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. 

\(\displaystyle \textup{Odd data set}: 9, 3, 2, 5, 6, 11, 4, 3, 2\)

\(\displaystyle \textup{Odd data set (ascending)}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

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

\(\displaystyle \textup{Even data set}: 11, 2, 4, 3, 8, 1, 2, 7, 4, 9\)

Rearrange into ascending order.

\(\displaystyle \textup{Even data set (ascending)}: 1, 2, 2, 3, 4, 4, 7, 8, 9, 11\)

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

\(\displaystyle \textup{Odd data set}: \not{2}, \not{2}, \not{3}, \not{3}, {\color{Red} 4}, \not{5}, \not{6}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, \not{2}, \not{3}, {\color{Red} 4}, {\color{Red} 4}, \not{7}, \not{8}, \not{9}, \not{11}\)

Find the average of the two centermost values.

\(\displaystyle \textup{Average}=\frac{4+4}{2}\)

\(\displaystyle \textup{Average}=\frac{8}{2}\)

\(\displaystyle \textup{Average}=4\)

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.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

Omit the centermost value.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid 5, 6, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Odd data set}: \not{2}, {\color{Red} 2}, {\color{Red} 3}, \not{3},\mid 5, 6, 9, 11\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{2+3}{2}\)

\(\displaystyle \textup{Average}=\frac{5}{2}\)

\(\displaystyle \textup{Average}=2.5\)

The median of the lower portion is \(\displaystyle 2.5\)

Find the median of the upper portion.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid \not{5}, {\color{Red} 6}, {\color{Red} 9}, \not{11}\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{6+9}{2}\)

\(\displaystyle \textup{Average}=\frac{15}{2}\)

\(\displaystyle \textup{Average}=7.5\)

The median of the upper potion is \(\displaystyle 7.5\)

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.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid 4, 7, 8, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, {\color{Red} 2}, \not{3}, \not{4}\mid 4, 7, 8, 9, 11\)

The median of the lower portion is two.

Find the median of the upper portion.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid \not{4}, \not{7}, {\color{Red} 8}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{IQR of the odd data set}=7.5-2.5\)

\(\displaystyle \textup{IQR}=5\)

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

\(\displaystyle \textup{IQR of the even data set}=8-2\)

\(\displaystyle \textup{IQR}=6\)

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 \(\displaystyle \small Q_{3}\) and \(\displaystyle \small Q_{1}\), because \(\displaystyle \small IQR = Q_{3} - Q_{1}\).

In data sets, \(\displaystyle \small Q_{3}\) is defined as the median of the top half of the data and \(\displaystyle \small Q_{1}\) is defined as the median of the bottom half of the data. In a previous problem, we placed the data pieces in numerical order: 

\(\displaystyle \small 3, 4, 5, 10, 10, 22, 23\),

and found \(\displaystyle \small 10\) to be the median or center of the data:

 \(\displaystyle \small 3, 4, 5, \textbf{10}, 10, 22, 23\).

Our upper half of our data set, the numbers above our median, now consists of \(\displaystyle \small 10, 22, 23\). The median, or middle number, of this upper half is \(\displaystyle 22\), our \(\displaystyle \small Q_{3}\). The lower half of data, numbers below our median, is \(\displaystyle \small 3, 4, 5\), with \(\displaystyle \small 4\) being our median, \(\displaystyle \small Q_{1}\).

We now have our \(\displaystyle \small \small Q_{3} = 22\) and our \(\displaystyle \small Q_{1} = 4\). 

\(\displaystyle \small 22 - 4 = 18.\) 

Thus our \(\displaystyle \small IQR = 18.\)

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. 

\(\displaystyle \textup{Odd data set}: 9, 3, 2, 5, 6, 11, 4, 3, 2\)

\(\displaystyle \textup{Odd data set (ascending)}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

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

\(\displaystyle \textup{Even data set}: 11, 2, 4, 3, 8, 1, 2, 7, 4, 9\)

Rearrange into ascending order.

\(\displaystyle \textup{Even data set (ascending)}: 1, 2, 2, 3, 4, 4, 7, 8, 9, 11\)

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

\(\displaystyle \textup{Odd data set}: \not{2}, \not{2}, \not{3}, \not{3}, {\color{Red} 4}, \not{5}, \not{6}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, \not{2}, \not{3}, {\color{Red} 4}, {\color{Red} 4}, \not{7}, \not{8}, \not{9}, \not{11}\)

Find the average of the two centermost values.

\(\displaystyle \textup{Average}=\frac{4+4}{2}\)

\(\displaystyle \textup{Average}=\frac{8}{2}\)

\(\displaystyle \textup{Average}=4\)

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.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

Omit the centermost value.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid 5, 6, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Odd data set}: \not{2}, {\color{Red} 2}, {\color{Red} 3}, \not{3},\mid 5, 6, 9, 11\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{2+3}{2}\)

\(\displaystyle \textup{Average}=\frac{5}{2}\)

\(\displaystyle \textup{Average}=2.5\)

The median of the lower portion is \(\displaystyle 2.5\)

Find the median of the upper portion.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid \not{5}, {\color{Red} 6}, {\color{Red} 9}, \not{11}\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{6+9}{2}\)

\(\displaystyle \textup{Average}=\frac{15}{2}\)

\(\displaystyle \textup{Average}=7.5\)

The median of the upper potion is \(\displaystyle 7.5\)

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.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid 4, 7, 8, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, {\color{Red} 2}, \not{3}, \not{4}\mid 4, 7, 8, 9, 11\)

The median of the lower portion is two.

Find the median of the upper portion.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid \not{4}, \not{7}, {\color{Red} 8}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{IQR of the odd data set}=7.5-2.5\)

\(\displaystyle \textup{IQR}=5\)

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

\(\displaystyle \textup{IQR of the even data set}=8-2\)

\(\displaystyle \textup{IQR}=6\)

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 \(\displaystyle \small Q_{3}\) and \(\displaystyle \small Q_{1}\), because \(\displaystyle \small IQR = Q_{3} - Q_{1}\).

In data sets, \(\displaystyle \small Q_{3}\) is defined as the median of the top half of data and \(\displaystyle \small Q_{1}\) is defined as the median of the bottom half of the data.

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

\(\displaystyle \small 1,2,3,4,5,6,7,8,9,10, 100\) 

and found \(\displaystyle \small 6\) to be the median or center of the data:  

\(\displaystyle \small \small 1,2,3,4,5,\mathbf{6},7,8,9,10, 100\).

Our upper half of our data set, the numbers above our median, now consists of \(\displaystyle \small \small 7,8,9,10,100\). The median, or middle number, of this upper half is \(\displaystyle \small 9\), our \(\displaystyle \small Q_{3}\). The lower half of data, numbers below our median, is \(\displaystyle \small \small 1,2,3,4,5\) with \(\displaystyle \small 3\) being our median, \(\displaystyle \small Q_{1}\).

We now have our \(\displaystyle \small \small \small Q_{3} = 9\) and our \(\displaystyle \small \small Q_{1} = 3\). 

\(\displaystyle \small \small 9-3=6\) 

Thus our \(\displaystyle \small \small IQR = 6\). Note that although we have an outlier of \(\displaystyle \small 100\), our \(\displaystyle \small IQR = 6\). Therefore, we can observe that an outlier's effect on a data set is not very strong when finding the interquartile range. 

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. 

\(\displaystyle \textup{Odd data set}: 9, 3, 2, 5, 6, 11, 4, 3, 2\)

\(\displaystyle \textup{Odd data set (ascending)}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

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

\(\displaystyle \textup{Even data set}: 11, 2, 4, 3, 8, 1, 2, 7, 4, 9\)

Rearrange into ascending order.

\(\displaystyle \textup{Even data set (ascending)}: 1, 2, 2, 3, 4, 4, 7, 8, 9, 11\)

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

\(\displaystyle \textup{Odd data set}: \not{2}, \not{2}, \not{3}, \not{3}, {\color{Red} 4}, \not{5}, \not{6}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, \not{2}, \not{3}, {\color{Red} 4}, {\color{Red} 4}, \not{7}, \not{8}, \not{9}, \not{11}\)

Find the average of the two centermost values.

\(\displaystyle \textup{Average}=\frac{4+4}{2}\)

\(\displaystyle \textup{Average}=\frac{8}{2}\)

\(\displaystyle \textup{Average}=4\)

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.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

Omit the centermost value.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid 5, 6, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Odd data set}: \not{2}, {\color{Red} 2}, {\color{Red} 3}, \not{3},\mid 5, 6, 9, 11\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{2+3}{2}\)

\(\displaystyle \textup{Average}=\frac{5}{2}\)

\(\displaystyle \textup{Average}=2.5\)

The median of the lower portion is \(\displaystyle 2.5\)

Find the median of the upper portion.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid \not{5}, {\color{Red} 6}, {\color{Red} 9}, \not{11}\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{6+9}{2}\)

\(\displaystyle \textup{Average}=\frac{15}{2}\)

\(\displaystyle \textup{Average}=7.5\)

The median of the upper potion is \(\displaystyle 7.5\)

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.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid 4, 7, 8, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, {\color{Red} 2}, \not{3}, \not{4}\mid 4, 7, 8, 9, 11\)

The median of the lower portion is two.

Find the median of the upper portion.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid \not{4}, \not{7}, {\color{Red} 8}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{IQR of the odd data set}=7.5-2.5\)

\(\displaystyle \textup{IQR}=5\)

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

\(\displaystyle \textup{IQR of the even data set}=8-2\)

\(\displaystyle \textup{IQR}=6\)

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 \(\displaystyle \small Q_{3}-Q_1\), which is \(\displaystyle \small 67 - 53 = 14\). 

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. 

\(\displaystyle \textup{Odd data set}: 9, 3, 2, 5, 6, 11, 4, 3, 2\)

\(\displaystyle \textup{Odd data set (ascending)}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

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

\(\displaystyle \textup{Even data set}: 11, 2, 4, 3, 8, 1, 2, 7, 4, 9\)

Rearrange into ascending order.

\(\displaystyle \textup{Even data set (ascending)}: 1, 2, 2, 3, 4, 4, 7, 8, 9, 11\)

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

\(\displaystyle \textup{Odd data set}: \not{2}, \not{2}, \not{3}, \not{3}, {\color{Red} 4}, \not{5}, \not{6}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, \not{2}, \not{3}, {\color{Red} 4}, {\color{Red} 4}, \not{7}, \not{8}, \not{9}, \not{11}\)

Find the average of the two centermost values.

\(\displaystyle \textup{Average}=\frac{4+4}{2}\)

\(\displaystyle \textup{Average}=\frac{8}{2}\)

\(\displaystyle \textup{Average}=4\)

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.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

Omit the centermost value.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid 5, 6, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Odd data set}: \not{2}, {\color{Red} 2}, {\color{Red} 3}, \not{3},\mid 5, 6, 9, 11\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{2+3}{2}\)

\(\displaystyle \textup{Average}=\frac{5}{2}\)

\(\displaystyle \textup{Average}=2.5\)

The median of the lower portion is \(\displaystyle 2.5\)

Find the median of the upper portion.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid \not{5}, {\color{Red} 6}, {\color{Red} 9}, \not{11}\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{6+9}{2}\)

\(\displaystyle \textup{Average}=\frac{15}{2}\)

\(\displaystyle \textup{Average}=7.5\)

The median of the upper potion is \(\displaystyle 7.5\)

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.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid 4, 7, 8, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, {\color{Red} 2}, \not{3}, \not{4}\mid 4, 7, 8, 9, 11\)

The median of the lower portion is two.

Find the median of the upper portion.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid \not{4}, \not{7}, {\color{Red} 8}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{IQR of the odd data set}=7.5-2.5\)

\(\displaystyle \textup{IQR}=5\)

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

\(\displaystyle \textup{IQR of the even data set}=8-2\)

\(\displaystyle \textup{IQR}=6\)

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 \(\displaystyle \small Q_{3}- Q_{1}\).

The first step is the find the median of the data set, which in this case is \(\displaystyle 15\). This number is what cuts the data set into two smaller sets, an upper quartile and lower quartile. 

\(\displaystyle Q_{3}\) is the median of the upper quartile, while \(\displaystyle Q_{1}\) is the median of the lower quartile.

For the upper quartile, if placed in numerical order 

\(\displaystyle 16,20, 30, 48\) 

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

\(\displaystyle \small 20 +30 = 50 \div2 = 25\), so our 

\(\displaystyle \small Q_{3} =25\).

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

\(\displaystyle \small Q_{1} = \frac{15}{2}\).

When we subtract \(\displaystyle Q_{1}\) from \(\displaystyle Q_{3}\) we end up with \(\displaystyle \frac{35}{2}\) as our IQR. 

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. 

\(\displaystyle \textup{Odd data set}: 9, 3, 2, 5, 6, 11, 4, 3, 2\)

\(\displaystyle \textup{Odd data set (ascending)}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

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

\(\displaystyle \textup{Even data set}: 11, 2, 4, 3, 8, 1, 2, 7, 4, 9\)

Rearrange into ascending order.

\(\displaystyle \textup{Even data set (ascending)}: 1, 2, 2, 3, 4, 4, 7, 8, 9, 11\)

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

\(\displaystyle \textup{Odd data set}: \not{2}, \not{2}, \not{3}, \not{3}, {\color{Red} 4}, \not{5}, \not{6}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, \not{2}, \not{3}, {\color{Red} 4}, {\color{Red} 4}, \not{7}, \not{8}, \not{9}, \not{11}\)

Find the average of the two centermost values.

\(\displaystyle \textup{Average}=\frac{4+4}{2}\)

\(\displaystyle \textup{Average}=\frac{8}{2}\)

\(\displaystyle \textup{Average}=4\)

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.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

Omit the centermost value.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid 5, 6, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Odd data set}: \not{2}, {\color{Red} 2}, {\color{Red} 3}, \not{3},\mid 5, 6, 9, 11\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{2+3}{2}\)

\(\displaystyle \textup{Average}=\frac{5}{2}\)

\(\displaystyle \textup{Average}=2.5\)

The median of the lower portion is \(\displaystyle 2.5\)

Find the median of the upper portion.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid \not{5}, {\color{Red} 6}, {\color{Red} 9}, \not{11}\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{6+9}{2}\)

\(\displaystyle \textup{Average}=\frac{15}{2}\)

\(\displaystyle \textup{Average}=7.5\)

The median of the upper potion is \(\displaystyle 7.5\)

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.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid 4, 7, 8, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, {\color{Red} 2}, \not{3}, \not{4}\mid 4, 7, 8, 9, 11\)

The median of the lower portion is two.

Find the median of the upper portion.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid \not{4}, \not{7}, {\color{Red} 8}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{IQR of the odd data set}=7.5-2.5\)

\(\displaystyle \textup{IQR}=5\)

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

\(\displaystyle \textup{IQR of the even data set}=8-2\)

\(\displaystyle \textup{IQR}=6\)

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 \(\displaystyle \small Q_{3}\) and \(\displaystyle \small Q_{1}\) .

The \(\displaystyle \small Q_{3}\) is the median of the upper quartile, the numbers above the median: 

\(\displaystyle \small 3,3,9\),

thus the \(\displaystyle \small Q_{3}\) is \(\displaystyle \small 3.\) 

\(\displaystyle \small Q_{1}\) is the median of the lower quartile: 

\(\displaystyle \small 1,1,2\),

thus the \(\displaystyle \small Q_{1}\) is \(\displaystyle \small 1\) .

So, \(\displaystyle \small IQR = 3-1 = 2\). 

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. 

\(\displaystyle \textup{Odd data set}: 9, 3, 2, 5, 6, 11, 4, 3, 2\)

\(\displaystyle \textup{Odd data set (ascending)}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

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

\(\displaystyle \textup{Even data set}: 11, 2, 4, 3, 8, 1, 2, 7, 4, 9\)

Rearrange into ascending order.

\(\displaystyle \textup{Even data set (ascending)}: 1, 2, 2, 3, 4, 4, 7, 8, 9, 11\)

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

\(\displaystyle \textup{Odd data set}: \not{2}, \not{2}, \not{3}, \not{3}, {\color{Red} 4}, \not{5}, \not{6}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, \not{2}, \not{3}, {\color{Red} 4}, {\color{Red} 4}, \not{7}, \not{8}, \not{9}, \not{11}\)

Find the average of the two centermost values.

\(\displaystyle \textup{Average}=\frac{4+4}{2}\)

\(\displaystyle \textup{Average}=\frac{8}{2}\)

\(\displaystyle \textup{Average}=4\)

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.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

Omit the centermost value.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid 5, 6, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Odd data set}: \not{2}, {\color{Red} 2}, {\color{Red} 3}, \not{3},\mid 5, 6, 9, 11\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{2+3}{2}\)

\(\displaystyle \textup{Average}=\frac{5}{2}\)

\(\displaystyle \textup{Average}=2.5\)

The median of the lower portion is \(\displaystyle 2.5\)

Find the median of the upper portion.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid \not{5}, {\color{Red} 6}, {\color{Red} 9}, \not{11}\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{6+9}{2}\)

\(\displaystyle \textup{Average}=\frac{15}{2}\)

\(\displaystyle \textup{Average}=7.5\)

The median of the upper potion is \(\displaystyle 7.5\)

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.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid 4, 7, 8, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, {\color{Red} 2}, \not{3}, \not{4}\mid 4, 7, 8, 9, 11\)

The median of the lower portion is two.

Find the median of the upper portion.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid \not{4}, \not{7}, {\color{Red} 8}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{IQR of the odd data set}=7.5-2.5\)

\(\displaystyle \textup{IQR}=5\)

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

\(\displaystyle \textup{IQR of the even data set}=8-2\)

\(\displaystyle \textup{IQR}=6\)

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:

When asked to find the IQR of a set of data, we must first put the numbers in numerical order: 

\(\displaystyle \small 12,24,35,46,57,68,79\).

Now we need to divide the data set into the upper quartile and lower quartile, we do so by finding the mean which is the center value of the data set.

In this data set, our median is \(\displaystyle \small 46\).

This means that our upper quartile consists of \(\displaystyle \small 57,68,79\).

So our \(\displaystyle \small Q_{3}\) =\(\displaystyle \small 68\), the median of the upper quartile.

Our \(\displaystyle \small Q_{1}\) = \(\displaystyle \small 24\), given that its the median of the lower quartile.

Thus, our IQR is \(\displaystyle \small 68-24 = 44\). 

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. 

\(\displaystyle \textup{Odd data set}: 9, 3, 2, 5, 6, 11, 4, 3, 2\)

\(\displaystyle \textup{Odd data set (ascending)}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

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

\(\displaystyle \textup{Even data set}: 11, 2, 4, 3, 8, 1, 2, 7, 4, 9\)

Rearrange into ascending order.

\(\displaystyle \textup{Even data set (ascending)}: 1, 2, 2, 3, 4, 4, 7, 8, 9, 11\)

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

\(\displaystyle \textup{Odd data set}: \not{2}, \not{2}, \not{3}, \not{3}, {\color{Red} 4}, \not{5}, \not{6}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, \not{2}, \not{3}, {\color{Red} 4}, {\color{Red} 4}, \not{7}, \not{8}, \not{9}, \not{11}\)

Find the average of the two centermost values.

\(\displaystyle \textup{Average}=\frac{4+4}{2}\)

\(\displaystyle \textup{Average}=\frac{8}{2}\)

\(\displaystyle \textup{Average}=4\)

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.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

Omit the centermost value.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid 5, 6, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Odd data set}: \not{2}, {\color{Red} 2}, {\color{Red} 3}, \not{3},\mid 5, 6, 9, 11\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{2+3}{2}\)

\(\displaystyle \textup{Average}=\frac{5}{2}\)

\(\displaystyle \textup{Average}=2.5\)

The median of the lower portion is \(\displaystyle 2.5\)

Find the median of the upper portion.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid \not{5}, {\color{Red} 6}, {\color{Red} 9}, \not{11}\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{6+9}{2}\)

\(\displaystyle \textup{Average}=\frac{15}{2}\)

\(\displaystyle \textup{Average}=7.5\)

The median of the upper potion is \(\displaystyle 7.5\)

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.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid 4, 7, 8, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, {\color{Red} 2}, \not{3}, \not{4}\mid 4, 7, 8, 9, 11\)

The median of the lower portion is two.

Find the median of the upper portion.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid \not{4}, \not{7}, {\color{Red} 8}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{IQR of the odd data set}=7.5-2.5\)

\(\displaystyle \textup{IQR}=5\)

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

\(\displaystyle \textup{IQR of the even data set}=8-2\)

\(\displaystyle \textup{IQR}=6\)

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:

When asked to find the IQR of a set of data, we must first put the numbers in numerical order: 

\(\displaystyle \small \small 21,35,46,57,68,79,80\).

Now we need to divide the data set into the upper quartile and lower quartile, we do so by finding the median which is the center value of the data set.

In this data set, our median is \(\displaystyle \small \small 57\).

This means that our upper quartile consists of \(\displaystyle \small \small 68,79,80\).

So our \(\displaystyle \small Q_{3}\) =\(\displaystyle \small \small 79\), the median of the upper quartile.

Our \(\displaystyle \small Q_{1}\) = \(\displaystyle \small \small 35\), given that its the median of the lower quartile.

Thus, our IQR is \(\displaystyle \small \small 79-35 = 44\). 

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. 

\(\displaystyle \textup{Odd data set}: 9, 3, 2, 5, 6, 11, 4, 3, 2\)

\(\displaystyle \textup{Odd data set (ascending)}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

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

\(\displaystyle \textup{Even data set}: 11, 2, 4, 3, 8, 1, 2, 7, 4, 9\)

Rearrange into ascending order.

\(\displaystyle \textup{Even data set (ascending)}: 1, 2, 2, 3, 4, 4, 7, 8, 9, 11\)

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

\(\displaystyle \textup{Odd data set}: \not{2}, \not{2}, \not{3}, \not{3}, {\color{Red} 4}, \not{5}, \not{6}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, \not{2}, \not{3}, {\color{Red} 4}, {\color{Red} 4}, \not{7}, \not{8}, \not{9}, \not{11}\)

Find the average of the two centermost values.

\(\displaystyle \textup{Average}=\frac{4+4}{2}\)

\(\displaystyle \textup{Average}=\frac{8}{2}\)

\(\displaystyle \textup{Average}=4\)

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.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

Omit the centermost value.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid 5, 6, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Odd data set}: \not{2}, {\color{Red} 2}, {\color{Red} 3}, \not{3},\mid 5, 6, 9, 11\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{2+3}{2}\)

\(\displaystyle \textup{Average}=\frac{5}{2}\)

\(\displaystyle \textup{Average}=2.5\)

The median of the lower portion is \(\displaystyle 2.5\)

Find the median of the upper portion.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid \not{5}, {\color{Red} 6}, {\color{Red} 9}, \not{11}\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{6+9}{2}\)

\(\displaystyle \textup{Average}=\frac{15}{2}\)

\(\displaystyle \textup{Average}=7.5\)

The median of the upper potion is \(\displaystyle 7.5\)

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.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid 4, 7, 8, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, {\color{Red} 2}, \not{3}, \not{4}\mid 4, 7, 8, 9, 11\)

The median of the lower portion is two.

Find the median of the upper portion.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid \not{4}, \not{7}, {\color{Red} 8}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{IQR of the odd data set}=7.5-2.5\)

\(\displaystyle \textup{IQR}=5\)

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

\(\displaystyle \textup{IQR of the even data set}=8-2\)

\(\displaystyle \textup{IQR}=6\)

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:

When asked to find the IQR of a set of data, we must first put the numbers in numerical order: 

\(\displaystyle \small \small \small 12,23,34,44,55,66,77,88,90,102,133\).

Now we need to divide the data set into the upper quartile and lower quartile, we do so by finding the median which is the center value of the data set.

In this data set, our median is \(\displaystyle \small \small \small 66\).

This means that our upper quartile consists of \(\displaystyle \small 77,88,90,102,133\).

So our \(\displaystyle \small Q_{3}\) =\(\displaystyle \small \small \small 90\), the median of the upper quartile.

Our \(\displaystyle \small Q_{1}\) = \(\displaystyle \small \small \small 34\), given that its the median of the lower quartile.

Thus, our IQR is \(\displaystyle \small 90-34 = 56\). 

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. 

\(\displaystyle \textup{Odd data set}: 9, 3, 2, 5, 6, 11, 4, 3, 2\)

\(\displaystyle \textup{Odd data set (ascending)}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

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

\(\displaystyle \textup{Even data set}: 11, 2, 4, 3, 8, 1, 2, 7, 4, 9\)

Rearrange into ascending order.

\(\displaystyle \textup{Even data set (ascending)}: 1, 2, 2, 3, 4, 4, 7, 8, 9, 11\)

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

\(\displaystyle \textup{Odd data set}: \not{2}, \not{2}, \not{3}, \not{3}, {\color{Red} 4}, \not{5}, \not{6}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, \not{2}, \not{3}, {\color{Red} 4}, {\color{Red} 4}, \not{7}, \not{8}, \not{9}, \not{11}\)

Find the average of the two centermost values.

\(\displaystyle \textup{Average}=\frac{4+4}{2}\)

\(\displaystyle \textup{Average}=\frac{8}{2}\)

\(\displaystyle \textup{Average}=4\)

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.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

Omit the centermost value.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid 5, 6, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Odd data set}: \not{2}, {\color{Red} 2}, {\color{Red} 3}, \not{3},\mid 5, 6, 9, 11\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{2+3}{2}\)

\(\displaystyle \textup{Average}=\frac{5}{2}\)

\(\displaystyle \textup{Average}=2.5\)

The median of the lower portion is \(\displaystyle 2.5\)

Find the median of the upper portion.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid \not{5}, {\color{Red} 6}, {\color{Red} 9}, \not{11}\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{6+9}{2}\)

\(\displaystyle \textup{Average}=\frac{15}{2}\)

\(\displaystyle \textup{Average}=7.5\)

The median of the upper potion is \(\displaystyle 7.5\)

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.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid 4, 7, 8, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, {\color{Red} 2}, \not{3}, \not{4}\mid 4, 7, 8, 9, 11\)

The median of the lower portion is two.

Find the median of the upper portion.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid \not{4}, \not{7}, {\color{Red} 8}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{IQR of the odd data set}=7.5-2.5\)

\(\displaystyle \textup{IQR}=5\)

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

\(\displaystyle \textup{IQR of the even data set}=8-2\)

\(\displaystyle \textup{IQR}=6\)

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:

This question provides us with all of the data necessary to answer its question, therefore no calculations are necessary.

To find the IQR, we simply use the formula \(\displaystyle \small Q_{3}-Q_{1}\), this means the 

\(\displaystyle \small IQR = 60-15\).

The median is the \(\displaystyle \small Q_{2}\), which is also given, \(\displaystyle 49\). 

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. 

\(\displaystyle \textup{Odd data set}: 9, 3, 2, 5, 6, 11, 4, 3, 2\)

\(\displaystyle \textup{Odd data set (ascending)}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

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

\(\displaystyle \textup{Even data set}: 11, 2, 4, 3, 8, 1, 2, 7, 4, 9\)

Rearrange into ascending order.

\(\displaystyle \textup{Even data set (ascending)}: 1, 2, 2, 3, 4, 4, 7, 8, 9, 11\)

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

\(\displaystyle \textup{Odd data set}: \not{2}, \not{2}, \not{3}, \not{3}, {\color{Red} 4}, \not{5}, \not{6}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, \not{2}, \not{3}, {\color{Red} 4}, {\color{Red} 4}, \not{7}, \not{8}, \not{9}, \not{11}\)

Find the average of the two centermost values.

\(\displaystyle \textup{Average}=\frac{4+4}{2}\)

\(\displaystyle \textup{Average}=\frac{8}{2}\)

\(\displaystyle \textup{Average}=4\)

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.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3, 4, 5, 6, 9, 11\)

Omit the centermost value.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid 5, 6, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Odd data set}: \not{2}, {\color{Red} 2}, {\color{Red} 3}, \not{3},\mid 5, 6, 9, 11\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{2+3}{2}\)

\(\displaystyle \textup{Average}=\frac{5}{2}\)

\(\displaystyle \textup{Average}=2.5\)

The median of the lower portion is \(\displaystyle 2.5\)

Find the median of the upper portion.

\(\displaystyle \textup{Odd data set}: 2, 2, 3, 3,\mid \not{5}, {\color{Red} 6}, {\color{Red} 9}, \not{11}\)

Calculate the average of the two values.

\(\displaystyle \textup{Average}=\frac{6+9}{2}\)

\(\displaystyle \textup{Average}=\frac{15}{2}\)

\(\displaystyle \textup{Average}=7.5\)

The median of the upper potion is \(\displaystyle 7.5\)

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.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid 4, 7, 8, 9, 11\)

Find the median of the lower portion.

\(\displaystyle \textup{Even data set}: \not{1}, \not{2}, {\color{Red} 2}, \not{3}, \not{4}\mid 4, 7, 8, 9, 11\)

The median of the lower portion is two.

Find the median of the upper portion.

\(\displaystyle \textup{Even data set}: 1, 2, 2, 3, 4\mid \not{4}, \not{7}, {\color{Red} 8}, \not{9}, \not{11}\)

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. 

\(\displaystyle \textup{IQR of the odd data set}=7.5-2.5\)

\(\displaystyle \textup{IQR}=5\)

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

\(\displaystyle \textup{IQR of the even data set}=8-2\)

\(\displaystyle \textup{IQR}=6\)

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:

Since the \(\displaystyle \small IQR = Q_{3} - Q_{1}\), then our formula should look like this: 

\(\displaystyle \small \small 3 = Q_{3}-4\).

When we solve for \(\displaystyle \small \small Q_{3}\) we get \(\displaystyle 7\).

This means that the median of the upper quartile of the set of data that this is for must be \(\displaystyle \small 7\).

The median for the lower quartile must be \(\displaystyle \small 4\).