The median in a data set is the number that lies directly in the middle. To determine the median, first list the numbers in ascending order:

\(\displaystyle 1,4,5,5,6,9,10,11,12,13,16,17,21,28,32,41,50\)

Then, count in from both sides to find the number that lies directly in the middle. Therefore the correct answer is "median".

Reorder the numbers in ascending order (from lowest to highest):

11, 13, 13, 13, 13, 14, 16, 19

Find the middle number. In this case, the middle number is the average of the 4th and 5th numbers. Because both the 4th and 5th number are 13, the answer is simply 13.

List the numbers in ascending order: 2,3,4,4,6,8,9,11,13,15,16

The median is the middle number, or 8.

To find the median of a set of values, simply place the numbers in order and find the value that is exactly "in the middle." Here, we can place the test scores in ascending order to get \(\displaystyle 61\), \(\displaystyle 72\), \(\displaystyle 85\), \(\displaystyle 85\), \(\displaystyle 92\). (Descending order would work just as well.) The median is the middle value, \(\displaystyle 85\). Make sure you don't confuse median with mean (average)! To get the mean value of this set, you would find the sum of the test scores and then divide by the number of values.

To find the median, first you arrange the numbers in order from least to greatest.

Then you count how many numbers you have and divide that number by two. In this case 12,15,16,21,22,32,93,108= 8 numbers.

So \(\displaystyle \frac{8}{2}=4\)

Then starting from the least side of the numbers count 4 numbers till you reach the median number of \(\displaystyle 21\)

Then starting from the greatest side count 4 numbers until you reach the other median number of \(\displaystyle 22\)

Finally find the mean of the two numbers by adding them together and dividing them by two \(\displaystyle \frac{(21+22)}{2}=\frac{43}{2}\)

to find the median number of \(\displaystyle 21.5\).

First, arrange all of the data in numerical order: \(\displaystyle 3,3,4,4,4,5,5,6,6,7,8\).

Then locate the middle number by using the formula

 \(\displaystyle \frac{n-1}{2}+1\), which gives you the location of the median in the ordered data set and where \(\displaystyle n\) is the number of terms in the data set.

Here, there are 11 terms.

So, \(\displaystyle \frac{11-1}{2}+1=\frac{10}{2}+1=5+1=6.\) 

Therefore, our number is the \(\displaystyle 6^{th}\) one in the list, which is \(\displaystyle 5\).

Since it's in increasing order, let's have some scenarios. Let's say \(\displaystyle a, b\) are both \(\displaystyle 5\), then that means \(\displaystyle e, f\) at can be \(\displaystyle 5, 15\) with \(\displaystyle f\) maxing out its value up to \(\displaystyle 10, 10\) with \(\displaystyle e\) maxing its value. That leaves \(\displaystyle c, d\) must be \(\displaystyle 5\) with \(\displaystyle f\) maxing out its value and \(\displaystyle c,d\) being in a range of \(\displaystyle 5-10\) with \(\displaystyle e\) maxing its value and not violating the set. Now, lets say \(\displaystyle a,b\) are \(\displaystyle 0, 10\)respectively, this leaves \(\displaystyle e, f\) only being \(\displaystyle 10\) and \(\displaystyle c, d\) being \(\displaystyle 10\) as well. Let's find the range of \(\displaystyle c,d\). If \(\displaystyle c,d\) were both \(\displaystyle 5\), that means the median of them is \(\displaystyle 5\). If \(\displaystyle c,d\) are both \(\displaystyle 10\), that means the median of them is \(\displaystyle 10\). \(\displaystyle 10\) is highest median of both \(\displaystyle c,d\) and \(\displaystyle 5\) is the lowest median of \(\displaystyle c,d\). We need to find a number in the answer choices that fit this description. Answer is \(\displaystyle 7.5.\)

Cubic numbers are numbers taken to the third power. The first six cubic numbers are: \(\displaystyle 1^3, 2^3, 3^3, 4^3, 5^3, 6^3\) or \(\displaystyle 1, 8, 27, 64, 125, 256\). 

Since, the numbers are inceasing, 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 \(\displaystyle 27\) and \(\displaystyle 64\). To find the middle number, just add both numbers and divide by two.

\(\displaystyle \frac{27+64}{2}=\frac{91}{2}=45.5\) That.s the final answer.

Let's interpret the problem. One number is \(\displaystyle x\). Another is two times square root of \(\displaystyle x\) or \(\displaystyle 2\sqrt{x}\). The last number is one less than \(\displaystyle x\) or \(\displaystyle x-1\). The sum is three which means the equation to set up is: \(\displaystyle x+2\sqrt{x}+x-1=3\). Let's solve for \(\displaystyle x\). 

\(\displaystyle 2x-1+2\sqrt{x}=3\)

\(\displaystyle 2x+2\sqrt{x}=4\) I want to have the square root on one side and the numbers and variable on the other.

\(\displaystyle 2\sqrt{x}=4-2x\) When  I square both sides, we get a quadratic equation. If I were to square the equation before, I still have a radical to get rid of.           

\(\displaystyle (2\sqrt{x})^2=(4-2x)^2\)

\(\displaystyle 4x=16-8x-8x+4x^{2}\)

Remember when foiling, you multiply the numbers/variables that first appear in each binomial, followed by multiplying the outer most numbers/variables, then multiplying the inner most numbers/variables and finally multiplying the last numbers/variables.

\(\displaystyle 0=16-4x-8x-8x+4x^2\)

\(\displaystyle 0=16-20x+4x^2\) Let's factor out a \(\displaystyle 4\) to reduce the quadratic.

\(\displaystyle 0=4(4-5x+x^2)\) If I divide both sides by \(\displaystyle 4\), I get:

\(\displaystyle 0=x^2-5x+4\)

 Remember, we need to find two terms that are factors of the c term that add up to the b term. We have: \(\displaystyle (x-4)(x-1)=0\)

Solve for \(\displaystyle x\). \(\displaystyle x=4, x=1\)

We are not done as the problem asks for median of the set. If we plug in \(\displaystyle 1\), we have: \(\displaystyle 1, 2\sqrt{x}, 1-1\) or \(\displaystyle 1, 2, 0\). Once we arrange in increasing order, we have \(\displaystyle 0, 1, 2\). By checking, the sum is \(\displaystyle 3\) and the middle number is \(\displaystyle 1\). Let's check when \(\displaystyle x\) is \(\displaystyle 4\). We have: \(\displaystyle 4, 2\sqrt{4}, 4-1\) or \(\displaystyle 4, 4, 3\). In increasing order we have \(\displaystyle 3, 4, 4\). The answer may be 4, HOWEVER, it doesn't satisfy the problem as the sum should be \(\displaystyle 3\) but instead we have \(\displaystyle 11\). Therefore the correct answer to this problem is \(\displaystyle 1.\)

When finding the median, you are looking for the middle number. Always arrange the numbers in ascending order. Since, the numbers are inceasing, count the numbers in the set. There are five. Then divide five by two. We do this because we will split the number set in half. Because five doesn't divide evenly into two, this means we can easily determine the median. Since five divided by two is \(\displaystyle 2.5\), we are going to eliminate two numbers from leftmost in number set toward the right direction and two numbers from rightmost in number set toward the left direction. The only number left is \(\displaystyle 7\) and therefore is the right answer.