Is it accurate that the median is a score where 50% of the observations are above it and 50% are below it? Lets say that the dataset has 3 observations namely 1, 2 and 3. The median is 2. Is 3, which is above the median, 50% of the dataset? How to correctly interpret the median?
 A: Sample median is defined as a value $\hat{x}$ that splits the samples $x_i$ of a data set $X$ such that at least 50% of the samples are greater or equal and at least 50% are less or equal than $\hat{x}$. Because you only have a finite sample set the median can only be approximated.
Assuming the samples are sorted, such a value can be computed by the following formula:
$$
\hat{x}=
\begin{cases}
x_{\frac{n+1}{2}},\hspace{3cm} \text{if n is odd}\\
\frac{1}{2}\left( x_{\frac{n}{2}} + x_{\frac{n}{2} + 1}  \right),\hspace{0.72cm} \text{if n is even}
\end{cases}
$$
where $n$ is the number of samples. In your example $X = \{1, 2, 3\}$, $n = 3$, which means case 1 applies: 
$$
\hat{x} = x_{\frac{3 + 1}{2}} = 2
$$
This means that there are two samples $x_i \in \{1, 2\}$ that satisfy  $x_i \leq \hat{x}$ and two samples $x_i \in \{2, 3\}$ that satisfy $x_i \geq \hat{x}$.
So it's exactly as we wanted it to be.
The concept can be generalized to quantiles. A $p$ quantile $\hat{x}_p$ is a a value that splits the samples such that at least $p \dot{}$ 100% of the samples are less or equal then $\hat{x}_p$ and at least $(1 - p) \dot{}$ 100% of the samples are greater or equal then $\hat{x}_p$. So the sample median is the $0.5$ quantile. 
The formula then becomes the following:
$$
\hat{x}_p=
\begin{cases}
x_{\lceil p\dot{}n \rceil},\hspace{3cm} \text{if $p \dot{} n$ is an not integer}\\
\frac{1}{2}\left(x_{p\dot{}n} + x_{p\dot{}n + 1}  \right),\hspace{0.72cm} \text{if $p \dot{} n$ is an integer}
\end{cases}
$$
where $n$ is the number of samples, $p$ the quantile, and $\lceil x \rceil$ is the ceiling function.
Some quantiles have names, because they are used often. Such as $0.5$ quantile is called median, $0.33$ quantile si the tercile, and so on.
A: Your statement is exactly accurate when the dataset has even number of observations (so that the median is calculated from the two "middle" ones) and almost accurate when you have an odd number of observations (because the middle one is neither above nor below the median). If you are concerned about precise terminology, you could say that median is a score where equal numbers of observations are above and below it, but your interpretation sounds fine in all practical situations.
