To find the median, the first step is to arrange the data in ascending order.
The given data set is: 89, 45, 56, 67, 89, 78, 45, 78, 45, 67.
Arranging in ascending order: 45, 45, 45, 56, 67, 67, 78, 78, 89, 89.
Next, count the number of observations, denoted by $n$.
In this data set, $n = 10$.
Since $n$ is an even number, the median is the average of the two middle values.
The positions of the middle values are $(\frac{n}{2})^{th}$ and $(\frac{n}{2} + 1)^{th}$.
For $n=10$, these are the $(\frac{10}{2})^{th} = 5^{th}$ term and the $(\frac{10}{2} + 1)^{th} = 6^{th}$ term.
From the ordered list, the $5^{th}$ term is 67 and the $6^{th}$ term is 67.
The median is the average of these two terms:
Median = $\frac{67 + 67}{2} = \frac{134}{2} = 67$.