To determine how much the median would increase when the number 5 is added to the given set of numbers \(\(\{6, 2, 2, 4, 6, 7\}\)\), we need to follow these steps:
1. Sort the original set of numbers.
2. Find the median of the original set.
3. Add the number 5 to the set and sort the new set.
4. Find the median of the new set.
5. Calculate the increase in the median.
### Step 1: Sort the Original Set
The original set is: \(\(\{6, 2, 2, 4, 6, 7\}\)\).
Sorting these numbers in ascending order:
\(\[ \{2, 2, 4, 6, 6, 7\} \]\)
### Step 2: Find the Median of the Original Set
To find the median of a set with an even number of elements (6 elements in this case), we take the average of the two middle numbers.
The sorted set is \(\(\{2, 2, 4, 6, 6, 7\}\)\).
The two middle numbers are the third and fourth numbers in this sorted list: 4 and 6.
So, the median of the original set is:
\(\[ \text{Median} = \frac{4 + 6}{2} = \frac{10}{2} = 5 \]\)
### Step 3: Add the Number 5 and Sort the New Set
The new set is: \(\(\{6, 2, 2, 4, 6, 7, 5\}\)\).
Sorting the new set in ascending order:
\(\[ \{2, 2, 4, 5, 6, 6, 7\} \]\)
### Step 4: Find the Median of the New Set
To find the median of a set with an odd number of elements (7 elements in this case), we take the middle number.
The sorted new set is \(\(\{2, 2, 4, 5, 6, 6, 7\}\)\).
The middle number is the fourth number in this sorted list: 5.
So, the median of the new set is:
\(\[ \text{Median} = 5 \]\)
### Step 5: Calculate the Increase in the Median
Now, we compare the medians of the original and new sets.
- Median of the original set: \(\( 5 \)\)
- Median of the new set: \(\( 5 \)\)
The increase in the median is:
\(\[ \text{Increase} = 5 - 5 = 0 \]\)
Therefore, the median would remain the same and the increase in the median would be \(\(0\)\).
