Median - ISEE Upper Level Mathematics Achievement

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Question

The following are the scores from a math test in a given classroom. What is the median score?

$\small \left { 70,89,67,77,92,83,68,75\right }$

Answer

To find the median you need to arrange the values in numerical order.

Starting with this:

Rearrange to look like this:

$\small \small \left { 67,68,70,75,77,83,89,92\right }$

If there are an odd number of values, the median is the middle value. In this case there are 8 values so the median is the average (or mean) of the two middle values.

$\small (75+77)\div2=76$

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Question

The median of nine consecutive integers is 604. What is the greatest integer?

Answer

The median of nine (an odd number) integers is the one in the middle when the numbers are arranged in ascending order; in this case, it is the fifth lowest. Since the nine integers are consecutive, the greatest integer is four more than the median, or .

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Question

Scores from a math test in a given classroom are as follows:

$\left { 80, 84, 86, 66, 57, 69, 92, 90 \right }$

What is the median score?

Answer

In order to find the median the data must first be ordered. So we have:

$\left { 57,66,69,80,84,86,90,92 \right }$

In this problem the number of values is even. We know that when the number of values is even, the median is the mean of the two middle values. So we get:

$Median=\frac{80+84}{2}=82$

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Question

Heights of a group of students in a high school are as follows (heights are given in ):

$\left { 176,180,181,182,168,174,169,188,165 \right }$

Find the median height.

Answer

In order to find the median the data must first be ordered. So we have:

$\left { 165,168,169,174,176,180,181,182,188 \right }$

When the number of values is odd, the median is the single middle value. In this problem we have nine values. So the median is th value which is .

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Question

Find the median in the following set of data:

$\left { 2,3,4,1,2,4,3,7,3,4 \right }$

Answer

In order to find the median, the data must first be ordered. So we should write:

$\left { 1,2,2,3,3,3,4,4,4,7 \right }$

When the number of values is even, the median is the mean of the two middle values. In this problem we have values, so the median would be the mean of the and values:

$Median=\frac{3+3}{2}=3$

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Question

The median of consecutive integers in a set of data is . What is the smallest integer in the set of data?

Answer

We know that the numbers should be arranged in ascending order to find the median. When the number of values is odd, the median is the single middle value. In this question we have consecutive integers with the median of . So the median is the number in the rearranged data set. Since the integers are consecutive, the smallest integer is five less than the median or it is equal to .

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Question

If is a real number, find the median in the following set of data in terms of .

$\left { t,t+4,t+2,t+5,t+1,t+8 \right }$

Answer

The data should first be ordered:

$\left { t, t+1, t+2, t+4, t+5, t+8 \right }$

When the number of values is even, the median is the mean of the two middle values. So in this problem we need to find the mean of the and values:

$Median=\frac{(t+2)+(t+4)}{2}=\frac{2t+6}{2}=t+3$

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Question

The heights of the members of a basketball team are inches. The mean of the heights is $78\ in$ . Give the median of the heights.

Answer

The mean is the sum of the data values divided by the number of values or as a formula we have:

$\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}$

Where:

$\bar{x}$ is the mean of a data set, $\sum$ indicates the sum of the data values $x_{i}$ and is the number of data values. So we can write:

$\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}=\frac{(64+78+76+80+82+83+75+x)}{8}$

$\Rightarrow \bar{x}=\frac{538+x}{8}=78\Rightarrow 538+x=78\times 8$

$\Rightarrow 538+x=624\Rightarrow x=624-538=86\ in$

In order to find the median, the data must first be ordered:

$\left { 64,75,76,78,80,82,83,86 \right }$

Since the number of values is even, the median is the mean of the two middle values. So we get:

$Median=\frac{78+80}{2}=79\ in$

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Question

Consider the data set

$\left { 24,61, 27, 57, 34, 37, 63, 50, N\right }$ .

For what value(s) of would this set have median ?

Answer

Arrange the eight known values from least to greatest.

$\left { 24, 27, 34, 37, 50, 57,61, 63\right }$

For to be the median of the nine elements, it muct be the fifth-greatest, This happens if $N \geq 50$ .

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Question

What is the median of the frequency distribution shown in the table:

$\begin{matrix} Data \ Value & Frequency \ 12 & 4\ 14& 7\ 15& 3\ 18& 2 \end{matrix}$

Answer

There are data values altogether. When the number of values is even, the median is the mean of the two middle values. So in this problem the median is the mean of the and largest values. So we can write:

$8th\ largest \ value=14$

$9th\ largest \ value=14$

So:

$Median=\frac{14+14}{2}=14$

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Question

Give the median of the frequency distribution shown in the following table:

$\begin{matrix} Data \ Value & Frequency \ 120 & 7\ 142& 3\ 160& 4\ 174& 6 \end{matrix}$

Answer

There are data values altogether. When the number of values is even, the median is the mean of the two middle values. So in this problem the median is the mean of the and largest values. So we can write:

$10th\ largest \ value=142$

$11th\ largest \ value=160$

So:

$Median=\frac{142+160}{2}=151$

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Question

Consider the data set: $\left { 12, 15, 18, 20, 20, 21, 23, 26, N\right }$

where is not known.

What are the possible values of the median of this set?

Answer

The median of this nine-element set is its fifth-highest element. Of the eight known elements, the fourth-highest and fifth-highest elements are both 20. Regardless of the value of , 20 is the fifth-highest element of the nine.

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Question

Determine the median of the following seven test scores:

Answer

To determine the median of a set of numbers, you first need to order them from least to greatest:

Since there is an odd number of scores, the median is the score that falls exactly in the middle of the new list. Thus, the median is 88.

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Question

Determine the median of the following set of numbers:

Answer

To determine the median of a set of numbers, you first need to order them from least to greatest:

Since there is an even amount of numbers, the median is determined by finding the average of the two numbers in the middle - 36 and 44.

$36+44 = 80 \div 2 = 40$

Thus, the median is 40.

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Question

Examine this stem-and-leaf display for a set of data:

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

What is the median of this data set?

Answer

The "stem" of this data set represents the tens digits of the data values; the "leaves" represent the units digits.

There are 22 elements, so the median is the arithmetic mean of the eleventh- and twelfth-highest elements, which are 64 and 65, the middle two "leaves". Their mean is $\left (64 + 65 \right )\div 2 = 64.5$ .

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Question

Find the median of the following numbers:

Answer

The median is the center number when the data points are listed in ascending or descending order. To find the median, reorder the values in numerical order:

In this problem, the middle number, or median, is the third number, which is

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Question

What is the median of the following set?

$\left ( 5.45, 3.12, 5.7,9.86, 14.78, 3.35\right )$

Answer

The first step towards solving for the set, $\left ( 5.45, 3.12, 5.7,9.86, 14.78, 3.35\right )$ is to reorder the numbers from smallest to largest.

This gives us:

$\left ( 3.12,3.35, 5.45, 5.7,9.86, 14.78\right )$

The median is equal to middle number in s a set. In since this set has 6 numbers, which is even, the average of the middle two numbers is the mean. The average can be found using the equation below:

$\frac{5.45+5.7}{2}$

$\frac{11.15}{2}$

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Question

Find the median of the following data set:

Answer

Find the median of the following data set:

Begin by putting your numbers in increasing order:

Next, identify the median by choosing the middle value:

$12,34,45,49,{\color{Green} 55},67,67,88,343$

So, our answer is 55

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Question

Find the median of the following data set:

Answer

Find the median of the following data set:

Let's begin by rearranging our terms from least to greatest:

Now, the median will be the middle term:

$5,9,12,33,34,{\color{DarkOrange} 55},67,76,76,89,109$

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Question

Find the median of the following data set:

Answer

Find the median of the following data set:

First, let's put our terms in increasing order:

Now, we can find our median simply by choosing the middle term.

$12,33,33,34,43,{\color{Blue} 56},76,77,93,98,99$

So, 56 is our median.

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