GMAT Quantitative - Calculating median

What is the median of the following numbers?

$\frac{1}{2}, \frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7}$

$\frac{9}{40}$

$\frac{1}{4}$

$\frac{223}{840}$

$\frac{1}{5}$

$\frac{2}{9}$

Explanation

The median of a data set with an even number of elements is the mean of its two middle elements, when ranked. The set is already ranked, so just find the mean of middle elements $\frac{1}{4}$ and $\frac{1}{5}$ :

$\frac{1}{2} \cdot \left (\frac{1}{4} + \frac{1}{5} \right ) = \frac{1}{2} \cdot \left (\frac{5}{20} + \frac{4}{20} \right )= \frac{1}{2} \cdot \frac{9}{20} = \frac{9}{40}$

Below is the stem-and-leaf display of a set of test scores.

$\left.\begin{matrix} 4\ 5\ 6\ 7\ 8 \end{matrix}\right|\begin{matrix} \textrm{2 5}; ; ; ; ; ; ; ; ; ; ;; ; ; ; ; ; ; ; ; ;\ \textrm{4 4 7 7}; ; ; ; ; ; ; ; ; ; ;; ; ; ; \ \textrm{0 1 2 2 4 5 8 8 9}\ \textrm{3 5 5 8} ; ; ; ; ; ; ; ;; ; ; ; ; ; ; \ \textrm{7 }; ; ; ; ; ; ; ; ; ;; ; ; ; ; ; ; ; ; ; ; ; ; \end{matrix}$

What is the first quartile of these test scores?

Explanation

The numbers in the "stem" of this display represent tens digits of the test scores, and the numbers in the "leaves" represent the units digits.

This stem-and-leaf display represents twenty scores. The first quartile is the median of the lower half, or the lower ten scores. This is the arithmetic mean of the fifth- and sixth-lowest scores. Both of these scores are the same, however - 57. Therefore, 57 is the first quartile.

Give the median of the following data set in terms of :

$\left {a -1, a +2, a-3, a+ 4, a -5, a+6, a-7, a+8 \right }$

$a + \frac{1}{2}$

$a - \frac{1}{2}$

Explanation

The data set can be arranged from least to greatest as follows:

$\left {a-7, a-5, a-3, a-1, a+2, a+4, a+6, a+8 \right }$

The median of a data set with eight elements is the mean of its fourth-highest and fourth-lowest elements, which are and . Add and divide by two:

$\frac{(a-1) + (a+2)}{2} = \frac{2a+1}{2} = a + \frac{1}{2}$

Sally is collecting information about the lengths of tree branches that fell in her back yard during a powerful storm the night before. With her trusty measuring stick, she measures out 10 different branches and finds their lengths in feet to be

Calculate the median of the branch lengths

6.5

6.4

9.5

Explanation

In order to find the median, we first have to order the data points from lowest to highest. This would be-

Since we have an even number of data points, the median is the mean average of the middle two numbers, .

Hence our answer is

Janice's course score in her statistics class depends on her six hourly tests. The mean of her best five scores and the median of her best five scores are both calculated, and she is assigned the better of the two.

Janice's first five tests were 90, 92, 80, 75, and 86. What score, at minimum, does she need to be assured a course score of 85 or better for the term?

Janice is already assured of a course average of at least 85 for the term.

Explanation

The median of her current five scores is the third-highest, or 86. Even if she does not take the sixth examination, this median will stand, and even if her mean is less, she has already achieved a score of 86 or better.

Find the median of the following data set:

Explanation

Find the median of the following data set:

To find median, we must first arrange our terms in ascending order.

So...

Now, our median will be the term directly in the middle of our data set. In this case, the median is 117.

What do you need to know in order to determine the median of a data set with one hundred elements?

Both the fiftieth-highest and fiftieth-lowest elements

Both the highest and lowest elements

The fiftieth-highest element

The fiftieth-lowest element

All one hundred elements

Explanation

The median of a dataset with an even number of elements is the arithmetic mean of the two elements that fall in the middle when the elements are arranged in ascending order. Since there are 100 elements and $100 \div 2 =50$ , this means the fiftieth-highest and fiftieth-lowest elements.

Consider the data set

$\left { 50, 55, 59, 66, 72, 74, 77, M, N\right }$

For the median to be 70, what must be true about and ?

One must be equal to 70; the other must be greater than or equal to 70.

One must be equal to 70; the other must be less than or equal to 70.

One must be equal to 70; the other must be greater than 70.

One must be equal to 70; the other must be less than 70.

Both must be equal to 70.

Explanation

There are nine elements, so the median is the fifth-highest element. For this fifth-highest element to be 70, first of all, 70 must be in the set; since none of the known elements are equal to 70, then one of the two unknowns must be 70.

Assume without loss of generality that . Then four of the elements are already known to be less than 70. Since four elements must be greater than or equal to 70, must be one of them.