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Correlation

NCERT Class 11 · Economics Based on NCERT Class 11 Economics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces the concept of correlation, which is a statistical tool used to examine the relationship between two variables. It exists in the syllabus to teach students how to measure the direction and intensity of a linear relationship, moving beyond simple summary measures of a single variable. The one big idea a student must walk away with is that correlation measures covariation, not causation, and that a high correlation does not imply that one variable causes the other to change.
Key Concepts (14)
Correlation
Correlation studies and measures the direction and intensity of relationship among variables.
e.g. As the summer heat rises, ice-cream sales become more brisk; temperature is related to the sale of ice-creams.
Covariation
Correlation measures covariation, not causation. It means when the value of one variable changes, the value of the other variable changes in a definite way.
e.g. The sale of ice-creams and deaths due to drowning are correlated because both are influenced by rising temperature, not because one causes the other.
Positive Correlation
The correlation is said to be positive when the variables move together in the same direction.
e.g. When income rises, consumption also rises. When income falls, consumption also falls.
Negative Correlation
The correlation is negative when the variables move in opposite directions.
e.g. When the price of apples falls, its demand increases. When the price rises, its demand decreases.
Linear Relationship
A relationship is said to be linear if it can be represented by a straight line on graph paper.
e.g. The scatter diagrams in Figures 6.1, 6.2, 6.4, and 6.5 show linear relationships.
Scatter Diagram
A scatter diagram is a useful technique for visually examining the form of relationship, without calculating any numerical value, by plotting the values of two variables as points on a graph paper.
e.g. Plotting years of schooling of farmers on X-axis and annual yield per acre on Y-axis to see the pattern of points.
Karl Pearson’s Coefficient of Correlation
Also known as product moment correlation coefficient or simple correlation coefficient, it gives a precise numerical value of the degree of linear relationship between two variables X and Y.
e.g. Calculating r between years of schooling of farmers and annual yield per acre, which was found to be 0.644.
Covariance
Covariance of X and Y is defined as Cov(X,Y) = Σ(X - X̄)(Y - Ȳ) / N, where x = X - X̄ and y = Y - Ȳ are deviations from their mean values.
e.g. In Table 6.1, Σxy = 42, so Cov(X,Y) = 42/7 = 6.
Spearman’s Rank Correlation
Spearman’s coefficient of correlation measures the linear association between ranks assigned to individual items according to their attributes, which are variables that cannot be numerically measured such as intelligence, physical appearance, honesty, etc.
e.g. Ranking five persons assessed by three judges in a beauty contest to find which pair of judges has the nearest approach to common perception of beauty.
Perfect Correlation
If all the points lie on a line, the correlation is perfect and is said to be in unity. If r = 1 or r = –1, the correlation is perfect and there is exact linear relation.
e.g. In Fig. 6.4, points lie on an upward rising line (perfect positive correlation). In Fig. 6.5, points lie on a downward falling line (perfect negative correlation).
Uncorrelated Variables
If r = 0, the two variables are uncorrelated. There is no linear relation between them, though other types of relation may exist.
e.g. In Fig. 6.3, there is no upward rising or downward sloping line around which the points are scattered, showing no correlation.
Attributes
Attributes are those variables which cannot be numerically measured such as intelligence of people, physical appearance, honesty, etc.
e.g. Ranking people according to beauty in a beauty contest.
Step Deviation Method
A method to calculate correlation coefficient by transforming variables using assumed means and common factors, based on the property that r is independent of change in origin and scale.
e.g. Transforming Price Index (X) and Money Supply (Y) using U = (X - 100)/10 and V = (Y - 1700)/100 to simplify calculation.
Correlation does not imply causation
Correlation should never be interpreted as implying cause and effect relation. The presence of correlation between two variables simply means they move together in a definite way, but one may not cause the other.
e.g. The relation between the arrival of migratory birds in a sanctuary and the birth rates in the locality cannot be given any cause and effect interpretation; it is a simple coincidence.
Formulas & Equations (8)
r = (Σxy) / (N * σx * σy)
Reads as: r equals sum of xy divided by N times sigma x times sigma y
Variables: Σxy = sum of products of deviations of X and Y from their means; N = number of observations; σx = standard deviation of X; σy = standard deviation of Y
Use when: When you have calculated deviations from mean and standard deviations, and need to find Karl Pearson's coefficient of correlation.
⚠ Forgetting to divide by N in the denominator, or using raw sums instead of deviations from mean.
r = Σ(X - X̄)(Y - Ȳ) / √[Σ(X - X̄)² * Σ(Y - Ȳ)²]
Reads as: r equals sum of (X minus X bar) times (Y minus Y bar) divided by the square root of [sum of (X minus X bar) squared times sum of (Y minus Y bar) squared]
Variables: X, Y = individual values; X̄ = mean of X; Ȳ = mean of Y
Use when: When you have raw data and need to calculate r directly using deviations from mean.
⚠ Forgetting to take the square root of the denominator, or incorrectly calculating the deviations.
r = [NΣXY - (ΣX)(ΣY)] / √[NΣX² - (ΣX)²] * √[NΣY² - (ΣY)²]
Reads as: r equals N times sum of XY minus sum of X times sum of Y, divided by the square root of [N times sum of X squared minus (sum of X) squared] times the square root of [N times sum of Y squared minus (sum of Y) squared]
Variables: N = number of observations; ΣXY = sum of products of X and Y; ΣX = sum of X; ΣY = sum of Y; ΣX² = sum of squares of X; ΣY² = sum of squares of Y
Use when: When you have raw data and want to avoid calculating deviations from mean; this is the most commonly used formula in board exams.
⚠ Mixing up the terms in the denominator, e.g., writing √[NΣX² - (ΣX)²] * √[NΣY² - (ΣY)²] incorrectly as √[NΣX² - (ΣX)² * NΣY² - (ΣY)²].
r = [ΣXY - (ΣX)(ΣY)/N] / √[ΣX² - (ΣX)²/N] * √[ΣY² - (ΣY)²/N]
Reads as: r equals sum of XY minus (sum of X times sum of Y divided by N), divided by the square root of [sum of X squared minus (sum of X) squared divided by N] times the square root of [sum of Y squared minus (sum of Y) squared divided by N]
Variables: Same as above, but expressed in a different algebraic form.
Use when: When you have raw data and prefer to use the formula that directly uses sums without the N factor outside the square roots.
⚠ Forgetting to divide (ΣX)(ΣY) by N in the numerator.
r_s = 1 - (6ΣD²) / (n³ - n)
Reads as: r subscript s equals 1 minus 6 times sum of D squared divided by (n cubed minus n)
Variables: D = deviation of ranks assigned to a variable from those assigned to the other variable; n = number of observations
Use when: When calculating Spearman's rank correlation coefficient for data where ranks are given or can be assigned, and there are no repeated ranks.
⚠ Forgetting to square the deviations D, or using n³ - n incorrectly as n³ - 1.
r_s = 1 - [6(ΣD² + (m₁³ - m₁)/12 + (m₂³ - m₂)/12 + ...)] / (n³ - n)
Reads as: r subscript s equals 1 minus 6 times [sum of D squared plus correction factors] divided by (n cubed minus n)
Variables: D = deviation of ranks; n = number of observations; m₁, m₂, ... = number of repetitions of ranks; (m³ - m)/12 = correction factor for each repeated rank
Use when: When calculating Spearman's rank correlation coefficient and there are repeated ranks in the data.
⚠ Forgetting to add the correction factors when ranks are repeated, or incorrectly calculating the correction factor.
σx² = Σ(X - X̄)² / N = Σx² / N
Reads as: sigma x squared equals sum of (X minus X bar) squared divided by N equals sum of x squared divided by N
Variables: σx² = variance of X; X = individual values; X̄ = mean of X; N = number of observations; x = deviation from mean
Use when: When calculating variance of a variable as a step toward finding standard deviation for correlation formula.
⚠ Forgetting to divide by N, or using N-1 instead of N (since this is population variance, not sample).
Cov(X,Y) = Σ(X - X̄)(Y - Ȳ) / N = Σxy / N
Reads as: Covariance of X and Y equals sum of (X minus X bar) times (Y minus Y bar) divided by N equals sum of xy divided by N
Variables: X, Y = individual values; X̄ = mean of X; Ȳ = mean of Y; N = number of observations; x, y = deviations from respective means
Use when: When calculating covariance as a step toward finding Karl Pearson's coefficient of correlation.
⚠ Forgetting to divide by N, or confusing covariance with the numerator of the correlation formula.
How It Works
Calculating Karl Pearson's Coefficient of Correlation using Formula 1 (Deviation Method)
  1. Step 1: Calculate the arithmetic means of X and Y: X̄ = ΣX/N, Ȳ = ΣY/N.
  2. Step 2: Calculate deviations from mean: x = X - X̄, y = Y - Ȳ.
  3. Step 3: Calculate Σxy, Σx², and Σy².
  4. Step 4: Calculate standard deviations: σx = √(Σx²/N), σy = √(Σy²/N).
  5. Step 5: Apply formula: r = (Σxy) / (N * σx * σy).
💡 The sign of covariance determines the sign of r; standard deviations are always positive, so r takes the sign of Σxy.
Calculating Karl Pearson's Coefficient of Correlation using Formula 4 (Direct Method)
  1. Step 1: Calculate ΣX, ΣY, ΣXY, ΣX², ΣY² from the raw data.
  2. Step 2: Apply formula: r = [NΣXY - (ΣX)(ΣY)] / √[NΣX² - (ΣX)²] * √[NΣY² - (ΣY)²].
  3. Step 3: Simplify the numerator and denominator separately.
  4. Step 4: Compute the final value of r.
💡 This formula avoids calculating deviations from mean and is the most efficient for raw data. Ensure you do not mix up the terms in the denominator.
Calculating Spearman's Rank Correlation (Ranks Given)
  1. Step 1: List the ranks assigned to each observation by the two variables.
  2. Step 2: Calculate the difference D between the two ranks for each observation.
  3. Step 3: Square each D to get D².
  4. Step 4: Sum all D² to get ΣD².
  5. Step 5: Apply formula: r_s = 1 - (6ΣD²) / (n³ - n).
💡 The sum of D should always be zero; if it is not, check your calculations.
Calculating Spearman's Rank Correlation (Ranks Not Given)
  1. Step 1: Arrange the data for each variable in descending or ascending order.
  2. Step 2: Assign rank 1 to the highest value, rank 2 to the next, and so on.
  3. Step 3: If there are repeated values, assign the average rank to each.
  4. Step 4: Proceed with steps 2-5 of the 'Ranks Given' process above.
  5. Step 5: If ranks are repeated, add correction factors: (m³ - m)/12 for each set of repeated ranks.
💡 When ranks are repeated, you must use the corrected formula; otherwise, the value of r_s will be incorrect.
Using Scatter Diagram to Determine Nature of Relationship
  1. Step 1: Plot the values of the two variables as points on a graph paper, with X on the horizontal axis and Y on the vertical axis.
  2. Step 2: Observe the overall pattern of the scatter points.
  3. Step 3: If points cluster around an upward rising line, it indicates positive correlation.
  4. Step 4: If points cluster around a downward sloping line, it indicates negative correlation.
  5. Step 5: If points are widely scattered with no clear line, it indicates no correlation.
  6. Step 6: If all points lie exactly on a line, the correlation is perfect.
💡 A scatter diagram only gives a visual idea; it does not provide a numerical value. It is advisable to first examine the scatter diagram before calculating Karl Pearson's coefficient to check if the relationship is linear.
Step Deviation Method for Calculating Correlation
  1. Step 1: Choose assumed means A for X and C for Y.
  2. Step 2: Choose common factors B for X and D for Y (same sign).
  3. Step 3: Transform variables: U = (X - A)/B, V = (Y - C)/D.
  4. Step 4: Calculate ΣU, ΣV, ΣU², ΣV², ΣUV.
  5. Step 5: Apply formula (3) using U and V values to get r_UV.
  6. Step 6: Since r_XY = r_UV, the result is the required correlation coefficient.
💡 The property that r is unaffected by change of origin and scale is used here. B and D must have the same sign.
Exam Traps — Don't Make These Mistakes
⚠ Confusing correlation with causation
✗ Since r = 0.9 between ice-cream sales and drowning deaths, eating ice-cream causes drowning.
→ Correlation measures covariation, not causation. A third variable (temperature) influences both, creating a spurious correlation.
✓ Correlation does not imply causation. The high correlation between ice-cream sales and drowning deaths is due to the common cause of rising temperature, which increases both ice-cream consumption and swimming activity.
⚠ Using Karl Pearson's coefficient for non-linear relationships
✗ Calculating r for data that follows a U-shaped curve and interpreting r = 0 as 'no relationship'.
→ Karl Pearson's coefficient measures only linear relationships. If the true relationship is non-linear (as in Figures 6.6 and 6.7), r may be close to zero even though a strong non-linear relationship exists.
✓ First examine the scatter diagram. If the relationship is non-linear, do not use Karl Pearson's coefficient. Use Spearman's rank correlation or describe the relationship visually.
⚠ Forgetting correction factors in Spearman's rank correlation with repeated ranks
✗ Using the formula r_s = 1 - (6ΣD²)/(n³ - n) even when ranks are repeated.
→ When ranks are repeated, the simple formula gives an incorrect value because it does not account for the ties. The corrected formula with (m³ - m)/12 must be used.
✓ When ranks are repeated, use the corrected formula: r_s = 1 - [6(ΣD² + (m₁³ - m₁)/12 + (m₂³ - m₂)/12 + ...)] / (n³ - n).
⚠ Misinterpreting r = 0 as independence
✗ Since r = 0, the two variables are independent.
→ r = 0 means there is no linear relationship, but there could be a non-linear relationship. Independence is a stronger condition that implies no relationship of any kind.
✓ r = 0 indicates that the two variables are uncorrelated, meaning there is no linear relationship between them. However, they may still have a non-linear relationship.
⚠ Ignoring the sign of covariance when interpreting r
✗ r = -0.8 means a weak negative relationship because the value is close to -1.
→ The magnitude of r indicates strength, not the sign. r = -0.8 indicates a strong negative relationship because it is close to -1 in absolute value.
✓ r = -0.8 indicates a strong negative linear relationship, meaning as one variable increases, the other decreases significantly.
⚠ Assuming r outside -1 to 1 is possible
✗ r = 1.2 indicates a very strong positive correlation.
→ The value of the correlation coefficient always lies between -1 and +1. A value outside this range indicates an error in calculation.
✓ The value of r must lie between -1 and +1. If r = 1.2, there is an error in calculation. Recheck the sums and formula application.
Compare & Contrast
Type of data used
Karl Pearson's Coefficient of Correlation
Spearman's Rank Correlation
Karl Pearson's coefficient uses actual numerical values of variables, while Spearman's rank correlation uses ranks assigned to the data.
Direction of movement of variables
Positive Correlation
Negative Correlation
In positive correlation, variables move in the same direction (both increase or both decrease); in negative correlation, variables move in opposite directions (one increases while the other decreases).
Nature of output
Scatter Diagram
Karl Pearson's Coefficient
A scatter diagram gives a visual presentation of the relationship without any numerical value, while Karl Pearson's coefficient gives a precise numerical measure of the degree and direction of linear relationship.
Interpretation of relationship
Correlation
Causation
Correlation measures the direction and intensity of covariation between two variables, but it does not imply that one variable causes the other to change. Causation requires additional evidence beyond correlation.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 Short Answer (3 marks) on properties or interpretation, 1 Long Answer (5 marks) on calculation of r or rank correlation
Most asked: Calculation of Karl Pearson's coefficient of correlation using the direct formula · Interpretation of the value of r (positive, negative, zero, perfect correlation) · Spearman's rank correlation with and without repeated ranks · Properties of correlation coefficient (range, unit, change of origin and scale) · Difference between correlation and causation
Don't skip: Long numerical problem to calculate Karl Pearson's coefficient of correlation using the direct formula (5 marks); Long numerical problem to calculate Spearman's rank correlation with repeated ranks (5 marks)
One-Line Revision (read aloud before the exam)
  1. Correlation studies and measures the direction and intensity of relationship among variables.
  2. Correlation measures covariation, not causation.
  3. Positive correlation means variables move in the same direction; negative correlation means they move in opposite directions.
  4. A scatter diagram gives a visual presentation of the relationship without any numerical value.
  5. Karl Pearson's coefficient of correlation gives a precise numerical value of the degree of linear relationship between two variables.
  6. The value of the correlation coefficient lies between minus one and plus one, –1 ≤ r ≤ 1.
  7. If r = 1 or r = –1, the correlation is perfect and there is exact linear relation.
  8. If r = 0, the two variables are uncorrelated; there is no linear relation between them.
  9. r has no unit; it is a pure number.
  10. The magnitude of r is unaffected by the change of origin and change of scale.
  11. Spearman's rank correlation measures the linear association between ranks assigned to individual items according to their attributes.
  12. Spearman's rank correlation is not affected by extreme values.
  13. When ranks are repeated, correction factors (m³ - m)/12 must be added to ΣD².
  14. Correlation does not imply causation; it only means covariation.
  15. A high value of r (close to +1 or –1) indicates a strong linear relationship.
  16. A low value of r (close to zero) indicates a weak linear relation, but there may be a non-linear relation.
  17. Karl Pearson's coefficient should be used only when there is a linear relation between the variables.
  18. It is advisable to first examine the scatter diagram before calculating the Karl Pearson's correlation coefficient.
  19. The sign of covariance between X and Y determines the sign of the correlation coefficient.
  20. If the covariance is zero, the correlation coefficient is always zero.

MCQs — 10 Questions with Answers

Q1. Which of the following is the PRIMARY advantage of using a scatter diagram to study correlation?

  • A. It provides an exact numerical value of the correlation coefficient
  • B. It visually displays the direction and intensity of relationship without calculation ✓
  • C. It can only be used when there is perfect correlation
  • D. It eliminates the need for Karl Pearson's coefficient entirely

Answer: B — A scatter diagram's key advantage is visual inspection of the pattern and closeness of points, showing direction and strength without numerical computation.

Q2. Which statement correctly distinguishes between positive and negative correlation?

  • A. Positive correlation occurs when both variables increase; negative correlation occurs when both decrease
  • B. Positive correlation means variables move in the same direction; negative correlation means they move in opposite directions ✓
  • C. Positive correlation is stronger than negative correlation
  • D. Negative correlation implies causation while positive correlation does not

Answer: B — Positive correlation means when X increases Y increases (or both decrease); negative correlation means when X increases Y decreases and vice versa.

Q3. A scatter diagram shows points scattered randomly with no clear pattern. This indicates:

  • A. Perfect positive correlation
  • B. Perfect negative correlation
  • C. No correlation or very weak correlation ✓
  • D. Strong negative correlation

Answer: C — Random scatter with no upward or downward trend indicates the variables have no consistent linear relationship or extremely weak correlation.

Q4. In a mandi, as the supply of tomatoes increases dramatically during harvest season, the price drops from Rs 40/kg to Rs 4/kg. What type of correlation exists between supply and price?

  • A. Positive correlation
  • B. Negative correlation ✓
  • C. Zero correlation
  • D. Perfect positive correlation

Answer: B — As supply increases (↑), price decreases (↓), showing variables move in opposite directions—this is negative correlation.

Q5. Which of the following is NOT a valid reason to reject the claim that correlation implies causation?

  • A. A third variable (like temperature) might cause both ice-cream sales and drowning deaths to increase
  • B. The correlation might be pure coincidence (e.g., migratory birds and birth rates)
  • C. The direction of causation could be reversed
  • D. One variable always has a higher numerical value than the other ✓

Answer: D — The magnitude of variable values has no bearing on whether correlation implies causation; lurking variables, coincidence, and reversed causation are valid reasons to reject causal claims.

Q6. Karl Pearson's coefficient of correlation should be applied ONLY when:

  • A. The relationship between variables is non-linear
  • B. The relationship between variables is linear ✓
  • C. Both variables are qualitative (non-numerical)
  • D. The sample size is very large (n > 100)

Answer: B — Karl Pearson's correlation coefficient measures linear relationships; applying it to non-linear data can give misleading results.

Q7. Which correlation measurement technique is best suited for analyzing the relationship between students' physical appearance and their academic performance?

  • A. Karl Pearson's coefficient of correlation
  • B. Scatter diagram with raw scores
  • C. Spearman's rank correlation ✓
  • D. Simple numerical mean of both variables

Answer: C — Physical appearance is a non-numerical attribute that must be ranked; Spearman's rank correlation is designed for ranking such qualitative variables.

Q8. Which of the following statements about perfect correlation is CORRECT? (A) Perfect positive correlation means r = +1 and all points lie on an upward-sloping line. (B) Perfect negative correlation means r = −1 and all points lie on a downward-sloping line. Choose the correct option:

  • A. Only (A) is correct
  • B. Only (B) is correct
  • C. Both (A) and (B) are correct ✓
  • D. Neither (A) nor (B) is correct

Answer: C — Both statements are accurate: r = +1 indicates perfect positive correlation (upward line) and r = −1 indicates perfect negative correlation (downward line).

Q9. Study the following scenario: A researcher finds a strong positive correlation (r = 0.92) between the number of firefighters at a fire scene and the total fire damage caused. Which is the MOST appropriate conclusion?

  • A. More firefighters cause greater fire damage
  • B. More fire damage causes more firefighters to be called to the scene
  • C. A third variable (size/severity of fire) causes both more firefighters and more damage ✓
  • D. There is no relationship between firefighters and damage; the correlation is meaningless

Answer: C — This is a classic lurking variable example: larger fires cause both more damage and attract more firefighters; correlation does not imply the firefighters cause damage.

Q10. In India's agricultural sector, a government economist observes that as rainfall increases, agricultural productivity increases (positive correlation r = +0.78). However, an economist argues that correlation here is misleading because rainfall might not be the true cause. Which factor could the second economist be hinting at?

  • A. The sample size is too small
  • B. Soil fertility, irrigation availability, or seed quality might be the actual drivers of productivity independent of rainfall ✓
  • C. Correlation values above 0.7 are always caused by one variable acting on the other
  • D. Productivity always increases linearly with rainfall in all regions

Answer: B — The economist is identifying lurking variables (soil fertility, irrigation, seeds) that could influence productivity independent of rainfall, showing why high correlation doesn't prove causation.

Flashcards

What does correlation measure?

Correlation measures the direction and intensity (strength) of the linear relationship between two variables, not causation.

Define positive correlation with an example.

When both variables move in the same direction (both increase or both decrease), like income and consumption rising together.

Define negative correlation with an example.

When variables move in opposite directions—as one increases, the other decreases, like price of apples and quantity demanded.

Why is the ice-cream and drowning death relationship NOT causal?

Both increase due to a third variable (rising temperature), not because ice-cream causes drowning; this is called a lurking variable.

What does a scatter diagram show about correlation?

It visually displays the direction and strength of a relationship: points on a line indicate strong/perfect correlation, scattered points indicate weak or no correlation.

What is perfect positive correlation?

All data points lie exactly on an upward-sloping straight line, showing a complete one-to-one positive relationship.

What is perfect negative correlation?

All data points lie exactly on a downward-sloping straight line, showing a complete one-to-one inverse relationship.

When should Karl Pearson's correlation coefficient be used?

Only when the relationship between two variables is linear (can be represented by a straight line).

What type of variables does Spearman's rank correlation measure?

It measures correlation between ranks of non-numerical attributes like intelligence, honesty, or physical appearance.

Why is correlation different from causation?

Correlation shows that two variables move together, but it does not prove one causes the other; there may be a third variable or pure coincidence.

Important Board Questions

Define correlation and explain why correlation cannot be interpreted as causation with one example from the study material. [2 marks]

State the definition of correlation (covariation, not causation). Use the ice-cream/drowning example or migratory birds/birth rates to show how a third variable or coincidence breaks the causal chain.

Explain the difference between positive and negative correlation. Draw or describe what a scatter diagram would look like for each type. Provide one economic example of each type of correlation. [5 marks]

Positive: both variables move in same direction (describe upward-sloping pattern on scatter). Negative: opposite directions (downward slope). Example for positive—income and consumption. Example for negative—price and demand. Show visual pattern clearly.

The following scatter plot shows the relationship between study hours (X-axis) and examination scores (Y-axis) for Class 11 students. The points form a clear upward-sloping line with one outlier. (a) Identify the type of correlation shown. (b) Explain why this correlation might be misleading or incomplete as evidence that more study hours CAUSE higher scores. (c) What third variable might actually be driving both study hours and examination scores? [6 marks]

Part (a): Positive correlation. Part (b): Correlation ≠ causation; lurking variables may exist. Part (c): Intelligence, aptitude, or prior subject knowledge could drive both effort and performance independently. Discuss the concept of covariation versus causation and why scatter diagrams alone cannot prove cause-effect.

Assertion-Reason Questions for Correlation

Q1. Assertion (A):

Correlation measures the relationship between two variables in terms of direction and intensity.

Reason (R):

Correlation coefficients quantify how one variable changes relative to another variable in a definite mathematical way.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both assertion and reason are true. The reason directly explains why correlation measures direction (positive/negative) and intensity (strength from 0 to ±1). Correlation quantifies covariation through standardised coefficients, validating the assertion.

Q2. Assertion (A):

Scatter diagrams provide a visual method to examine correlation without calculating numerical values.

Reason (R):

Degree of closeness and overall direction of scatter points enable examination of the nature and intensity of relationship.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both assertion and reason are true, but the reason does NOT explain why scatter diagrams avoid calculation—it explains what scatter diagrams show. The reason describes the visual interpretation method, not the distinction of being non-numerical. The assertion stands on its own merit without this reason being its causal explanation.

Q3. Assertion (A):

Karl Pearson's coefficient of correlation is more suitable than Spearman's rank correlation for measuring variables like intelligence and honesty.

Reason (R):

Intelligence and honesty are attributes that cannot be numerically measured, requiring ranking methods instead.

Answer: C — A is true but R is false — The assertion is true—Spearman's rank correlation, not Karl Pearson's, should be used for non-numerical attributes. The reason is false; it correctly identifies that these attributes cannot be measured numerically and thus require Spearman's method. The assertion contradicts the correct method, making it false.

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Case-Based Questions (CBQ) for Correlation

Case Study:

A market research team in Mumbai collected data on ice-cream sales (in litres per day) and daily maximum temperature (in °C) over 30 days in summer. They created a scatter diagram and calculated Karl Pearson's correlation coefficient as r = +0.88. The team also noted that on the hottest day (42°C), ice-cream sales were 450 litres, and on the coolest day (28°C), sales were 120 litres. The manager suggested that heat waves cause people to crave ice-cream, so the company should invest heavily in cold storage to boost sales during summer.

Q1. What does the correlation coefficient r = +0.88 indicate about the relationship between temperature and ice-cream sales?

  • A. Strong positive linear relationship; as temperature increases, sales tend to increase ✓
  • B. Weak positive relationship; temperature has minimal impact on sales
  • C. Strong negative relationship; high temperature reduces sales
  • D. Perfect relationship; temperature is the only factor determining sales

Answer: A

Q2. The manager's conclusion that 'heat waves cause people to crave ice-cream, so invest in cold storage to boost sales' demonstrates which misunderstanding?

  • A. Ignoring that correlation does not prove causation ✓
  • B. Overestimating the weakness of the correlation
  • C. Assuming scatter diagrams always show perfect relationships
  • D. Confusing negative correlation with positive correlation

Answer: A

Q3. Which factor could explain why the temperature-sales correlation might be weaker in winter months when temperature data is collected?

  • A. Cold storage capacity decreases in winter
  • B. Consumer preference shifts away from ice-cream in cold months, weakening the temperature-sales relationship ✓
  • C. Karl Pearson's method becomes invalid below 30°C
  • D. Scatter diagrams cannot be drawn during winter

Answer: B

Q4. Besides temperature, which third variable might explain the strong correlation between heat and ice-cream sales in Mumbai during summer?

  • A. The number of rainy days in July
  • B. School holidays coinciding with peak summer heat, increasing foot traffic to markets and shops ✓
  • C. The price of sugar used in ice-cream production
  • D. The opening of new ice-cream shops in Mumbai

Answer: B

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Every formula from the Statistics and Microeconomics chapters on one page, checked against the NCERT textbook and the CBSE 2026-27 syllabus. Free PDF download.

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