Chapter 7 Correlation Solutions
Question - 11 : - Does zero correlation mean independence?
Answer - 11 : -
Correlation measures the linear relationship between the two variables. So, r being 0 implies the absence of linear relationship. But they may be non-linearly related. Hence, if two variables are not correlated, it does not necessarily follow that they are independent.
Question - 12 : - Can simple correlation coefficient measure any type of relationship?
Answer - 12 : -
No, the simple correlation coefficient cannot measure any type of relationship. The simple correlation coefficient can measure only the direction and magnitude of linear relationship between the two variables. It cannot measure non-linear relationship like quadratic, trigonometric, cubic, etc. Therefore, in such cases, the purview of simple correlation coefficient falls short. For example, the simple correlation coefficient may depict that X and Y are not correlated in the equation X= Y2, hence it may be concluded that both the variables are independent, but such conclusion may be wrong.
Question - 13 : - Collect the price of five vegetables from your local market every day for a week. Calculate their correlation coefficients. Interpret the result.
Answer - 13 : -
This question is about multivariate correlation that is out of syllabus
Question - 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.
Answer - 14 : -
| Height of Classmate X | Height of Benchmate Y |
| 67 | 65 |
| 56 | 66 |
| 65 | 57 |
| 68 | 67 |
| 72 | 68 |
| 72 | 69 |
| 69 | 70 |
| 71 | 72 |
Question - 15 : - List some variables where accurate measurement is difficult.
Answer - 15 : -
The following are the some variables where the accurate measurement is difficult.
1. Temperature and number of people falling ill.
2. Change in temperature with the height of mountain.
3. Low rainfall and agricultural productivity
4. High population growth and degree of poverty
5. Number of tourists and change in the political atmosphere in India.
Question - 16 : - Interpret the values of r as 1, –1 and 0.
Answer - 16 : -
The value of r being 1 implies that there is a perfect positive correlation between the two variables involved. A high value of r (i.e. close to 1) represents a strong positive linear relationship between the two variables.
If r = –1, then the correlation is perfectly negative. A negative value of r indicates an inverse relation. A low value of r (i.e. close to –1) represents a strong negative linear relationship between the variables. On the other hand, if the value of r = 0, then it implies that the two variables are uncorrelated to each other. But this should not be misunderstood as the variables are independent of each other. The value of r equals zero confirms only the non-existence of any linear relation but the variables may be non-linearly related to each other.
Question - 17 : - Why does rank correlation coefficient differ from Pearsonian correlation coefficient?
Answer - 17 : -
Generally, all the properties of Karl Pearson’s coefficient of correlation are similar to that of the rank correlation coefficient. However, rank correlation coefficient is generally lower or equal to Karl Pearson’s coefficient. Rank correlation coefficient is generally preferred to measure the correlation between the two qualitative variables. These variable attributes are given the ranks on the basis of preferences. The difference between the two coefficients is due to the fact that the rank correlation coefficient uses ranks instead of the full set of observations that leads to some loss of information. If the precisely measured data are available, then both the coefficients will be identical. Secondly, if extreme values are present in the data, then the rank correlation coefficient is more precise and reliable and consequently its value differs from that of the Karl Pearson’s coefficient.
Question - 18 : - Calculate the correlation coefficient between the heights of fathers in inches (X) and their sons (Y)
Answer - 18 : - 
Question - 19 : - Calculate the correlation coefficient between X and Y and comment on their relationship:
Answer - 19 : - 
As the value of r is zero, so there is no linear correlation between X and Y
Question - 20 : - Calculate the correlation coefficient between X and Y and comment on their relationship
Answer - 20 : - 
As the correlation coefficient between the two variables is +1, so the two variables are perfectly positive correlated.