ISEE Upper Level Quantitative Reasoning - How to find median

Find the median of the following data set:

Explanation

Find the median of the following data set:

To find the median, first put the numbers in increasing order

Now, identify the median by choosing the middle term

$12,12,12,33,44,{\color{Green} 44},55,65,65,79,99$

In this case, it is 44, because 44 is in the middle of all our terms.

Use the following data set to answer the question:

Find the median.

Explanation

To find the median of a data set, we will first arrange the numbers in ascending order. Then, we will locate the number in the center of the data set.

So, given the data set

we will arrange the numbers in ascending order. To do that, we will arrange them from smallest to largest. So, we get

Now, we will locate the number in the center of the data set.

$3, 3, 4, 5, {\color{Red} 6}, 6, 6, 8, 8$

We can see that it is 6.

Therefore, the median of the data set is 6.

Which is the greater quantity?

(a) The mean of the first ten prime numbers

(b) The median of the first ten prime numbers

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Explanation

The first ten primes form the data set:

$\left { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29\right }$

(a) Add these primes, and divide by :

$\left ( 2 + 3+5+ 7+ 11+13+17+19+ 23+29 \right )\div 10 = 129\div 10= 12.9$

(b) The median of a data set with ten elements is the arithmetic mean of the fifth-highest and sixth-highest elements. These are and , so the median is

$\left (11 + 13 \right ) \div 2 = 12$ .

(a) is the greater quantity.

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 ?

Any number greater than or equal to

Any number greater than

Any number less than or equal to

Any number less than

Any number except

Explanation

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 $N \geq 50$ .

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

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

Explanation

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:

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

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

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

Explanation

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:

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

Find the median of the following numbers:

Explanation

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

Use the following data set to answer the question:

Find the median.

Explanation

To find the median of a data set, we will first arrange the data set in ascending order. Then, we will find the number that is located in the middle of the set.

So, given the set

we will arrange the set in ascending order (from smallest to largest). We get

Now, we will locate the number in the middle of the set.

$2, 4, 5, 6, {\color{Red} 6}, 6, 7, 9, 9$

We can see that it is 6.

Therefore, the median of the data set is 6.

Which is the greater quantity?

(a) The median of the data set $\left {8, 8, 9, 9, 10, 10, 11, 11,12, 12\right }$

(b) The median of the data set $\left {1,1, 2,2, 10, 10, 11, 11,12, 12\right }$

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

Explanation

Each data set has ten elements, so the median in each case is the arithmetic mean of the fifth-highest and sixth-highest elements. In each data set, these elements are 10 and 10, so the median of each set is 10. Therefore, both quantities are equal.

In the following set of data the mean and the mode are equal. Find the median.

$\left { 1,3,3,3,N,4,8 \right }$

Explanation

The mode of a set of data is the value which occurs most frequently which is in this problem (it is not dependent on in this problem). So the mean is also equal to .