Class 11 Economics Chapter 7 Question Answer | Correlation | English Medium | ASSEB
Welcome to HSLC Guru. In this lesson we explore Chapter 7 — Correlation from the ASSEB Class 11 Statistics for Economics syllabus. Correlation is one of the most important statistical tools used by economists to study relationships between variables such as price and demand, income and consumption, or rainfall and yield. This chapter introduces the meaning of correlation, its types, methods of measurement (scatter diagram, Karl Pearson’s coefficient, and Spearman’s rank correlation), the properties of the correlation coefficient, and the difference between correlation and causation. Complete textbook questions, solved numerical problems, MCQs, fill in the blanks, true/false, glossary, and a formula table are provided.
Summary of the Chapter
Meaning of Correlation: Correlation is a statistical technique that studies the degree and direction of relationship between two or more variables. When changes in one variable are accompanied by changes in another variable, the two variables are said to be correlated. For example, as price falls, demand rises; as income increases, savings tend to increase. Correlation does not establish a cause-and-effect relationship; it only measures the extent to which two variables move together.
Types of Correlation: (i) Positive and Negative — when both variables move in the same direction it is positive correlation (e.g., income and consumption); when they move in opposite directions it is negative correlation (e.g., price and demand). (ii) Linear and Non-linear — when the ratio of change between the two variables is constant, the correlation is linear; otherwise it is non-linear (curvilinear). (iii) Simple, Multiple and Partial — simple correlation involves only two variables; multiple correlation involves three or more variables studied together; partial correlation studies the relationship between two variables while keeping other variables constant.
Methods of Measurement: The three principal methods used to measure correlation are: (1) Scatter Diagram — a graphical method where pairs of values of X and Y are plotted as points; the closeness and direction of the dots indicate the type and degree of correlation. (2) Karl Pearson’s Coefficient of Correlation (r) — a mathematical method that gives a precise numerical value of correlation, computed as r = Σxy / √(Σx² × Σy²) where x and y are deviations from their respective means. (3) Spearman’s Rank Correlation Coefficient (rk) — used when data are qualitative or arranged in ranks, computed as rk = 1 − [6ΣD² / N(N²−1)] where D is the difference between the ranks of paired observations.
Properties and Interpretation: The value of Karl Pearson’s coefficient r always lies between −1 and +1, that is −1 ≤ r ≤ +1. r = +1 indicates perfect positive correlation, r = −1 indicates perfect negative correlation, and r = 0 indicates no linear correlation. r is independent of change of origin and scale, and it is a pure number with no unit. Correlation vs Causation: Correlation between two variables does not mean that one causes the other. Two variables may be correlated due to (i) pure chance, (ii) influence of a common third factor, or (iii) mutual dependence. Hence, statistical correlation should be interpreted carefully in the light of economic theory.
Textbook Questions and Answers
A. Very Short Answer Type Questions (1 Mark)
Q1. What is correlation?
Answer: Correlation is a statistical measure that expresses the degree and direction of relationship between two or more variables.
Q2. Give one example of positive correlation.
Answer: The relationship between income and consumption expenditure is an example of positive correlation, because both rise together.
Q3. Give one example of negative correlation.
Answer: The relationship between price and demand of a normal good shows negative correlation, since demand falls when price rises.
Q4. What is the range of Karl Pearson’s coefficient of correlation?
Answer: The value of r lies between −1 and +1, that is −1 ≤ r ≤ +1.
Q5. What does r = 0 mean?
Answer: r = 0 indicates the absence of any linear correlation between the two variables.
Q6. Who developed the rank correlation method?
Answer: The rank correlation method was developed by the British psychologist Charles Edward Spearman in 1904.
Q7. What is a scatter diagram?
Answer: A scatter diagram is a graphical method of showing correlation in which pairs of values of two variables are plotted as dots on a graph paper.
Q8. Does correlation imply causation?
Answer: No. Correlation only measures association between variables; it does not necessarily mean that one variable causes the change in the other.
Q9. What is perfect positive correlation?
Answer: When two variables move in the same direction in the same proportion, the correlation between them is perfect positive and r = +1.
Q10. Name one merit of the scatter diagram method.
Answer: It is a simple, easy and non-mathematical method that gives a quick visual idea of the nature and direction of correlation.
B. Short Answer Type Questions (2-3 Marks)
Q1. Distinguish between positive and negative correlation.
Answer: When two variables move in the same direction — both increasing together or both decreasing together — the correlation is called positive. Examples include income and consumption, or height and weight. When the variables move in opposite directions — one rising while the other falls — the correlation is called negative. Examples include price and demand of a normal commodity, or supply of a good and its price under restrictive conditions.
Q2. Distinguish between linear and non-linear correlation.
Answer: Correlation is said to be linear when the amount of change in one variable bears a constant ratio to the change in the other variable, so that the plotted points lie close to a straight line. Correlation is non-linear (curvilinear) when the ratio of change is not constant, and the plotted points lie on a curve rather than a straight line.
Q3. What are the merits of Karl Pearson’s coefficient of correlation?
Answer: The merits of Karl Pearson’s r are: (i) it gives a precise numerical measure of the degree of correlation, (ii) it indicates both the direction (sign) and the magnitude of the relationship, (iii) it is independent of the change of origin and scale, and (iv) it provides a basis for further statistical analysis such as regression.
Q4. State any three properties of the correlation coefficient r.
Answer: (i) The value of r always lies between −1 and +1. (ii) r is a pure number, independent of the units of measurement of the variables. (iii) r is independent of the change of origin and scale; that is, adding or subtracting a constant from each value, or multiplying/dividing by a constant, does not change the value of r.
Q5. Distinguish between simple, multiple and partial correlation.
Answer: Simple correlation studies the relationship between two variables only, e.g., between price and demand. Multiple correlation studies the joint relationship of one variable with two or more other variables together, e.g., yield of rice with rainfall and use of fertilizer. Partial correlation studies the relationship between two variables while keeping the effect of other variables constant.
Q6. Why is correlation not the same as causation?
Answer: Correlation only measures the strength of association between two variables. Two variables may be correlated because (i) one influences the other, (ii) they are both influenced by a common third factor, or (iii) the relationship is purely accidental. Therefore, a high correlation between two series does not by itself prove a cause-and-effect relationship.
C. Long Answer Type Questions (5-6 Marks)
Q1. Explain the different types of correlation with examples.
Answer: Correlation can be classified on three different bases:
(i) On the basis of direction — Positive and Negative correlation: Correlation is positive when both variables move in the same direction. For example, an increase in income generally leads to an increase in consumption. Correlation is negative when the two variables move in opposite directions, e.g., as the price of a normal commodity rises, its demand falls.
(ii) On the basis of ratio of change — Linear and Non-linear correlation: Correlation is linear when the ratio of change between the two variables is constant, so that the points on a scatter diagram lie close to a straight line. It is non-linear when the ratio of change is not constant, and the points lie along a curve.
(iii) On the basis of number of variables — Simple, Multiple and Partial correlation: Simple correlation involves two variables only. Multiple correlation involves three or more variables studied together. Partial correlation measures the relationship between two variables, treating others as constant. Together, these classifications help economists choose the right statistical tool for analysis.
Q2. Calculate Karl Pearson’s coefficient of correlation between X and Y from the following data:
| X | 2 | 4 | 6 | 8 | 10 |
|---|---|---|---|---|---|
| Y | 4 | 6 | 8 | 10 | 12 |
Answer: Mean of X = (2+4+6+8+10)/5 = 30/5 = 6. Mean of Y = (4+6+8+10+12)/5 = 40/5 = 8.
| X | Y | x = X−6 | y = Y−8 | x² | y² | xy |
|---|---|---|---|---|---|---|
| 2 | 4 | −4 | −4 | 16 | 16 | 16 |
| 4 | 6 | −2 | −2 | 4 | 4 | 4 |
| 6 | 8 | 0 | 0 | 0 | 0 | 0 |
| 8 | 10 | 2 | 2 | 4 | 4 | 4 |
| 10 | 12 | 4 | 4 | 16 | 16 | 16 |
| Total | 0 | 0 | 40 | 40 | 40 |
Applying the formula r = Σxy / √(Σx² × Σy²) = 40 / √(40 × 40) = 40 / 40 = +1. Hence the two series have a perfect positive correlation.
Q3. Calculate Karl Pearson’s coefficient of correlation between price (X) and demand (Y):
| X (Price) | 10 | 20 | 30 | 40 | 50 |
|---|---|---|---|---|---|
| Y (Demand) | 50 | 40 | 30 | 20 | 10 |
Answer: Mean of X = 150/5 = 30. Mean of Y = 150/5 = 30.
| X | Y | x = X−30 | y = Y−30 | x² | y² | xy |
|---|---|---|---|---|---|---|
| 10 | 50 | −20 | 20 | 400 | 400 | −400 |
| 20 | 40 | −10 | 10 | 100 | 100 | −100 |
| 30 | 30 | 0 | 0 | 0 | 0 | 0 |
| 40 | 20 | 10 | −10 | 100 | 100 | −100 |
| 50 | 10 | 20 | −20 | 400 | 400 | −400 |
| Total | 0 | 0 | 1000 | 1000 | −1000 |
r = Σxy / √(Σx² × Σy²) = −1000 / √(1000 × 1000) = −1000 / 1000 = −1. Hence price and demand show perfect negative correlation.
Q4. Calculate Spearman’s rank correlation coefficient from the following marks of 5 students in Statistics (X) and Economics (Y):
| Student | A | B | C | D | E |
|---|---|---|---|---|---|
| X (Statistics) | 85 | 60 | 73 | 40 | 90 |
| Y (Economics) | 78 | 65 | 80 | 50 | 92 |
Answer: Assigning ranks (1 = highest):
| Student | X | Rank R₁ | Y | Rank R₂ | D = R₁−R₂ | D² |
|---|---|---|---|---|---|---|
| A | 85 | 2 | 78 | 3 | −1 | 1 |
| B | 60 | 4 | 65 | 4 | 0 | 0 |
| C | 73 | 3 | 80 | 2 | 1 | 1 |
| D | 40 | 5 | 50 | 5 | 0 | 0 |
| E | 90 | 1 | 92 | 1 | 0 | 0 |
| Total | 0 | 2 |
N = 5, ΣD² = 2. Applying Spearman’s formula rk = 1 − [6ΣD² / N(N²−1)] = 1 − [6 × 2 / 5 × (25−1)] = 1 − [12 / 120] = 1 − 0.1 = 0.9. There is a high degree of positive rank correlation between marks in Statistics and Economics.
Q5. Explain the properties of Karl Pearson’s coefficient of correlation.
Answer: The important properties of Karl Pearson’s coefficient of correlation r are:
(i) Limits: r always lies between −1 and +1, i.e., −1 ≤ r ≤ +1. r = +1 means perfect positive correlation, r = −1 means perfect negative correlation, and r = 0 means no linear correlation.
(ii) Pure number: r has no unit; it is a pure number that can be compared across different sets of data measured in different units.
(iii) Independence of origin and scale: r is unaffected by changing the origin (adding or subtracting a constant) or scale (multiplying or dividing by a constant) of either variable.
(iv) Symmetry: The correlation between X and Y is the same as the correlation between Y and X, i.e., rxy = ryx.
(v) Geometric mean of regression coefficients: r is the geometric mean of the two regression coefficients bxy and byx, i.e., r = √(bxy × byx).
Additional Multiple Choice Questions (MCQs)
Q1. The value of correlation coefficient r lies between:
(a) 0 and 1 (b) −1 and 0 (c) −1 and +1 (d) −∞ and +∞
Answer: (c) −1 and +1.
Q2. When two variables move in the same direction, the correlation is called:
(a) Negative (b) Positive (c) Zero (d) Non-linear
Answer: (b) Positive.
Q3. Karl Pearson’s coefficient of correlation is also known as:
(a) Rank correlation (b) Product moment correlation (c) Concurrent deviation method (d) None of these
Answer: (b) Product moment correlation.
Q4. Spearman’s rank correlation coefficient was developed by:
(a) Karl Pearson (b) R. A. Fisher (c) C. E. Spearman (d) Bowley
Answer: (c) C. E. Spearman.
Q5. If r = +1, the correlation is:
(a) Perfect negative (b) Perfect positive (c) Zero (d) Imperfect
Answer: (b) Perfect positive.
Q6. The correlation between price and demand is generally:
(a) Positive (b) Zero (c) Negative (d) Perfect
Answer: (c) Negative.
Q7. Which method gives a quick visual idea of correlation?
(a) Karl Pearson method (b) Spearman method (c) Scatter diagram (d) None of these
Answer: (c) Scatter diagram.
Q8. Correlation coefficient is independent of:
(a) Origin only (b) Scale only (c) Both origin and scale (d) Neither origin nor scale
Answer: (c) Both origin and scale.
Q9. When two variables are not at all related, the value of r is:
(a) +1 (b) −1 (c) 0 (d) Infinity
Answer: (c) 0.
Q10. Spearman’s rank correlation is most useful when data are:
(a) Quantitative (b) Qualitative or in ranks (c) Continuous (d) Time series
Answer: (b) Qualitative or in ranks.
Fill in the Blanks
Q1. Correlation measures the ________ and ________ of relationship between two variables.
Answer: degree, direction.
Q2. The value of r ranges from ________ to ________.
Answer: −1, +1.
Q3. When r = 0, there is ________ linear correlation between the variables.
Answer: no.
Q4. The graphical method of measuring correlation is called the ________ diagram.
Answer: scatter.
Q5. Spearman’s formula for rank correlation is rk = 1 − ________.
Answer: 6ΣD² / N(N²−1).
True or False
Q1. Correlation always implies causation.
Answer: False. Correlation does not necessarily imply a cause-and-effect relationship.
Q2. The coefficient of correlation can be greater than +1.
Answer: False. r always lies between −1 and +1.
Q3. Karl Pearson’s r is independent of the change of origin and scale.
Answer: True.
Q4. Scatter diagram gives a precise numerical value of correlation.
Answer: False. It only gives a visual indication; it does not give a numerical value.
Q5. Spearman’s rank correlation method can be used for qualitative data.
Answer: True.
Glossary
| Term | Meaning |
|---|---|
| Correlation | Statistical relationship between two or more variables. |
| Positive Correlation | Both variables move in the same direction. |
| Negative Correlation | Variables move in opposite directions. |
| Linear Correlation | Ratio of change between variables is constant. |
| Non-linear Correlation | Ratio of change is not constant; relationship is curvilinear. |
| Simple Correlation | Relationship between two variables only. |
| Multiple Correlation | Joint relationship of one variable with two or more others. |
| Partial Correlation | Relationship between two variables, others held constant. |
| Scatter Diagram | Graphical method of plotting paired values of X and Y. |
| Karl Pearson’s r | Mathematical measure of linear correlation. |
| Spearman’s rk | Rank correlation coefficient for ranked or qualitative data. |
| Causation | Cause-and-effect relationship between variables. |
Important Formulae
| Concept | Formula |
|---|---|
| Karl Pearson’s r (deviations from mean) | r = Σxy / √(Σx² × Σy²) |
| Karl Pearson’s r (actual values) | r = [NΣXY − ΣXΣY] / √{[NΣX² − (ΣX)²] × [NΣY² − (ΣY)²]} |
| Spearman’s Rank Correlation | rk = 1 − [6ΣD² / N(N² − 1)] |
| Range of r | −1 ≤ r ≤ +1 |
| r and regression coefficients | r = √(bxy × byx) |
| Coefficient of determination | r² |
This completes the study material for Class 11 Economics Chapter 7 — Correlation as per the ASSEB syllabus. Students should practise both Karl Pearson’s and Spearman’s methods on a variety of numerical problems, and remember that correlation is only a measure of association, not of causation. Continue learning with HSLC Guru.