When finding the median, you are looking for the middle number. Always arrange the numbers in ascending order. Since, the numbers are not increasing, let's organize it.

The new set is

$\displaystyle -6, -5, -4, -4, -3, -2$.

Remember, for negative numbers, the bigger the negative value, the smaller the number is since it's further away in the number line. Now, let's count the numbers in the set. There are six. Then divide six by two. We do this because we will split the number set in half. Because six does divide evenly into two, this means we can't easily determine the median. Since six divided by two is three, we are going to eliminate three numbers from leftmost in number set toward the right direction and three numbers from rightmost in number set toward the left direction. The last number crossed out in both direction are $\displaystyle -4$ and $\displaystyle -4$. To find the middle number, just add both numbers and divide by two.

$\displaystyle \frac{-4+(-4)}{2}=\frac{-8}{2}=-4$ That's the final answer.

Let's look at each statement. 

I. Always search for the middle number

This is false, because what happens if the number set is jumbled. To find median, it's important to oragnize in increasing or decreasing order.

II. Always arrange in increasing or decreasing order before searching for the middle number

As explained in statement one explanation, this is true.

III. Once arranged, if the set has an even number, just take the two middle numbers and subtract them and divide by two

This s false, because once there is an even number set, you need to ADD the middle numbers and divide it by two. Essentially, the new value represents the middle of the set.

IV. Once arranged, if the set has an even number, just take the two middle numbers and add them and divide by two

This is true based on statement three explanation. 

The set is already in increasing order. We have six numbers in the set and we need to ensure the set will have a median of $\displaystyle 4$. When there is an even number in the set, we need to take the two middle numbers by adding them then dividing by two. The two middle numbers represent $\displaystyle x$ and $\displaystyle y$.  Let's set up an equation.

$\displaystyle \frac{x+y}{2}=4$.

The numerator represents the two middle numbers being added and divided by the denominator. If we multiply both sides by $\displaystyle 2$ we get the sum of the variables to be $\displaystyle 8$. So we need to find the sum of $\displaystyle x$ and $\displaystyle y$ to be $\displaystyle 8$. The only choices are $\displaystyle x=2$, $\displaystyle y=6$ and $\displaystyle x=-4$, $\displaystyle y=12$. However, $\displaystyle x=-4$, $\displaystyle y=12$ doesn't work because $\displaystyle 12$ is bigger than both $\displaystyle 7$ and $\displaystyle 10$ and thus changing the median. $\displaystyle x=2$, $\displaystyle y=6$ is good because both of the values are les than $\displaystyle 7$ but greater than $\displaystyle -5$. 

The set is already in increasing order. We have six numbers in the set and we need to ensure the set will have a median of $\displaystyle 6$. When there is an even number in the set, we need to take the two middle numbers by adding them then dividing by two. The two middle numbers represent $\displaystyle x$ and $\displaystyle 7$.  Let's set up an equation.

$\displaystyle \frac{x+7}{2}=6$.

The numerator represents the two middle numbers being added and divided by the denominator. If we multiply both sides by $\displaystyle 2$, and subtract $\displaystyle 7$ on both sides, we get $\displaystyle x$ to be $\displaystyle 5$. Finally, to find $\displaystyle y$, we need a number that is greater than or equal to $\displaystyle 9$ and less than or equal to $\displaystyle 10$. Answer $\displaystyle x=5$, $\displaystyle y=9$ satisfies all conditions. 

The set is already in increasing order. We have five numbers in the set, however, we need to add another number to ensure the set will have a median of $\displaystyle 15$. This will make the set have six numbers. When there is an even number in the set, we need to take the two middle numbers by adding them then dividing by two. Let's say this number is $\displaystyle x$. Let's setup an equation.

$\displaystyle \frac{8+x}{2}=15$.

The numerator represents the two middle numbers being added and divided by the denominator. If we multiply both sides by $\displaystyle 2$ and subtract $\displaystyle 8$ on both sides, $\displaystyle x$ is $\displaystyle 22$. Make sure this answer doesn't violate the set. $\displaystyle 22$ is less than $\displaystyle 25$ but greater than $\displaystyle 8$, so therefore $\displaystyle 22$ is the correct answer.

The median is the middle number of the set, when it is listed in order from smallest to largest or vice versa. In this case we have an even amount of numbers in the set meaning there are two "middle numbers". But for this set, both of those middle numbers are 89, meaning we don't need to take an average. 

This gives us the final answer of 89 for the median.

The median is the middle number of the set, when it is listed in order from smallest to largest or vice versa. In this case we have an even amount of numbers in the set meaning there are two "middle numbers"- 23 and 25. In order to find the median we take the average of 23 and 25:

$\displaystyle \frac{23+25}{2}=24$

The median is the middle number of the set, when it is listed in order from smallest to largest or vice versa. In this case we have an even amount of numbers in the set meaning there are two "middle numbers"- 22 and 23. In order to find the median we take the average of 22 and 23:

$\displaystyle \frac{22+23}{2}=22.5$

The median is the middle number of the set, when it is listed in order from smallest to largest or vice versa. In this case we have an even amount of numbers in the set meaning there are two "middle numbers"- 11 and 12. In order to find the median we take the average of 11 and 12:

$\displaystyle \frac{11+12}{2}=11.5$

The median is the middle number of the set, when it is listed in order from smallest to largest or vice versa. In this case we have an even amount of numbers in the set meaning there are two "middle numbers"- 8 and 13. In order to find the median we take the average of 8 and 13:

$\displaystyle \frac{8+13}{2}=10.5$