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SAT II Math I : Quartiles and Interquartile Range

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Example Questions

Example Question #1 : Quartiles And Interquartile Range

$\left \{ 1,2,3,3,4,5,5,5,6,6,7\right \}$

What is the 3rd quartile of this set?

Possible Answers:

Correct answer:

Explanation:

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

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

For this set those numbers would be $\left \{ 5,5,6,6,7\right \}$ .

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

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Example Question #1 : How To Find Interquartile Range

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

$25,\; 32,\; 49,\; 21,\; 37,\; 43,\; 27,\; 45,\; 31$

Possible Answers:

Correct answer:

Explanation:

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 2.5

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 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.

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

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.

How to find iqr boxplot 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.

$21,\; 25,\; 27,\; 31,\; 32,\; 37,\; 43,\; 45,\; 49$

$Q_{2}$ = 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 21, 25, 27, 31 as the lower half of the data and 37, 43, 45, 49 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:

2*18=36

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Example Question #1 : How To Find Interquartile Range

Determine the interquartile range of the following numbers:

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

Possible Answers:

None of these

Correct answer:

Explanation:

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

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

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

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

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 2.5

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 7.5

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

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.

How to find iqr boxplot 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:

Q3-Q1=52.5-42.5=10

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Example Question #1 : How To Find Interquartile Range

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

Find the interquartile range for the data set.

5, 7, 3, 7, 4, 9, 7, 10, 2, 3, 8

Possible Answers:

4.5

5.5

Correct answer:

Explanation:

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

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

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

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

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 2.5

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 7.5

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

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.

How to find iqr boxplot 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:

5, 7, 3, 7, 4, 9, 7, 10, 2, 3, 8 --> 2, 3, 3, 4, 5, 7, 7, 7, 8, 9, 10

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.

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Example Question #1 : How To Find Interquartile Range

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

Find the interquartile range of the following data set:

4, 1, 5, 6, 12, 8, 2, 5, 5, 3, 2, 8

Possible Answers:

4.5

6.25

5.25

Correct answer:

5.25

Explanation:

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

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

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

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

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 2.5

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 7.5

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

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.

How to find iqr boxplot 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:

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

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

x(b-a) +a , 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:

7.5-2.25 = 5.25

Thus, our interquartile range is 5.25 .

Report an Error

Example Question #1 : How To Find Interquartile Range

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

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

Possible Answers:

$\small 13$

$\small 14$

$\small 10$

$\small 3$

Correct answer:

Explanation:

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

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

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

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

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 2.5

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 7.5

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

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.

How to find iqr boxplot 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 $\small IQR = 18.$

Report an Error

Example Question #1 : How To Find Interquartile Range

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

Using the data above, what is the interquartile range?

Possible Answers:

$\small 132$

$\small 6$

$\small 56$

$\small 3$

$\small 33$

Correct answer:

$\small 6$

Explanation:

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

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

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

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

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 2.5

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 7.5

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

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.

How to find iqr boxplot 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.

Report an Error

Example Question #2 : How To Find Interquartile Range

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

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

Possible Answers:

Correct answer:

Explanation:

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

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

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

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

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 2.5

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 7.5

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

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.

How to find iqr boxplot 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 $\small 67 - 53 = 14$ .

Report an Error

Example Question #1 : How To Find Interquartile Range

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

Using the data above, find the interquartile range.

Possible Answers:

$\small \frac{35}{2}$

$\small \frac{34}{2}$

$\small 16$

Correct answer:

$\small \frac{35}{2}$

Explanation:

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

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

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

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

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 2.5

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 7.5

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

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.

How to find iqr boxplot 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

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

$\small 20 +30 = 50 \div2 = 25$ , 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.

Report an Error

Example Question #1 : How To Find Interquartile Range

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

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

Possible Answers:

$\small 2$

$\small 4$

$\small 1$

$\small 5$

Correct answer:

$\small 2$

Explanation:

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

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

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

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

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 2.5

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 7.5

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

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.

How to find iqr boxplot 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, $\small IQR = 3-1 = 2$ .

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SAT II Math I : Quartiles and Interquartile Range

Study concepts, example questions & explanations for SAT II Math I

All SAT II Math I Resources

Example Questions

Example Question #1 : Quartiles And Interquartile Range

Example Question #1 : How To Find Interquartile Range

How do you find the interquartile range?

Step 1: Order the data

Step 2: Calculate the median

Step 3: Upper and lower medians

Step 4: Calculate the difference

Solution:

Example Question #1 : How To Find Interquartile Range

How do you find the interquartile range?

Step 1: Order the data

Step 2: Calculate the median

Step 3: Upper and lower medians

Step 4: Calculate the difference

Solution:

Example Question #1 : How To Find Interquartile Range

How do you find the interquartile range?

Step 1: Order the data

Step 2: Calculate the median

Step 3: Upper and lower medians

Step 4: Calculate the difference

Solution:

Example Question #1 : How To Find Interquartile Range

How do you find the interquartile range?

Step 1: Order the data

Step 2: Calculate the median

Step 3: Upper and lower medians

Step 4: Calculate the difference

Solution:

Example Question #1 : How To Find Interquartile Range

How do you find the interquartile range?

Step 1: Order the data

Step 2: Calculate the median

Step 3: Upper and lower medians

Step 4: Calculate the difference

Solution:

Example Question #1 : How To Find Interquartile Range

How do you find the interquartile range?

Step 1: Order the data

Step 2: Calculate the median

Step 3: Upper and lower medians

Step 4: Calculate the difference

Solution:

Example Question #2 : How To Find Interquartile Range

How do you find the interquartile range?

Step 1: Order the data

Step 2: Calculate the median

Step 3: Upper and lower medians

Step 4: Calculate the difference

Solution:

Example Question #1 : How To Find Interquartile Range

How do you find the interquartile range?

Step 1: Order the data

Step 2: Calculate the median

Step 3: Upper and lower medians

Step 4: Calculate the difference

Solution:

Example Question #1 : How To Find Interquartile Range

How do you find the interquartile range?

Step 1: Order the data

Step 2: Calculate the median

Step 3: Upper and lower medians

Step 4: Calculate the difference

Solution:

All SAT II Math I Resources

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