For the data 3, 5, 1, 6, 5, 9, 5, 2, 8, 6 the mean median and mode are x. y and a respectively. Which one of the following is correct?

Options

A. `x=y ne z`
B. `x ne y=z`
C. `x ne y ne z`
D. `x =y =z`

Answer

To solve the problem, we need to find the mean, median, and mode of the given data set: 3, 5, 1, 6, 5, 9, 5, 2, 8, 6. We will denote the mean as \( x \), the median as \( y \), and the mode as \( z \).

### Step 1: Calculate the Mean
The mean is calculated by adding all the numbers together and dividing by the total count of numbers.

1. **Add the numbers**:
\[ 3 + 5 + 1 + 6 + 5 + 9 + 5 + 2 + 8 + 6 = 50 \]

2. **Count the total numbers**:
There are 10 numbers in total.

3. **Calculate the mean**:
\[ \text{Mean} (x) = \frac{\text{Total Sum}}{\text{Count}} = \frac{50}{10} = 5 \]

### Step 2: Calculate the Median
To find the median, we first need to arrange the data in ascending order.

1. **Arrange the data**:
\[ 1, 2, 3, 5, 5, 5, 5, 6, 6, 8, 9 \]

2. **Find the median**:
Since there are 10 numbers (even count), the median is the average of the 5th and 6th terms.
- The 5th term is 5.
- The 6th term is also 5.
\[ \text{Median} (y) = \frac{5 + 5}{2} = \frac{10}{2} = 5 \]

### Step 3: Calculate the Mode
The mode is the number that appears most frequently in the data set.

1. **Identify the frequency of each number**:
- 1 appears 1 time
- 2 appears 1 time
- 3 appears 1 time
- 5 appears 4 times
- 6 appears 2 times
- 8 appears 1 time
- 9 appears 1 time

2. **Determine the mode**:
The number that appears most frequently is 5.
\[ \text{Mode} (z) = 5 \]

### Conclusion
Now we have:
- Mean \( x = 5 \)
- Median \( y = 5 \)
- Mode \( z = 5 \)

Thus, we can conclude that:
\[ x = y = z \]

### Final Answer
The correct relation is \( x = y = z \).