SAT II Math I : Quartiles and Interquartile Range

Example Questions

Example Question #1 : Quartiles And Interquartile Range

What is the 3rd quartile of this set?

Explanation:

This first step is to find the median which is  since it is the middle number of the 11 number set.

To find the 3rd quartile, you find the middle number of the set of numbers above the median.

For this set those numbers would be .

The middle number for this new set, which is the 3rd quartile, is .

Example Question #1 : How To Find Interquartile Range

Given the following set of data, what is twice the interquartile range?

Explanation:

How do you find the interquartile range?

We can find the interquartile range or IQR in four simple steps:

1.      Order the data from least to greatest
2.      Find the median
3.      Calculate the median of both the lower and upper half of the data
4.      The IQR is the difference between the upper and lower medians

Step 1: Order the data

In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest).

Let's sort an example data set with an odd number of values into ascending order.

Now, let's perform this task with another example data set that is comprised of an even number of values.

Rearrange into ascending order.

Step 2: Calculate the median

Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two.

First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point

The median of the odd valued data set is four.

Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values.

Find the average of the two centermost values.

The median of the even valued set is four.

Step 3: Upper and lower medians

Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data.

Omit the centermost value.

Find the median of the lower portion.

Calculate the average of the two values.

The median of the lower portion is

Find the median of the upper portion.

Calculate the average of the two values.

The median of the upper potion is

If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions.

Find the median of the lower portion.

The median of the lower portion is two.

Find the median of the upper portion.

The median of the upper portion is eight.

Step 4: Calculate the difference

Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR.

Let's find the IQR of the odd data set.

Finally, we will find the IQR of the even data set.

In order to better illustrate these values, their positions in a box plot have been labeled in the provided image.

Now that we have solved a few examples, let's use this knowledge to solve the given problem.

Solution:

First, we need to put the data in order from smallest to largest.

= median of the overall data set

= median of the lower half of the data

= median of the upper half of the data

is the overall median, leaving as the lower half of the data and as the upper half of the data.

The median of the lower half falls between two values.

The median of the upper half falls between two values.

The interquartile range is the difference between the third and first quartiles.

Multiply by  to find the answer:

Example Question #1 : How To Find Interquartile Range

Determine the interquartile range of the following numbers:

42, 51, 62, 47, 38, 50, 54, 43

None of these

Explanation:

How do you find the interquartile range?

We can find the interquartile range or IQR in four simple steps:

1.      Order the data from least to greatest
2.      Find the median
3.      Calculate the median of both the lower and upper half of the data
4.      The IQR is the difference between the upper and lower medians

Step 1: Order the data

In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest).

Let's sort an example data set with an odd number of values into ascending order.

Now, let's perform this task with another example data set that is comprised of an even number of values.

Rearrange into ascending order.

Step 2: Calculate the median

Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two.

First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point

The median of the odd valued data set is four.

Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values.

Find the average of the two centermost values.

The median of the even valued set is four.

Step 3: Upper and lower medians

Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data.

Omit the centermost value.

Find the median of the lower portion.

Calculate the average of the two values.

The median of the lower portion is

Find the median of the upper portion.

Calculate the average of the two values.

The median of the upper potion is

If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions.

Find the median of the lower portion.

The median of the lower portion is two.

Find the median of the upper portion.

The median of the upper portion is eight.

Step 4: Calculate the difference

Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR.

Let's find the IQR of the odd data set.

Finally, we will find the IQR of the even data set.

In order to better illustrate these values, their positions in a box plot have been labeled in the provided image.

Now that we have solved a few examples, let's use this knowledge to solve the given problem.

Solution:

First reorder the numbers in ascending order:

38, 42, 43, 47, 50, 51, 54, 62

Then divide the numbers into 2 groups, each containing an equal number of values:

(38, 42, 43, 47)(50, 51, 54, 62)

Q1 is the median of the group on the left, and Q3 is the median of the group on the right. Because there is an even number in each group, we'll need to find the average of the 2 middle numbers:

The interquartile range is the difference between Q3 and Q1:

Example Question #2 : How To Find Interquartile Range

The interquartile range is the difference in value between the upper quartile and lower quartile.

Find the interquartile range for the data set.

Explanation:

How do you find the interquartile range?

We can find the interquartile range or IQR in four simple steps:

1.      Order the data from least to greatest
2.      Find the median
3.      Calculate the median of both the lower and upper half of the data
4.      The IQR is the difference between the upper and lower medians

Step 1: Order the data

In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest).

Let's sort an example data set with an odd number of values into ascending order.

Now, let's perform this task with another example data set that is comprised of an even number of values.

Rearrange into ascending order.

Step 2: Calculate the median

Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two.

First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point

The median of the odd valued data set is four.

Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values.

Find the average of the two centermost values.

The median of the even valued set is four.

Step 3: Upper and lower medians

Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data.

Omit the centermost value.

Find the median of the lower portion.

Calculate the average of the two values.

The median of the lower portion is

Find the median of the upper portion.

Calculate the average of the two values.

The median of the upper potion is

If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions.

Find the median of the lower portion.

The median of the lower portion is two.

Find the median of the upper portion.

The median of the upper portion is eight.

Step 4: Calculate the difference

Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR.

Let's find the IQR of the odd data set.

Finally, we will find the IQR of the even data set.

In order to better illustrate these values, their positions in a box plot have been labeled in the provided image.

Now that we have solved a few examples, let's use this knowledge to solve the given problem.

Solution:

As always, rearranging the data set helps us immensely:

-->

To find , we use the equation , where  is the number of points in our data set.

, so our lower quartile is the  number in the set, or .

To find , we use the equation  instead.

, so our upper quartile is the  number in the set, or .

Lastly, our interquartile range is , so  is our interquartile range.

Example Question #2 : How To Find Interquartile Range

The interquartile range is the difference in value between the upper quartile and lower quartile.

Find the interquartile range of the following data set:

Explanation:

How do you find the interquartile range?

We can find the interquartile range or IQR in four simple steps:

1.      Order the data from least to greatest
2.      Find the median
3.      Calculate the median of both the lower and upper half of the data
4.      The IQR is the difference between the upper and lower medians

Step 1: Order the data

In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest).

Let's sort an example data set with an odd number of values into ascending order.

Now, let's perform this task with another example data set that is comprised of an even number of values.

Rearrange into ascending order.

Step 2: Calculate the median

Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two.

First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point

The median of the odd valued data set is four.

Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values.

Find the average of the two centermost values.

The median of the even valued set is four.

Step 3: Upper and lower medians

Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data.

Omit the centermost value.

Find the median of the lower portion.

Calculate the average of the two values.

The median of the lower portion is

Find the median of the upper portion.

Calculate the average of the two values.

The median of the upper potion is

If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions.

Find the median of the lower portion.

The median of the lower portion is two.

Find the median of the upper portion.

The median of the upper portion is eight.

Step 4: Calculate the difference

Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR.

Let's find the IQR of the odd data set.

Finally, we will find the IQR of the even data set.

In order to better illustrate these values, their positions in a box plot have been labeled in the provided image.

Now that we have solved a few examples, let's use this knowledge to solve the given problem.

Solution:

The first step (as with most data set problems) is to rearrange the data set from least to greatest value:

To find the lower quartile ()'s position, we use the equation , where n is the number of data points in the set.

Thus, our lower quartile is at  position. Since this is a non-integer, we must include a further equation.

Since our 3rd number is 2, and our 4th number is 3, we need to find 1/4 of the way between 2 and 3. We will use the equation

, where  is our lower value,  is our upper value, and  is the fractional distance between them we need to find. Since  is  here, our equation becomes

Thus, our

We can repeat the process above to find the upper quartile (), but we must multiply our original equation by 3. Thus, our equation becomes

, where  is still the number of data points in the set.  will be at the

position, so we must find the value  of the way between the  and  data point in the set. With  and  as our values for the  and  data point, our equation becomes

So, our .

The last step is easy by comparison. Subtract  from  to get our interquartile range:

Thus, our interquartile range is .

Example Question #1 : How To Find Interquartile Range

Using the data provided above, what is the interquartile range (IQR)?

Explanation:

How do you find the interquartile range?

We can find the interquartile range or IQR in four simple steps:

1.      Order the data from least to greatest
2.      Find the median
3.      Calculate the median of both the lower and upper half of the data
4.      The IQR is the difference between the upper and lower medians

Step 1: Order the data

In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest).

Let's sort an example data set with an odd number of values into ascending order.

Now, let's perform this task with another example data set that is comprised of an even number of values.

Rearrange into ascending order.

Step 2: Calculate the median

Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two.

First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point

The median of the odd valued data set is four.

Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values.

Find the average of the two centermost values.

The median of the even valued set is four.

Step 3: Upper and lower medians

Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data.

Omit the centermost value.

Find the median of the lower portion.

Calculate the average of the two values.

The median of the lower portion is

Find the median of the upper portion.

Calculate the average of the two values.

The median of the upper potion is

If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions.

Find the median of the lower portion.

The median of the lower portion is two.

Find the median of the upper portion.

The median of the upper portion is eight.

Step 4: Calculate the difference

Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR.

Let's find the IQR of the odd data set.

Finally, we will find the IQR of the even data set.

In order to better illustrate these values, their positions in a box plot have been labeled in the provided image.

Now that we have solved a few examples, let's use this knowledge to solve the given problem.

Solution:

To find the IQR, we first must find the  and , because .

In data sets,  is defined as the median of the top half of the data and  is defined as the median of the bottom half of the data. In a previous problem, we placed the data pieces in numerical order:

,

and found  to be the median or center of the data:

.

Our upper half of our data set, the numbers above our median, now consists of . The median, or middle number, of this upper half is , our . The lower half of data, numbers below our median, is , with  being our median, .

We now have our  and our

Thus our

Example Question #1 : How To Find Interquartile Range

Using the data above, what is the interquartile range?

Explanation:

How do you find the interquartile range?

We can find the interquartile range or IQR in four simple steps:

1.      Order the data from least to greatest
2.      Find the median
3.      Calculate the median of both the lower and upper half of the data
4.      The IQR is the difference between the upper and lower medians

Step 1: Order the data

In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest).

Let's sort an example data set with an odd number of values into ascending order.

Now, let's perform this task with another example data set that is comprised of an even number of values.

Rearrange into ascending order.

Step 2: Calculate the median

Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two.

First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point

The median of the odd valued data set is four.

Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values.

Find the average of the two centermost values.

The median of the even valued set is four.

Step 3: Upper and lower medians

Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data.

Omit the centermost value.

Find the median of the lower portion.

Calculate the average of the two values.

The median of the lower portion is

Find the median of the upper portion.

Calculate the average of the two values.

The median of the upper potion is

If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions.

Find the median of the lower portion.

The median of the lower portion is two.

Find the median of the upper portion.

The median of the upper portion is eight.

Step 4: Calculate the difference

Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR.

Let's find the IQR of the odd data set.

Finally, we will find the IQR of the even data set.

In order to better illustrate these values, their positions in a box plot have been labeled in the provided image.

Now that we have solved a few examples, let's use this knowledge to solve the given problem.

Solution:

To find the IQR, we first must find the  and , because .

In data sets,  is defined as the median of the top half of data and  is defined as the median of the bottom half of the data.

In a previous problem, we placed the data pieces in numerical order:

and found  to be the median or center of the data:

.

Our upper half of our data set, the numbers above our median, now consists of . The median, or middle number, of this upper half is , our . The lower half of data, numbers below our median, is  with  being our median, .

We now have our  and our

Thus our . Note that although we have an outlier of , our . Therefore, we can observe that an outlier's effect on a data set is not very strong when finding the interquartile range.

Example Question #1 : How To Find Interquartile Range

, Median , Highest value is , Lowest value is

Using the data provided, find the Interquartile range, IQR.

Explanation:

How do you find the interquartile range?

We can find the interquartile range or IQR in four simple steps:

1.      Order the data from least to greatest
2.      Find the median
3.      Calculate the median of both the lower and upper half of the data
4.      The IQR is the difference between the upper and lower medians

Step 1: Order the data

In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest).

Let's sort an example data set with an odd number of values into ascending order.

Now, let's perform this task with another example data set that is comprised of an even number of values.

Rearrange into ascending order.

Step 2: Calculate the median

Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two.

First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point

The median of the odd valued data set is four.

Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values.

Find the average of the two centermost values.

The median of the even valued set is four.

Step 3: Upper and lower medians

Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data.

Omit the centermost value.

Find the median of the lower portion.

Calculate the average of the two values.

The median of the lower portion is

Find the median of the upper portion.

Calculate the average of the two values.

The median of the upper potion is

If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions.

Find the median of the lower portion.

The median of the lower portion is two.

Find the median of the upper portion.

The median of the upper portion is eight.

Step 4: Calculate the difference

Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR.

Let's find the IQR of the odd data set.

Finally, we will find the IQR of the even data set.

In order to better illustrate these values, their positions in a box plot have been labeled in the provided image.

Now that we have solved a few examples, let's use this knowledge to solve the given problem.

Solution:

The data set provided is called a five number summary.

These data values allow us to find the median, IQR, and range.

This question is asking for the IQR which is , which is

Example Question #1 : How To Find Interquartile Range

Using the data above, find the interquartile range.

Explanation:

How do you find the interquartile range?

We can find the interquartile range or IQR in four simple steps:

1.      Order the data from least to greatest
2.      Find the median
3.      Calculate the median of both the lower and upper half of the data
4.      The IQR is the difference between the upper and lower medians

Step 1: Order the data

In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest).

Let's sort an example data set with an odd number of values into ascending order.

Now, let's perform this task with another example data set that is comprised of an even number of values.

Rearrange into ascending order.

Step 2: Calculate the median

Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two.

First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point

The median of the odd valued data set is four.

Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values.

Find the average of the two centermost values.

The median of the even valued set is four.

Step 3: Upper and lower medians

Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data.

Omit the centermost value.

Find the median of the lower portion.

Calculate the average of the two values.

The median of the lower portion is

Find the median of the upper portion.

Calculate the average of the two values.

The median of the upper potion is

If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions.

Find the median of the lower portion.

The median of the lower portion is two.

Find the median of the upper portion.

The median of the upper portion is eight.

Step 4: Calculate the difference

Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR.

Let's find the IQR of the odd data set.

Finally, we will find the IQR of the even data set.

In order to better illustrate these values, their positions in a box plot have been labeled in the provided image.

Now that we have solved a few examples, let's use this knowledge to solve the given problem.

Solution:

The Interquartile range, or IQR, is defined as the .

The first step is the find the median of the data set, which in this case is . This number is what cuts the data set into two smaller sets, an upper quartile and lower quartile.

is the median of the upper quartile, while  is the median of the lower quartile.

For the upper quartile, if placed in numerical order

we see that there is an even number, thus we must take the center two numbers and find the average to find the true center of this data set, giving us

, so our

.

We do the same for the lower quartile, giving us a

.

When we subtract  from  we end up with  as our IQR.

Example Question #1 : How To Find Interquartile Range

Using the data above, find the IQR. (interquartile range)

Explanation:

How do you find the interquartile range?

We can find the interquartile range or IQR in four simple steps:

1.      Order the data from least to greatest
2.      Find the median
3.      Calculate the median of both the lower and upper half of the data
4.      The IQR is the difference between the upper and lower medians

Step 1: Order the data

In order to calculate the IQR, we need to begin by ordering the values of the data set from the least to the greatest. Likewise, in order to calculate the median, we need to arrange the numbers in ascending order (i.e. from the least to the greatest).

Let's sort an example data set with an odd number of values into ascending order.

Now, let's perform this task with another example data set that is comprised of an even number of values.

Rearrange into ascending order.

Step 2: Calculate the median

Next, we need to calculate the median. The median is the "center" of the data. If the data set has an odd number of data points, then the mean is the centermost number. On the other hand, if the data set has an even number of values, then we will need to take the arithmetic average of the two centermost values. We will calculate this average by adding the two numbers together and then dividing that number by two.

First, we will find the median of a set with an odd number of values. Cross out values until you find the centermost point

The median of the odd valued data set is four.

Now, let's find the mean of the data set with an even number of values. Cross out values until you find the two centermost points and then calculate the average the two values.

Find the average of the two centermost values.

The median of the even valued set is four.

Step 3: Upper and lower medians

Once we have found the median of the entire set, we can find the medians of the upper and lower portions of the data. If the data set has an odd number of values, we will omit the median or centermost value of the set. Afterwards, we will find the individual medians for the upper and lower portions of the data.

Omit the centermost value.

Find the median of the lower portion.

Calculate the average of the two values.

The median of the lower portion is

Find the median of the upper portion.

Calculate the average of the two values.

The median of the upper potion is

If the data set has an even number of values, we will use the two values used to calculate the original median to divide the data set. These values are not omitted and become the largest value of the lower data set and the lowest values of the upper data set, respectively. Afterwards, we will calculate the medians of both the upper and lower portions.

Find the median of the lower portion.

The median of the lower portion is two.

Find the median of the upper portion.

The median of the upper portion is eight.

Step 4: Calculate the difference

Last, we need to calculate the difference of the upper and lower medians by subtracting the lower median from the upper median. This value equals the IQR.

Let's find the IQR of the odd data set.

Finally, we will find the IQR of the even data set.

In order to better illustrate these values, their positions in a box plot have been labeled in the provided image.

Now that we have solved a few examples, let's use this knowledge to solve the given problem.

Solution:

To find the IQR, we must first find the  and  .

The  is the median of the upper quartile, the numbers above the median:

,

thus the  is

is the median of the lower quartile:

,

thus the  is  .

So,