Quartiles and Interquartile Range - SAT Math

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Question

Above is the stem-and-leaf display for a group of test scores. What is the first quartile?

Answer

To find the first quartile, first find the median of the scores. There are 53 scores represented, so, 53 being odd, look for the score in the center. This is the score in position

$\frac{53+1}{2} = 27$ .

As can be seen in this diagram, the score is 74.

The first quartile is the median of the lower half of the scores - that is, the lower 26 scores. Since 26 is even, the median is the mean of the scores in positions $\frac{26}{2} = 13$ and from the bottom.

As can be seen in this diagram, these scores are 60 and 62.

The first quartile of the scores is therefore $\frac{60 + 62 }{2} = 61$ .

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Question

Above is the stem-and-leaf display for a group of test scores. Give the interquartile range of the scores represented.

Answer

The interquartile range is the difference between the third and first quartiles. To find these quartiles, first find the median of the scores. There are 53 scores represented, so look for the score in position

$\frac{53+1}{2} = 27$ .

As can be seen in this diagram, the score is 74.

The first quartile is the median of the lower half of the scores - that is, the lower 26 scores. Since 26 is even, the median is the mean of the scores in positions $\frac{26}{2} = 13$ and .

As can be seen in this diagram, these scores are 60 and 62.

The first quartile of the scores is therefore $\frac{60 + 62 }{2} = 61$

The third quartile, similarly, can be found by finding the mean of the 13th and 14th elements in the top half of the scores:

As can be seen in this diagram, these scores are 81 and 81, so the third quartile is 81.

The interquartile range is the difference:

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Question

Above is the stem-and-leaf display for a group of test scores. What score is at the third quartile?

Answer

To find the third quartile, first find the median of the scores. There are 53 scores represented, so look for the score in position

$\frac{53+1}{2} = 27$ .

As can be seen in this diagram, the score is 74.

The third quartile is the median of the upper half of the scores - that is, the higher 26 scores. Since 26 is even, the median is the mean of the scores in positions $\frac{26}{2} = 13$ and from the top.

As can be seen in this diagram, both of these scores are 81, so the third quartile is 81.

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Question

Consider the following numbers: . What is the value of the first quartile?

Answer

The first quartile is Q1.

Reorganize the numbers in chronological order.

$[5, 8, 1, 4, 8, 9, 1]\rightarrow [1,1,4,5,8,8,9]$

The Q1 is the median of , which is the three numbers left of the median of the entire set of numbers, 5.

The answer is:

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Question

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

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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:

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

$Q_{2}$ = median of the overall data set

$Q_{1}$ = median of the lower half of the data

$Q_{3}$ = median of the upper half of the data

$Q_{2} = 32$

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.

$Q_{1} = \frac{25 + 27}{2} = 26$

The median of the upper half falls between two values.

$Q_{3} = \frac{43 + 45}{2} = 44$

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

$Q_{3} - Q_{1} = 44 - 26 = 18$

Multiply by to find the answer:

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Question

Determine the interquartile range of the following numbers:

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

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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:

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:

$Q1=\frac{42+43}{2}=\frac{85}{2}=42.5$

$Q3=\frac{51+54}{2}=\frac{105}{2}=52.5$

The interquartile range is the difference between Q3 and Q1:

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Question

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

Find the interquartile range of the following data set:

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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

$4, 1, 5, 6, 12, 8, 2, 5, 5, 3, 2, 8\rightarrow$

To find the lower quartile ( $Q_{1}$ )'s position, we use the equation $\frac{n+1}{4}$ , where n is the number of data points in the set.

Thus, our lower quartile is at $\frac{12+1}{4} = (3\frac{1}{4})^{th}$ 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 $\frac{1}{4}$ here, our equation becomes

$(\frac{1}{4})(3-2)+2=\frac{1}{4} + 2 = 2.25$

Thus, our $Q_{1} = 2.25.$

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

$3(\frac{n+1}{4})$ , where is still the number of data points in the set. $Q_{3}$ will be at the

$3(\frac{13}{4}) = \frac{39}{4} = (9\frac{3}{4})^{th}$ position, so we must find the value $\frac{3}{4}$ of the way between the $9^{th}$ and $10^{th}$ data point in the set. With and as our values for the $9^{th}$ and $10^{th}$ data point, our equation becomes

$(\frac{3}{4})(8-6)+6= \frac{6}{4}+6 = 7.5$

So, our $Q_{3} = 7.5$ .

The last step is easy by comparison. Subtract $Q_{1}$ from $Q_{3}$ to get our interquartile range:

Thus, our interquartile range is .

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Question

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

Find the interquartile range for the data set.

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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:

As always, rearranging the data set helps us immensely:

-->

To find $Q_{1}$ , we use the equation $\frac{n+1}{4}$ , where is the number of points in our data set.

$\frac{11+1}{4} = 3$ , so our lower quartile is the $3^{rd}$ number in the set, or .

To find $Q_{3}$ , we use the equation $3(\frac{n+1}{4})$ instead.

$3(\frac{11+1}{4}) = 9$ , so our upper quartile is the $9^{th}$ number in the set, or .

Lastly, our interquartile range is $Q_{3}-Q_{1} = 5$ , so is our interquartile range.

Compare your answer with the correct one above

Question

$\small 10, 23, 4, 10, 3, 5,22$

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

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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 $\small Q_{3}$ and $\small Q_{1}$ , because $\small IQR = Q_{3} - Q_{1}$ .

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

$\small 3, 4, 5, 10, 10, 22, 23$ ,

and found $\small 10$ to be the median or center of the data:

$\small 3, 4, 5, \textbf{10}, 10, 22, 23$ .

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

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

$\small 22 - 4 = 18.$

Thus our $\small IQR = 18.$

Compare your answer with the correct one above

Question

$\small 1,2,3,4,5,6,7,8,9,10,100$

Using the data above, what is the interquartile range?

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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 $\small Q_{3}$ and $\small Q_{1}$ , because $\small IQR = Q_{3} - Q_{1}$ .

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

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

$\small 1,2,3,4,5,6,7,8,9,10, 100$

and found $\small 6$ to be the median or center of the data:

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

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

We now have our $\small \small \small Q_{3} = 9$ and our $\small \small Q_{1} = 3$ .

$\small \small 9-3=6$

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

Compare your answer with the correct one above

Question

$\small Q _{1} = 53$ , $\small Q_{3}= 67$ , Median $\small = 55$ , Highest value is , Lowest value is $\small 32$ .

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

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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

Compare your answer with the correct one above

Question

$\small 6,6,9,10,15,16,20,30,48$

Using the data above, find the interquartile range.

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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 $\small Q_{3}- Q_{1}$ .

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.

$Q_{3}$ is the median of the upper quartile, while $Q_{1}$ 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

$\small 20 +30 = 50 \div2 = 25$ , so our

$\small Q_{3} =25$ .

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

$\small Q_{1} = \frac{15}{2}$ .

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

Compare your answer with the correct one above

Question

$\small 9,3,1,2,2,1,3$

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

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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 $\small Q_{3}$ and $\small Q_{1}$ .

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

$\small 3,3,9$ ,

thus the $\small Q_{3}$ is $\small 3.$

$\small Q_{1}$ is the median of the lower quartile:

$\small 1,1,2$ ,

thus the $\small Q_{1}$ is $\small 1$ .

So, $\small IQR = 3-1 = 2$ .

Compare your answer with the correct one above

Question

$\small 12,24,35,46,57,68,79$

Find the interquartile range of the data set above.

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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:

$\small 12,24,35,46,57,68,79$ .

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 $\small 46$ .

This means that our upper quartile consists of $\small 57,68,79$ .

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

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

Thus, our IQR is $\small 68-24 = 44$ .

Compare your answer with the correct one above

Question

$\small 21,35,46,57,68,79,80$

Find the interquartile range given the data set above.

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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:

$\small \small 21,35,46,57,68,79,80$ .

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 $\small \small 57$ .

This means that our upper quartile consists of $\small \small 68,79,80$ .

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

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

Thus, our IQR is $\small \small 79-35 = 44$ .

Compare your answer with the correct one above

Question

$\small 12,23,34,44,55,66,77,88,90,102,133$

Using the data set above, find the interquartile range.

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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:

$\small \small \small 12,23,34,44,55,66,77,88,90,102,133$ .

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 $\small \small \small 66$ .

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

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

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

Thus, our IQR is $\small 90-34 = 56$ .

Compare your answer with the correct one above

Question

Given the following data, find the range, mode, and mean:

$\small \small 1,2,3,4,5,6,7,8,9$

Answer

Luckily, we are provided with a data set that is already in numerical order:

$\small 1,2,3,4,5,6,7,8,9$ .

To find the range, we subtract the lowest value from the highest,

$\small 9-1 = 8$ .

To find the mean, we add all of the data pieces and divide by the number of data pieces,

$\small \frac{1+2+3+4+5+6+7+8+9}{9} = \frac{45}{9} = 5$ .

To find the mode, we search for numbers tha appear more than once. Since none of these data values appear more than once, we have no mode.

Compare your answer with the correct one above

Question

Provided with the following information, find the IQR and state the median.

$\small Q_{1} = 15 , Q_{2} = 49, Q_{3} = 60$ .

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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 $\small Q_{3}-Q_{1}$ , this means the

$\small IQR = 60-15$ .

The median is the $\small Q_{2}$ , which is also given, .

Compare your answer with the correct one above

Question

If the $\small \small IQR = 3$ and the $\small \small Q_{1} = 4$ , what must $\small Q_{3} = ?$

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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 $\small IQR = Q_{3} - Q_{1}$ , then our formula should look like this:

$\small \small 3 = Q_{3}-4$ .

When we solve for $\small \small Q_{3}$ we get .

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

The median for the lower quartile must be $\small 4$ .

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Question

$\small 3,5,6,8,9,25,46$

Using the data provided, find the interquartile range.

Answer

How do you find the interquartile range?

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

Order the data from least to greatest

Find the median

Calculate the median of both the lower and upper half of the data

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.

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

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

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

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

Rearrange into ascending order.

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

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

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

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.

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

Find the average of the two centermost values.

$\textup{Average}=\frac{4+4}{2}$

$\textup{Average}=\frac{8}{2}$

$\textup{Average}=4$

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.

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

Omit the centermost value.

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

Find the median of the lower portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{2+3}{2}$

$\textup{Average}=\frac{5}{2}$

$\textup{Average}=2.5$

The median of the lower portion is

Find the median of the upper portion.

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

Calculate the average of the two values.

$\textup{Average}=\frac{6+9}{2}$

$\textup{Average}=\frac{15}{2}$

$\textup{Average}=7.5$

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.

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

Find the median of the lower portion.

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

The median of the lower portion is two.

Find the median of the upper portion.

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

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.

$\textup{IQR of the odd data set}=7.5-2.5$

$\textup{IQR}=5$

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

$\textup{IQR of the even data set}=8-2$

$\textup{IQR}=6$

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:

Using the data provided:

$\small 3,5,6,8,9,25,46$ , we are asked to find the interquartile range. We first must find the median of the data set, and to do that we first must put the data set in numerical order, which it luckily already is. Now we can being crossing out numbers bilaterally and simultaneously, so first we cross out $\small 3$ and $\small 46$ , then and $\small 25$ , etc Until we are left with , which is our median.

$\small 3,5,6,{\color{Red} 8},9,25,46$

Once we have the median divide the set into an upper quartile and lower quartile, we can use the same method to find the medians of those mini-groups, giving us an upper quartile median, $\small \small Q_{3} = 25$ , and lower quartile $\small \small Q_{1} = 5$ .

The IQR is defined as

$\small Q_{3}-Q_{1} = 25-5=20$ .

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