Step 1: Count and Sort the Data

First, count the total number of observations, which is $n = 10$ (even). Then, arrange the data in ascending order: $12, 12, 12, 23, 34, 45, 56, 67, 89, 90$.

$n = 10$

Step 2: Identify the Middle Terms

Since $n$ is even, the median is the average of the $\left(\frac{n}{2}\right)^{\text{th}}$ and $\left(\frac{n}{2} + 1\right)^{\text{th}}$ terms. Here, $\frac{10}{2} = 5^{\text{th}}$ term and the $6^{\text{th}}$ term.

$5^{\text{th}} \text{ term} = 34, \quad 6^{\text{th}} \text{ term} = 45$

Step 3: Calculate the Median

Substitute the values of the 5th and 6th terms into the median formula for an even number of observations.

$\text{Median} = \frac{34 + 45}{2} = \frac{79}{2} = 39.5$