Finding the Value of x using Median

The problem provides a dataset arranged in ascending order and its median value. We need to find the value of the unknown variable '$x$'.

The given dataset is: $8, 10, 12, 15, x, x + 2, 20, 25, 30, 32$.

There are 10 observations in the dataset. Since 10 is an even number, the median is calculated as the average of the two middle terms.

The positions of the middle terms are $\frac{n}{2}$ and $\frac{n}{2} + 1$. In this case, $\frac{10}{2} = 5$ and $\frac{10}{2} + 1 = 6$. So, the median is the average of the 5th and 6th terms.

From the dataset, the 5th term is '$x$' and the 6th term is '$x + 2$'.

The median is given as 17. Therefore, we can set up the equation:

$ \text{Median} = \frac{5^{\text{th}} \text{ term} + 6^{\text{th}} \text{ term}}{2} $

$ 17 = \frac{x + (x + 2)}{2} $

Solving for x

Now, we solve the equation:

1. Multiply both sides by 2: $ 17 \times 2 = x + x + 2 $ $ 34 = 2x + 2 $
2. Subtract 2 from both sides: $ 34 - 2 = 2x $ $ 32 = 2x $
3. Divide by 2: $ x = \frac{32}{2} $ $ x = 16 $

Thus, the value of $x$ is 16.