If mean and variance of 2, 3, 16, 20, 13, 7, x, y are 10 and 25 respectively then find xy

Explanation

To solve the problem, we need to find the values of \(x\) and \(y\) given the mean and variance of the numbers \(2, 3, 16, 20, 13, 7, x, y\).

### Step-by-Step Solution:

1. **Calculate the Mean:**
The mean of a set of numbers is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]
Here, the mean is given as \(10\) and the number of observations is \(8\). Therefore:
\[ \frac{2 + 3 + 16 + 20 + 13 + 7 + x + y}{8} = 10 \]
Simplifying the left side:
\[ 2 + 3 + 16 + 20 + 13 + 7 = 61 \]
Thus, we have:
\[ \frac{61 + x + y}{8} = 10 \]
Multiplying both sides by \(8\):
\[ 61 + x + y = 80 \]
Rearranging gives us:
\[ x + y = 80 - 61 = 19 \quad \text{(Equation 1)} \]

2. **Calculate the Variance:**
The variance is given by the formula:
\[ \text{Variance} = \frac{\sum (x_i^2)}{n} - \left(\text{Mean}\right)^2 \]
The variance is given as \(25\). Therefore:
\[ \frac{2^2 + 3^2 + 16^2 + 20^2 + 13^2 + 7^2 + x^2 + y^2}{8} - 10^2 = 25 \]
Calculating the squares:
\[ 2^2 = 4, \quad 3^2 = 9, \quad 16^2 = 256, \quad 20^2 = 400, \quad 13^2 = 169, \quad 7^2 = 49 \]
Summing these:
\[ 4 + 9 + 256 + 400 + 169 + 49 = 887 \]
Thus, we have:
\[ \frac{887 + x^2 + y^2}{8} - 100 = 25 \]
Rearranging gives:
\[ \frac{887 + x^2 + y^2}{8} = 125 \]
Multiplying both sides by \(8\):
\[ 887 + x^2 + y^2 = 1000 \]
Rearranging gives us:
\[ x^2 + y^2 = 1000 - 887 = 113 \quad \text{(Equation 2)} \]

3. **Use the Equations to Find \(xy\):**
We have two equations:
- \(x + y = 19\) (Equation 1)
- \(x^2 + y^2 = 113\) (Equation 2)

We can use the identity:
\[ (x + y)^2 = x^2 + y^2 + 2xy \]
Substituting the known values:
\[ 19^2 = 113 + 2xy \]
Calculating \(19^2\):
\[ 361 = 113 + 2xy \]
Rearranging gives:
\[ 2xy = 361 - 113 = 248 \]
Therefore:
\[ xy = \frac{248}{2} = 124 \]

### Final Answer:
\[ xy = 124 \]