Find the Spearmen 's rank correlation coeffcient ,given : `n = 10, sum |d_x -d_y| ^(2) = 30`

Answer

To find the Spearman's rank correlation coefficient, we will follow these steps:

### Step 1: Write down the formula for Spearman's rank correlation coefficient.
The formula for Spearman's rank correlation coefficient (p) is given by:

\[ p = 1 - \frac{6 \sum (d_x - d_y)^2}{n(n^2 - 1)} \]

### Step 2: Identify the given values.
From the problem, we have:
- \( n = 10 \)
- \( \sum (d_x - d_y)^2 = 30 \)

### Step 3: Substitute the values into the formula.
Now, we will substitute the values into the formula:

\[ p = 1 - \frac{6 \times 30}{10 \times (10^2 - 1)} \]

### Step 4: Calculate \( n^2 - 1 \).
First, calculate \( n^2 - 1 \):
\[ n^2 = 10^2 = 100 \quad \Rightarrow \quad n^2 - 1 = 100 - 1 = 99 \]

### Step 5: Substitute \( n^2 - 1 \) back into the formula.
Now substitute back into the formula:

\[ p = 1 - \frac{6 \times 30}{10 \times 99} \]

### Step 6: Calculate the numerator and denominator.
Calculate the numerator:
\[ 6 \times 30 = 180 \]

Calculate the denominator:
\[ 10 \times 99 = 990 \]

### Step 7: Substitute the calculated values into the formula.
Now substitute these values back into the formula:

\[ p = 1 - \frac{180}{990} \]

### Step 8: Simplify the fraction.
To simplify \( \frac{180}{990} \):
\[ \frac{180 \div 90}{990 \div 90} = \frac{2}{11} \]

### Step 9: Final calculation.
Now substitute this back into the equation for \( p \):

\[ p = 1 - \frac{2}{11} = \frac{11}{11} - \frac{2}{11} = \frac{9}{11} \]

### Conclusion:
Thus, the Spearman's rank correlation coefficient \( p \) is:

\[ \boxed{\frac{9}{11}} \]