To find the difference between the two medians, we first calculate the median of the original data and then the median of the data after the replacement.

Median Calculation: Original Data

The original data set is:

80, 90, 40, 30, 20, 10, 70, 60, 50

First, arrange the data in ascending order:

10, 20, 30, 40, 50, 60, 70, 80, 90

There are $n=9$ observations. The median is the middle value, which is the $(\frac{n+1}{2})^{th}$ term.

Median position = $(\frac{9+1}{2}) = 5^{th}$ term.

The 5th term in the sorted list is 50. So, the original median is 50.

Median Calculation: Replaced Data

Now, replace the number 30 with 100 in the original data set:

80, 90, 40, 100, 20, 10, 70, 60, 50

Arrange this new data set in ascending order:

10, 20, 40, 50, 60, 70, 80, 90, 100

The number of observations is still $n=9$. The median is again the $(\frac{n+1}{2})^{th}$ term.

Median position = $(\frac{9+1}{2}) = 5^{th}$ term.

The 5th term in this sorted list is 60. So, the new median is 60.

Difference Calculation: Final Step

The question asks for the difference between the two medians.

Difference = (New Median) - (Original Median)

Difference = $60 - 50$

Difference = $10$

The difference between the two medians is 10.