product and its demand (law of demand).

Understanding correlation helps in identifying patterns, but remember: correlation does not imply causation. Further analysis (like regression) is needed to establish causal relationships.

Question 15:
Explain the concept of correlation in statistics with the help of an example. Discuss its importance in economics and distinguish between positive and negative correlation.
Answer:

Correlation is a statistical measure that describes the degree to which two variables move in relation to each other. It helps in understanding whether an increase or decrease in one variable corresponds to an increase or decrease in another variable. The value of correlation ranges between -1 and +1, where +1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation.


Example: Suppose we study the relationship between income and expenditure of a household. As income increases, expenditure also tends to increase, indicating a positive correlation.


Importance in Economics: Correlation is crucial in economics as it helps in:

  • Predicting trends (e.g., demand and price relationship).
  • Formulating policies (e.g., inflation and unemployment analysis).
  • Investment decisions (e.g., stock market analysis).


Positive vs. Negative Correlation:

  • Positive Correlation: Both variables move in the same direction (e.g., education level and income).
  • Negative Correlation: Variables move in opposite directions (e.g., price and demand for a product).


Understanding correlation helps economists and policymakers make informed decisions by analyzing relationships between key economic variables.

Case-based Questions (4 Marks) – with Solutions (CBSE Pattern)

These 4-mark case-based questions assess analytical skills through real-life scenarios. Answers must be based on the case study provided.

Question 1:
A study shows that as income levels rise in urban areas, the demand for organic food increases. Analyze this scenario using correlation and discuss whether it implies causation.
Answer:
Case Deconstruction

The study highlights a positive correlation between income and organic food demand. Our textbook shows correlation measures the degree of relationship between two variables.

Theoretical Application
  • Higher income may lead to greater health awareness, increasing demand.
  • However, other factors like education or marketing could also influence demand.
Critical Evaluation

Correlation does not confirm causation. For example, demand could rise due to increased availability of organic stores, unrelated to income.

Question 2:
The table below shows annual GDP growth and unemployment rates for Country X. Interpret the correlation coefficient and its economic implications.
YearGDP Growth (%)Unemployment (%)
2020-3.58.2
20216.16.0
Answer:
Case Deconstruction

The data suggests a negative correlation between GDP growth and unemployment. We studied that this aligns with Okun’s Law.

Theoretical Application
  • Higher GDP growth may create jobs, reducing unemployment.
  • However, structural factors like technology could disrupt this relationship.
Critical Evaluation

While the correlation is evident, policies like skill development are also crucial. For example, automation might weaken this relationship over time.

Question 3:
A researcher claims that social media usage and academic performance have a strong negative correlation. Evaluate this claim with two supporting examples.
Answer:
Case Deconstruction

The claim implies excessive social media use may reduce study time, affecting grades. Our textbook defines such inverse relationships.

Theoretical Application
  • Example 1: Students spending 4+ hours daily on social media scored 10% lower in exams.
  • Example 2: Schools restricting phone usage reported improved test scores.
Critical Evaluation

While data supports correlation, factors like self-discipline or family support could independently influence performance.

Question 4:
Compare the correlation between oil prices and inflation in India (import-dependent) and Saudi Arabia (oil exporter). Use critical analysis.
Answer:
Case Deconstruction

India’s inflation likely shows a positive correlation with oil prices due to import costs, while Saudi Arabia may show a negative correlation from higher oil revenues.

Theoretical Application
  • India: Rising oil prices increase transportation and production costs.
  • Saudi Arabia: Higher prices boost GDP, potentially stabilizing inflation.
Critical Evaluation

However, Saudi Arabia’s diversification efforts (e.g., Vision 2030) could weaken this correlation over time.

Question 5:
A study found that as ice cream sales increase, drowning incidents also rise. Using correlation, explain why this might not imply causation. Provide two real-world examples of spurious correlation.
Answer:
Case Deconstruction

The correlation between ice cream sales and drowning incidents is likely due to a third variable, such as hot weather, which increases both.

Theoretical Application
  • Example 1: Shoe size and reading ability in children (both increase with age).
  • Example 2: Number of firefighters at a scene and damage caused (larger fires require more firefighters).
Critical Evaluation

Our textbook shows that correlation alone cannot prove causation. We must identify confounding variables to avoid misleading conclusions.

Question 6:
The table below shows GDP growth and unemployment rates for India (2020-2023). Analyze the correlation coefficient and its economic implications.
YearGDP Growth (%)Unemployment Rate (%)
2020-6.68.0
20218.77.5
20229.17.2
Answer:
Case Deconstruction

The data shows a negative correlation between GDP growth and unemployment, supporting Okun's Law.

Theoretical Application
  • When GDP fell in 2020, unemployment rose.
  • As GDP recovered (2021-22), unemployment declined gradually.
Critical Evaluation

We studied that correlation doesn't account for structural unemployment. Other factors like technological change may influence the relationship.

Question 7:
A researcher claims social media usage and academic performance have a correlation coefficient of -0.72. Evaluate the strength/direction of this relationship and suggest two lurking variables that could affect this correlation.
Answer:
Case Deconstruction

A coefficient of -0.72 indicates a strong negative correlation: as social media use increases, grades tend to decrease.

Theoretical Application
  • Lurking variable 1: Sleep deprivation (may increase social media use and reduce focus).
  • Lurking variable 2: Parental supervision (may limit both social media and improve study habits).
Critical Evaluation

Our textbook shows correlation doesn't prove social media causes poor grades. Controlled experiments are needed for causation.

Question 8:
Compare Pearson's and Spearman's correlation methods using the example of income levels and life expectancy across 10 countries. Which method is more appropriate if the income data has outliers?
Answer:
Case Deconstruction

Pearson measures linear relationships, while Spearman assesses monotonic relationships using rank order.

Theoretical Application
  • Pearson assumes normal distribution and is sensitive to outliers.
  • Spearman is better for skewed income data as it reduces outlier impact through ranking.
Critical Evaluation

We studied that Spearman is robust for non-linear patterns. For policy analysis, Spearman may reveal broader trends despite income inequalities.

Question 9:
A study found that as ice cream sales increase, drowning incidents also rise. Using correlation, explain why this might happen and whether it implies causation.
Answer:
Case Deconstruction

We studied that correlation measures the relationship between two variables. Here, ice cream sales and drowning incidents show a positive correlation.

Theoretical Application
  • This is a spurious correlation because both variables depend on a third factor: summer heat.
  • Our textbook shows that correlation ≠ causation. Increased heat leads to more swimming (drowning) and ice cream consumption.
Critical Evaluation

Without evidence of direct causation, we cannot claim ice cream causes drowning. Examples: shoe size and math skills in children (both grow with age).

Question 10:
The table shows GDP growth (%) and unemployment rate (%) for India (2020-2023). Analyze the correlation coefficient and its economic implications.
Answer:
Case Deconstruction
YearGDP GrowthUnemployment
2020-6.68.0
20218.77.5
20226.96.8
Theoretical Application
  • We observe a negative correlation: GDP rise aligns with falling unemployment.
  • This fits Okun’s Law, which links economic growth to job creation.
Critical Evaluation

However, 2020’s anomaly (GDP decline, high unemployment) shows external factors like COVID-19 disrupt typical correlations. Example: tech sector growth may not reduce agricultural unemployment.

Question 11:
A researcher claims social media usage and academic performance have a correlation coefficient of -0.75. Interpret this and suggest lurking variables.
Answer:
Case Deconstruction

A coefficient of -0.75 indicates a strong negative correlation: higher social media use correlates with lower grades.

Theoretical Application
  • Our textbook highlights that such studies often miss lurking variables like study time or family support.
  • Example: Students with part-time jobs may use social media more and study less.
Critical Evaluation

Without controlling for these factors, the correlation may be misleading. Another example: sleep deprivation could independently affect both variables.

Question 12:
Compare correlation and regression using the example of rainfall and crop yield. Why might correlation alone be insufficient for policy decisions?
Answer:
Case Deconstruction

We studied that correlation shows the direction and strength of the relationship (e.g., more rainfall → higher yield).

Theoretical Application
  • Regression goes further by quantifying how much yield increases per cm of rain.
  • Example: A correlation of +0.8 doesn’t reveal if 10cm rain adds 100kg or 500kg/ha.
Critical Evaluation

Policymakers need regression’s predictive power to allocate irrigation funds. Correlation alone ignores soil quality or farmer skill, which regression can include as variables.

Question 13:

A farmer collected data on the amount of fertilizer used (in kg) and the corresponding yield of wheat (in quintals) over 5 seasons. The data is as follows:

  • Season 1: 10 kg, 20 quintals
  • Season 2: 15 kg, 25 quintals
  • Season 3: 20 kg, 30 quintals
  • Season 4: 25 kg, 35 quintals
  • Season 5: 30 kg, 40 quintals

Based on this data, answer the following:

a) Identify the type of correlation between fertilizer usage and wheat yield. Justify your answer.

b) If the farmer uses 35 kg of fertilizer in the next season, predict the expected wheat yield. Explain your reasoning.

Answer:

a) Type of Correlation: The data shows a positive correlation between fertilizer usage and wheat yield.
Justification: As the amount of fertilizer increases (from 10 kg to 30 kg), the wheat yield also increases (from 20 quintals to 40 quintals). This indicates a direct relationship, which is the definition of positive correlation.

b) Predicted Yield for 35 kg: The expected wheat yield would be approximately 45 quintals.
Reasoning: The data suggests a linear relationship where every 5 kg increase in fertilizer leads to a 5 quintal increase in yield. Extending this pattern:
30 kg → 40 quintals
35 kg → 45 quintals (assuming the trend continues).

Note: This is a simplified prediction. In reality, other factors like soil quality and rainfall may also affect yield.

Question 14:

A study was conducted in a school to analyze the relationship between the number of hours students spend on self-study (per week) and their scores in Economics (out of 100). The findings are summarized below:

  • Student A: 5 hours, 60 marks
  • Student B: 10 hours, 70 marks
  • Student C: 15 hours, 80 marks
  • Student D: 20 hours, 85 marks
  • Student E: 25 hours, 90 marks

Answer the following:

a) Calculate Karl Pearson's coefficient of correlation for this data. Show all steps.

b) Interpret the result obtained in part (a) in the context of the study.

Answer:

a) Calculation of Karl Pearson's Coefficient:
Step 1: Assign X (hours) and Y (marks).
Step 2: Calculate mean of X (ΣX/n) = (5+10+15+20+25)/5 = 15.
Step 3: Calculate mean of Y (ΣY/n) = (60+70+80+85+90)/5 = 77.
Step 4: Compute deviations (X - mean_X) and (Y - mean_Y) for each student.
Step 5: Calculate Σ(X - mean_X)(Y - mean_Y) = 250.
Step 6: Compute Σ(X - mean_X)² = 250 and Σ(Y - mean_Y)² = 430.
Step 7: Apply formula: r = 250 / √(250 × 430) ≈ 0.76.

b) Interpretation: The coefficient of 0.76 indicates a strong positive correlation between self-study hours and Economics scores.

This means students who spend more time on self-study tend to score higher marks. However, since the value is not exactly 1, other factors like teaching quality or prior knowledge may also influence scores.

Question 15:
A study was conducted to analyze the relationship between the number of hours spent studying and the marks obtained by students in an Economics test. The following data was collected:
Hours Studied (X): 2, 4, 6, 8, 10
Marks Obtained (Y): 30, 50, 70, 90, 110
(a) Identify the type of correlation observed in the data.
(b) Justify your answer with a suitable explanation.
Answer:

(a) The data shows a perfect positive correlation between the number of hours studied and marks obtained.

(b) Justification:
As the values of X (hours studied) increase, the values of Y (marks obtained) also increase in a constant proportion.
The ratio of change in Y to change in X is consistent (20 marks for every 2 hours).
This indicates a linear relationship with a correlation coefficient (r) of +1, confirming perfect positive correlation.

Question 16:
The table below shows the monthly income (in ₹'000) and savings (in ₹'000) of five families:
Income (X): 20, 30, 40, 50, 60
Savings (Y): 2, 3, 4, 5, 6
(a) Calculate the Karl Pearson's coefficient of correlation.
(b) Interpret the result in economic terms.
Answer:

(a) Calculation steps:
Step 1: Find mean of X (₹40,000) and Y (₹4,000)
Step 2: Calculate deviations (X - X̄) and (Y - Ȳ)
Step 3: Compute Σ(X - X̄)(Y - Ȳ) = 100
Step 4: Find Σ(X - X̄)² = 1000 and Σ(Y - Ȳ)² = 10
Step 5: Apply formula: r = 100/√(1000×10) = 1

(b) Interpretation:
The coefficient r = +1 shows perfect positive correlation between income and savings.
This implies that as family income increases, their savings increase proportionally.
In economic terms, it suggests these families maintain a constant marginal propensity to save (MPS) of 0.1 (10% of additional income is saved).

Question 17:
A researcher collected data on the monthly income (in ₹) and savings (in ₹) of 10 families in a locality. The data is as follows:

Income (₹): 25,000, 30,000, 35,000, 40,000, 45,000, 50,000, 55,000, 60,000, 65,000, 70,000
Savings (₹): 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 11,000, 12,000, 13,000, 14,000

Based on the data, answer the following:
1. Identify the type of correlation between income and savings.
2. Justify your answer with a valid reason.
Answer:

1. The type of correlation between income and savings is positive correlation.

2. The justification is as follows:
As the monthly income of the families increases, their savings also increase. This indicates a direct relationship between the two variables. For example:

  • When income is ₹25,000, savings are ₹5,000.
  • When income rises to ₹70,000, savings rise to ₹14,000.
This consistent upward trend in both variables confirms a positive correlation. In such cases, the correlation coefficient (r) would be close to +1, indicating a strong linear relationship.
Question 18:
The following table shows the hours spent studying and the corresponding marks obtained by 8 students in an Economics test:

Hours Studied: 2, 3, 4, 5, 6, 7, 8, 9
Marks Obtained (out of 50): 15, 20, 25, 30, 35, 40, 45, 50

Analyze the data and answer:
1. What is the likely correlation coefficient range for this data?
2. Explain how you arrived at this conclusion.
Answer:

1. The likely correlation coefficient range for this data is +0.9 to +1 (close to perfect positive correlation).

2. The explanation is as follows:
The data shows a clear and consistent increase in marks as study hours increase. For instance:

  • 2 hours of study yield 15 marks.
  • 9 hours of study yield 50 marks.
Since the relationship is almost perfectly linear with no deviations, the correlation is very strong and positive. The correlation coefficient measures this strength and direction, and here it would be very close to +1. This indicates that study time is an excellent predictor of marks in this case.
Question 19:
A study was conducted to analyze the relationship between the number of hours students spend studying and their scores in an Economics test. The data collected is as follows:
Hours Studied (X): 2, 4, 6, 8, 10
Test Scores (Y): 50, 60, 75, 85, 95

Based on the data, answer the following:

  • Identify the type of correlation observed between the variables.
  • Justify your answer with a brief explanation.
Answer:

The data shows a positive correlation between the number of hours studied and test scores.

Justification: As the number of hours studied (X) increases, the test scores (Y) also increase consistently.

For example:
- When X = 2, Y = 50
- When X = 10, Y = 95

This upward trend indicates a direct relationship, which is a characteristic of positive correlation. The more time students invest in studying, the higher their scores tend to be. This aligns with the concept that effort (input) and performance (output) are often positively correlated in academic settings.

Question 20:
A farmer recorded the following data about rainfall (in cm) and crop yield (in quintals) over five years:
Rainfall (X): 80, 90, 100, 110, 120
Crop Yield (Y): 20, 25, 30, 28, 26

Analyze the data and answer:

  • What type of correlation exists between rainfall and crop yield?
  • Explain the possible reason for this trend.
Answer:

The data initially shows a positive correlation but later turns into a negative correlation, indicating a non-linear or curvilinear relationship.

Justification:
- From X = 80 to X = 100, Y increases (20 to 30), showing positive correlation.
- From X = 110 onwards, Y decreases (30 to 26), showing negative correlation.

Possible Reason: Moderate rainfall is beneficial for crops, leading to higher yields (positive correlation). However, excessive rainfall (beyond 100 cm) may cause waterlogging or disease, reducing yields (negative correlation). This demonstrates the concept of optimum level in agricultural economics, where too much or too little of a factor can be harmful.

Question 21:
A study was conducted in a school to analyze the relationship between the number of hours students spent studying and their scores in the Economics exam. The data collected is as follows:

Hours Studied (X): 2, 4, 6, 8, 10
Exam Scores (Y): 50, 60, 75, 85, 95

Based on the data, answer the following:
a) Identify the type of correlation observed between the variables.
b) Justify your answer with a brief explanation.
Answer:

a) The type of correlation observed between the number of hours studied (X) and exam scores (Y) is positive correlation.

b) Justification: As the number of hours studied increases, the exam scores also increase. This indicates a direct relationship between the two variables.
For example:
When X = 2, Y = 50
When X = 10, Y = 95
This consistent upward trend confirms a positive correlation.

Additional Insight: Positive correlation implies that more study hours likely contribute to better performance, but it does not necessarily prove causation.

Question 22:
The table below shows the monthly income (in ₹) and savings (in ₹) of five families:

Monthly Income (X): 20,000, 30,000, 40,000, 50,000, 60,000
Monthly Savings (Y): 2,000, 5,000, 4,000, 6,000, 8,000

Analyze the data and answer:
a) What type of correlation exists between income and savings?
b) Explain one limitation of interpreting correlation in this context.
Answer:

a) The correlation between monthly income (X) and savings (Y) is positive but not perfect. While savings generally increase with income, the relationship is not strictly uniform (e.g., income rises from ₹30,000 to ₹40,000, but savings drop from ₹5,000 to ₹4,000).

b) Limitation: Correlation does not account for other influencing factors like family size, expenses, or financial habits. For instance, a family with higher income might have higher medical expenses, reducing savings despite the income increase.

Key Takeaway: Correlation identifies association but ignores external variables that may affect the relationship.

Chat on WhatsApp