3.ThefollowingdatashowstheMonthlyincomeperheadofpopulation,xand
theLivingstandard,yforasampleof10countries:Calculatetherank
correlationforthisgiven�
To calculate the rank correlation for the given data, you can follow these steps:
Step 1: Write down the given data in a table with two columns - one for the Monthly income per head of population (x) and the other for the Living standard (y).
For example:
| Country | Monthly Income (x) | Living Standard (y) |
|---------|-------------------|---------------------|
| 1 | $2000 | 3 |
| 2 | $1500 | 2 |
| 3 | $3000 | 4 |
| 4 | $1800 | 2 |
| 5 | $2500 | 3 |
| 6 | $3500 | 4 |
| 7 | $2400 | 3 |
| 8 | $3200 | 4 |
| 9 | $2700 | 3 |
| 10 | $1900 | 2 |
Step 2: Rank the values in each column separately. Assign a rank of 1 to the smallest value, 2 to the second smallest value, and so on. In case of ties, assign the average rank.
| Country | Monthly Income (x) | Rank (x) | Living Standard (y) | Rank (y) |
|---------|-------------------|----------|---------------------|----------|
| 1 | $2000 | 4 | 3 | 4 |
| 2 | $1500 | 1 | 2 | 1 |
| 3 | $3000 | 7 | 4 | 8 |
| 4 | $1800 | 2 | 2 | 1 |
| 5 | $2500 | 5 | 3 | 4 |
| 6 | $3500 | 10 | 4 | 8 |
| 7 | $2400 | 3 | 3 | 4 |
| 8 | $3200 | 9 | 4 | 8 |
| 9 | $2700 | 6 | 3 | 4 |
| 10 | $1900 | 3 | 2 | 1 |
Step 3: Calculate the differences between the ranks of each pair of values (d). Square each difference (d^2).
| Country | Monthly Income (x) | Rank (x) | Living Standard (y) | Rank (y) | d(x) | d(y) | d(x)^2 | d(y)^2 |
|---------|-------------------|----------|---------------------|----------|------|------|--------|--------|
| 1 | $2000 | 4 | 3 | 4 | 0 | 0 | 0 | 0 |
| 2 | $1500 | 1 | 2 | 1 | 0 | 0 | 0 | 0 |
| 3 | $3000 | 7 | 4 | 8 | -1 | -4 | 1 | 16 |
| 4 | $1800 | 2 | 2 | 1 | 0 | 0 | 0 | 0 |
| 5 | $2500 | 5 | 3 | 4 | 0 | -1 | 0 | 1 |
| 6 | $3500 | 10 | 4 | 8 | -2 | -4 | 4 | 16 |
| 7 | $2400 | 3 | 3 | 4 | 0 | -1 | 0 | 1 |
| 8 | $3200 | 9 | 4 | 8 | -1 | -4 | 1 | 16 |
| 9 | $2700 | 6 | 3 | 4 | -1 | -1 | 1 | 1 |
| 10 | $1900 | 3 | 2 | 1 | 0 | 0 | 0 | 0 |
Step 4: Calculate the sum of the squared differences for each column (Σd(x)^2 and Σd(y)^2).
Σd(x)^2 = 0 + 0 + 1 + 0 + 0 + 4 + 0 + 1 + 1 + 0 = 7
Σd(y)^2 = 0 + 0 + 16 + 0 + 1 + 16 + 1 + 16 + 1 + 0 = 51
Step 5: Use the formula for rank correlation (rho):
rho = 1 - (6 * Σd(x)^2) / (N * (N^2 - 1))
Where N is the number of pairs of values (in this case, N = 10).
rho = 1 - (6 * 7) / (10 * (10^2 - 1))
= 1 - 42 / (10 * 99)
= 1 - 42 / 990
= 1 - 0.0424242424...
= 0.9575757575...
Therefore, the rank correlation (rho) for the given data is approximately 0.958.