Quartiles and Interquartile Range

Find the lower quartile of the data set 9, 12, 3, 5, 8, 3, 4.

Solution

Step 1.

We rearrange the data values in ascending order to get

3, 3, 4, 5, 8, 9, 12

Step 2.

We identify that 5 is the median in the entire data set. However, that means that the lower half of the data is now left with 3, 3, 4.

Step 3.

The median for that is 3. Therefore, the lower quartile is

$Q_1=3$.

Find the lower quartile of the given data set 78, 62, 46, 89, 98, 23, 45, 77.

Solution

Step 1.

We rearrange the data values in ascending order to get,

23, 45, 46, 62, 77, 78, 89, 98

Step 2.

Since the number of the data values is even, we can split them into two equal parts with the lower half being,

23, 45, 46, 62

Step 3.

To find the median for these values, we will need to find the average of the two values in the middle, since this data set is also even. Thus, the lower quartile is given by,

$\frac{45+46}{2}=45.5$

$Q_1=45.5$

Find the second quartile of the data set 9, 12, 3, 5, 8, 3, 4

Solution

Step 1.

We rearrange data values in ascending order, to get

3, 3, 4, 5, 8, 9, 12

Step 2.

5 here is identified as the middle value in the data set. Therefore, the second quartile is

$Q_2=5$

Find the second quartile of the given data set 78, 62, 46, 89, 98, 23, 45, 77.

Solution

Step 1.

We rearrange the data values in ascending order

23, 45, 46, 62, 77, 78, 89, 98

Step 2.

Since the number of data set is even, two numbers can be identified as the middle values. These are 62 and 77. We will find the average of these values, to get

$\frac{62+77}{2}=69.5$

$Q_2=69.5$

Find the upper quartile of the data set 9, 12, 3, 5, 8, 3, 4

Solution

Step 1. We rearrange the data values in ascending order to get,

3, 3, 4, 5, 8, 9, 12

Step 2.

We identify that 5 is the median in the entire data set. However, that means that the upper half of the data is now left with

8, 9, 12.

The median for that is 9. Therefore,

$Q_3=9$

Find the upper quartile of the given data set 78, 62, 46, 89, 98, 23, 45, 77.

Solution

Step 1.

We rearrange the data values in ascending order to get,

23, 45, 46, 62, 77, 78, 89, 98

Step 2.

Since the number of the data values is even, we can split them into two equal parts with the upper half being,

77, 78, 89, 98

Step 3.

To find the median for these values, we will need to find the average of the two values in the middle, since this data set is also even.

$\frac{78+89}{2}=83.5$

$Q_3=83.5$

Find the interquartile range for the data set 6, 47, 49, 15, 43, 41, 7, 39, 43, 41, 36.

Solution

Step 1.

We rearrange the data set in order from lowest to highest, to get

6, 7, 15, 36, 39, 41, 41, 43, 43, 47, 49

Step 2.

We find the median by locating the middle data point, which is 41. This is also known as the second quartile,

$Q_2=41$

Step 3.

With finding the median for both halves, we need to understand that the point where the median is located divides the data points into two.

Hence, the median for the first half will be the first quartile, whist the median for the second half will be the third quartile. Let us find the median for the first half first.

The first half is 6, 7, 15, 36, 39 . The median is 15. Thus$Q_1=15$

We find the median for the second half too, which is 41, 43, 43, 47, 49 . The median is 43. Thus$Q_3=43$.

Now, we can proceed to calculate the interquartile range,

$IQR=Q_3-Q_1=43-15=28$

The table below is the data of basketball players' points scored per game over a seven-game span. Visualize this on a box plot.

Solution

Step 1.

We rearrange the values in the data set from lowest to highest.

5, 10, 16, 17, 18, 20, 32.

Step 2.

Now identify the highest and lowest values in the data set

$$\begin{gathered} Highestvalue=32 \\ Lowestvalue=5 \end{gathered}$$

Step 3.

We can now identify the midpoint value (median) of the data set,

$Median=17$

Step 4.

We will now find the upper and lower quartiles.

The lower quartile is the median for the first half of the data set. That means that we are finding the median for 5, 10, 16

$Lowerquartile=10$

The upper quartile is the median for the second half of the data set. That means that we are finding the median for 18, 20, 32

$Upperquartile=20$

Step 5.

We can now find the inter-quartile range by the formula,

$IQR=Upperquartile\left(Q_3\right)-lowerquartile\left(Q_1\right)=20-10=10$

Step 6.

Now that we have all our necessary values, we will construct our box and whisker plot.

$Lowestvalue=5$

$Median=17$

$Upperquartile=20$

$Lowerquartile=10$

We will first draw a number line that fits the data, and plot all the necessary values we found.

Construct a rectangle that encloses the median of the entire data set that its vertical lines pass through the upper and lower quartiles. Now construct a vertical line through the median that hits both ends of the rectangle.

What will be the quartile deviation for the data set 6, 9, 3, 6, 6, 5, 2, 3, 8?

Solution

Step 1.

We rearrange the data set in order from lowest to highest,

2, 3, 3, 5, 6, 6, 6, 8, 9

Step 2.

We find the median by locating the middle data point, which is 6. This means the second quartile is 6.

$Q_2=6$

Step 3.

We find the median for both halves. Let us start with the first.

2, 3, 3, 5

We have both values are 3, therefore the first quartile is 3.

$Q_1=\frac{3+3}{2}=\frac{6}{2}=3$

Now we will find the median for the second half

6, 6, 8, 9

Since we have two figures here, we will find the average of them.

$Q_3=\frac{6+8}{2}=\frac{14}{2}=7$

Now that we have found the lower and upper quartiles, we want to know how much these values deviate from the median (which is the middle point value of the data set). Finding the quartile deviation means we will subtract the first quartile from the third quartile, and divide it by 2,

$Quartiledeviation=\frac{Q_3-Q_1}{2}=\frac{7-3}{2}=\frac{4}{2}=2$