Question

When an observation that is much larger than the rest of the data is added to a data​ set, the value of the median will increase substantially?

Answer 1

No. To compute the median, you arrange the whole dataset in increasing order, and pick the element in the middle. So, you have
`N` observations so far, you write the dataset as

`x_1,\ x_2,\ x_3,\ldots,x_{(N)/(2)-1},\ x_{(N)/(2)},\ x_{(N)/(2)+1},\ldots, x_N`

And the median is the element in the middle, i.e
`x_{(N)/(2)}`

Now, assume you add the last observation,
`x_(N+1)`. This is much larger than the rest of the dataset, so if we arrange the dataset in increasing order, it will be the last:

`x_1,\ x_2,\ x_3,\ldots,x_{(N)/(2)-1},\ x_{(N)/(2)},\ x_{(N)/(2)+1},\ldots, x_N,\ x_(N+1)`

But the median is always the element in the middle, so it is again either
`x_{(N)/(2)}` or
`x_{(N+1)/(2)}`