The mean of 2,12,4,9, 5 and 16 is x . The median of 4,3, x, x-1, 12 and 16 is y. Find the value of x and y.

Answer

To solve the problem step by step, we will first find the mean \( x \) of the given numbers, and then use that value to find the median \( y \) of another set of numbers.

### Step 1: Calculate the Mean \( x \)

The numbers given are: \( 2, 12, 4, 9, 5, 16 \).

1. **Sum the numbers**:
\[ 2 + 12 + 4 + 9 + 5 + 16 = 48 \]

2. **Count the number of observations**:
There are 6 numbers.

3. **Use the formula for mean**:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} = \frac{48}{6} = 8 \]
Thus, \( x = 8 \).

### Step 2: Calculate the Median \( y \)

Now we need to find the median of the numbers: \( 4, 3, x, x-1, 12, 16 \).

1. **Substitute the value of \( x \)**:
\[ x = 8 \implies x - 1 = 7 \]
So the numbers are: \( 4, 3, 8, 7, 12, 16 \).

2. **Arrange the numbers in ascending order**:
\[ 3, 4, 7, 8, 12, 16 \]

3. **Count the number of observations**:
There are 6 numbers (even).

4. **Use the formula for median for an even number of observations**:
The median is given by:
\[ \text{Median} = \frac{\text{(n/2)th term} + \text{(n/2 + 1)th term}}{2} \]
Here, \( n = 6 \), so:
\[ \text{Median} = \frac{\text{3rd term} + \text{4th term}}{2} \]

5. **Identify the 3rd and 4th terms**:
From the ordered list \( 3, 4, 7, 8, 12, 16 \):
- 3rd term = 7
- 4th term = 8

6. **Calculate the median**:
\[ y = \frac{7 + 8}{2} = \frac{15}{2} = 7.5 \]

### Final Results
Thus, the values are:
- \( x = 8 \)
- \( y = 7.5 \)