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NCERT Class 11 Economics Chapter 7 Correlation
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Correlation
Chapter: 7
PART – (A) STATISTICS FOR ECONOMICS
TEXTUAL QUESTION ANSWERS
1. The unit of correlation coefficient between height in feet and weight in kgs is:
(a) Kg/feet.
(b) Percentage.
(c) Non-existent.
Ans: (c) Non-existent.
2. The range of simple correlation coefficient is:
(a) 0 to infinity.
(b) Minus one to plus one.
(c) Minus infinity to infinity.
Ans: (b) Minus one to plus one.
3. If r xy is positive, the relation between X and Y is of the type:
(a) When Y increases X increases.
(b) When Y decreases X increases.
(c) When Y increases X does not change.
Ans: (a) When Y increases X increases.
4. If r xy = 0, the variable X and Y are:
(a) Linearly related.
(b) Not linearly related.
(c) Independent.
Ans: (b) Not linearly related.
5. Of the following three measures which can measure any type of relationship?
(a) Karl Pearson’s coefficient of correlation.
(b) Spearman’s rank correlation
(c) Scatter diagram.
Ans: (b) Spearman’s rank correlation.
6. If precisely measured data are available, the simple correlation coefficient is:
(a) More accurate than rank correlation coefficient.
(b) less accurate than correlation coefficient rank.
(c) As accurate as the rank correlation coefficient.
Ans: (c) As accurate as the rank correlation coefficient.
7. Why is r preferred to covariance as a measure of association?
Ans: r (the correlation coefficient) is preferred over covariance because it standardizes the relationship between two variables by removing the unit of measurement. While covariance measures the degree to which two variables change together, its value depends on the units of the variables, making it difficult to interpret. In contrast, r is dimensionless and lies between -1 and +1, making it easier to understand and compare across different datasets.
8. Can r lie outside the -1 and +1 range depending on the type of data?
Ans: No, the value of the correlation coefficient lies between minus one and plus one, -1 ≤ r ≤ +1. If the value of r is outside this range in any type of data, it indicates error in calculation.
9. Does correlation imply causation?
Ans: This fallacy occurs when a causal connection is assumed without proof. All too often claims to a causal connection are based on a mere correlation. The occurrence of one event after the other or the occurrence of events simultaneously is not proof of a causal connection.
10. When is rank correlation more precise than simple correlation coefficient?
Ans: Rank correlation is more precise than simple correlation when there are extreme values in the data, or when there is precisely measured data.
11. Does zero correlation mean independence?
Ans: No, zero correlation does not necessarily mean independence. Zero correlation indicates no linear relationship between two variables, but they could still be non-linearly related.
12. Can a simple correlation coefficient measure any type of relationship?
Ans: Correlation coefficients are used to measure the strength of the linear relationship between two variables. A correlation coefficient greater than zero indicates a positive relationship while a value less than zero signifies a negative relationship.
13. Collect the price of five vegetables from your local market every day for a week. Calculate their correlation coefficients. Interpret the result.
Ans:
(i) Correlation between potato and cabbage:
(ii) Correlation between cabbage and tomato:
(iii) Correlation between tomato and peas:
(iv) Correlation between peas and carrot:
(v) Correlation between carrot potato:
Interpretation of Result: It is clear from the above result that every vegetable is correlated. There is a positive correlation among these vegetables. Correlation of peas and carrots is 0.48 which is highest and correlation of potato and cabbage is 0.089 which is lowest.
14. Measure the height of your classmates. Ask them the height of their benchmate. Calculate the correlation coefficient of these two variables. Interpret the result.
Ans:
| Roll No. of Classmate | Height as asked from classmate (in inches) X | Measured height (in inches) Y |
| 1 | 60 | 62 |
| 2 | 62 | 64 |
| 3 | 58 | 53 |
| 4 | 56 | 60 |
| 5 | 59 | 72 |
| 6 | 63 | 56 |
| 7 | 65 | 68 |
| 8 | 66 | 60 |
| 9 | 68 | 69 |
| 10 | 60 | 65 |
It is clear from the above analysis that there is a positive correlation between the asked height and measured height of the classmates.
15. List some variables where accurate measurement is difficult.
Ans: Qualitative variables such as beauty, intelligence, honesty, etc.
It is also difficult to measure subjective variables such as poverty, development, etc which are interpreted differently by different people.
16. Interpret the values of r as 1, -1 and 0.
Ans: r > 0 indicates a positive association.
r < 0 indicates a negative association. Values of r near 0 indicate a very weak linear relationship. The strength of the linear relationship increases as r moves away from 0 toward -1 or 1.
17. Why does the rank correlation coefficient differ from Pearsonian correlation coefficient?
Ans: The rank correlation coefficient is used to measure the relationship between ordinal (ranked) variables, while Pearson’s correlation coefficient measures the linear relationship between continuous (quantitative) variables.
18. Calculate the correlation coefficient between the heights of fathers in inches (X) and their sons (Y).
| X | 65 | 66 | 57 | 67 | 68 | 69 | 70 | 72 |
| Y | 67 | 56 | 65 | 68 | 72 | 72 | 69 | 71 |
Ans:
| X | Y | XY | X² | Y² |
| 65 | 67 | 4355 | 4225 | 4489 |
| 66 | 56 | 3696 | 4356 | 3136 |
| 57 | 65 | 3705 | 3249 | 4225 |
| 67 | 68 | 4556 | 4489 | 4624 |
| 68 | 72 | 4896 | 4624 | 5184 |
| 69 | 72 | 4968 | 4761 | 5184 |
| 70 | 69 | 4830 | 4900 | 4761 |
| 72 | 71 | 5112 | 5184 | 5041 |
| ∑X = 534 | ∑Y = 540 | ∑XY = 36118 | ∑X² = 35788 | ∑Y² = 36644 |
19. Calculate the correlation coefficient between X and Y and comment on their relationship:
| X | -3 | -2 | -1 | 1 | 2 | 3 |
| Y | 9 | 4 | 1 | 1 | 4 | 9 |
Ans:
| X | Y | XY | X² | Y² |
| -3 | 9 | -27 | 9 | 81 |
| -2 | 4 | -8 | 4 | 16 |
| -1 | 1 | -1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 |
| 2 | 4 | 8 | 4 | 16 |
| 3 | 9 | 27 | 9 | 81 |
| ∑X = 0 | ∑X = 28 | ∑XY = 0 | ∑X² = 28 | ∑Y² = 196 |
As the value of r is zero, so there is no linear correlation between the variables X and Y.
20. Calculate the correlation coefficient between X and Y and comment on their relationship.
| X | 1 | 3 | 4 | 5 | 7 | 8 |
| Y | 2 | 6 | 8 | 10 | 14 | 16 |
Ans:
| X | Y | XY | X² | Y² |
| 1 | 2 | 2 | 1 | 4 |
| 3 | 6 | 18 | 9 | 36 |
| 4 | 8 | 32 | 16 | 64 |
| 5 | 10 | 50 | 25 | 100 |
| 7 | 14 | 98 | 49 | 196 |
| 8 | 16 | 128 | 64 | 256 |
| ∑X = 28 | ∑Y = 56 | ∑XY = 328 | ∑X² = 164 | ∑X² = 656 |
As the correlation coefficient between the two variables is + 1, so the two variables are perfectly positively correlated.
