Impact on Median

The median is the middle of the ordered data.
When you add a value, the middle position can move, which shifts the median.
Where the new value lands decides the direction of the shift.

A value added on the low end pushes the middle toward the lower numbers, lowering the median.
A value added on the high end pushes it toward the higher numbers, raising the median.
A value right at the median can leave it unchanged.

Take $1$, $2$, $3$, with median $2$.
Add an $8$: the ordered list is $1$, $2$, $3$, $8$, and the median is the average of $2$ and $3$, which is $2.5$.
The median shifted right, but only a little, even though $8$ is far out.

Unlike the mean, the median barely moves for a far-out value.
Whether the extra number is $8$ or $800$, the middle of $1$, $2$, $3$, and that value is still $2.5$.
That is why the median describes the center well when there are outliers.

The scores $73$, $78$, $82$, $85$, $88$, $91$ have a median of $83.5$. A new score of $65$ is added. What happens to the median?
The list becomes $65$, $73$, $78$, $82$, $85$, $88$, $91$, with $82$ in the middle.
The median shifts down from $83.5$ to $82$.

The set $2$, $4$, $6$ has a median of $4$. What is the median after adding $20$?
The ordered set is $2$, $4$, $6$, $20$, and the median is the average of $4$ and $6$, which is $5$.
The median shifts up from $4$ to $5$.

The set $10$, $20$, $30$ has a median of $20$. You add another $20$. What happens to the median?
The ordered set is $10$, $20$, $20$, $30$, and the two middle values are both $20$.
So the median stays at $20$.