Quartiles and Interquartile Range

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Assuming a company put together a team of junior staff to work on a project together and they were assigned senior staff to supervise the project.

Visualizing their salary using a range may not account for the salary gap between the junior staff and the senior staff.

Quartiles and interquartile range are what account for the data values between the two extreme ends of the data set.

In this article, we will be learning about quartiles and the inter-quartile range.

What is a quartile?

Quartiles are the values that divide a set into quarters (four parts).

Though quartiles divide the data set into four parts, we thus have three quartiles: the first quartile, the second quartile, and the third quartile.

The lower quartile

The lower quartile, also known as the first quartile is what accounts for the data under 25%. It is technically the middle point value between the lowest data point and the median of the data set. It is also denoted by $Q_{1}$ .

We recall that the median of a data set is the midpoint value. The lower quartile is the median of the set of values from the lowest data point to the median of the entire data set.

To find the lower quartile, we use the median as a reference point.

If the number of data values in the data set is odd, disregard the middle number. The lower quartile is the median of the lower half of the data set.
If the number of data values in the data set is even, the lower quartile is still the median of the lower half of the data set.

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,

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

$Q_{1} = 45.5$

The second quartile

The second quartile denoted by $Q_{2}$ is the median of the data set. This is the middle point value of the entire data set.

To find the second quartile, we identify the middle value of the given data set if the number of data values is odd. If the number of the data values in the given data set is even, we find the average of the two middle values. That average is the second quartile.

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

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

$Q_{2} = 69.5$

The third quartile

The third quartile, also known as the upper quartile is the value under which 75% of the data are found when arranged in increasing order. It is denoted by $Q_{3}$ . This value is the middle point value between the median and the highest data value.

To find the upper quartile, use the median as a reference point.

If the number of data values in the data set is odd, disregard the middle number. The upper quartile is the median of the upper half of the data set.
If the number of data values in the data set is even, the upper quartile is still the median of the upper half of the data set.

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.

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

$Q_{3} = 83.5$

Importance of quartiles in statistics

There are significant uses of finding quartiles in statistics. These are discussed below.

Quartiles easily identify a dataset's central tendency and its variability.
Quartiles help identify outliers in a dataset.
Quartiles give information on the shape of the distribution of data.
They summarize large data sets.
Quartiles are the primary elements used to calculate interquartile ranges.

Interquartile range and quartile deviation

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

This means that to find the interquartile range of any given data successfully, you will need to know the upper and lower quartiles.

Interquartile range formula

The interquartile range formula is given by

$I Q R = Q_{3} - Q_{1}$

where $Q_{3} = t h i r d q u a r t i l e$ , $Q_{1} = f i r s t q u a r t i l e$

To find the quartiles and interquartile range of a given data set you can proceed as follows,

Order the values in ascending order.
Find the median. This is always labeled as the second quartile ( $Q_{2}$ ).
Now find the median of both halves of the data set. The lowest half is labelled $Q_{1}$ , and the highest half is labelled $Q_{3}$ .
Find the interquartile range (IQR) by subtracting $Q_{1}$ from $Q_{3}$ .

Quartiles and interquartile range calculation

In this section, we are going to take an example of how quartiles and interquartile range are calculated.

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,

$I Q R = Q_{3} - Q_{1} = 43 - 15 = 28$

Plotting interquartile ranges

Plotting interquartile ranges on a graph means you would be drawing a box plot. To construct one, we follow the following steps,

Rearrange the values in the data set from lowest to highest.
Identify the highest and lowest values in the data set.
Identify the data set's midpoint value (median).
Find the upper and lower quartiles.
Find the inter-quartile range.
Construct the box plot with the necessary values found.

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

Game	Points
1	10
2	17
3	5
4	32
5	16
6	18
7	20

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

$H i g h e s t v a l u e = 32 L o w e s t v a l u e = 5$

Step 3.

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

$M e d i a n = 17$

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

$L o w e r q u a r t i l e = 10$

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

$U p p e r q u a r t i l e = 20$

Step 5.

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

$I Q R = U p p e r q u a r t i l e (Q_{3}) - l o w e r q u a r t i l e (Q_{1}) = 20 - 10 = 10$

Step 6.

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

H i g h e s t v a l u e = 32

$L o w e s t v a l u e = 5$

$M e d i a n = 17$

$U p p e r q u a r t i l e = 20$

$L o w e r q u a r t i l e = 10$

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.

Quartile deviation

Quartile deviation is defined as half of the difference between the upper and lower quartile.

Quartile deviation is one of the measures that measure dispersion in a data set. Mathematically, this measures the extent to which the lower and upper quartiles differ from the median. It is calculated by dividing a data set's inter-quartile range by 2.

The quartile deviation is also known as the semi-inter-quartile range. Its formula is defined by,

$Q u a r t i l e d e v i a t i o n = \frac{T h i r d q u a r t i l e - f i r s t q u a r t i l e}{2} = \frac{Q_{3} - Q_{1}}{2}$

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,

$Q u a r t i l e d e v i a t i o n = \frac{Q_{3} - Q_{1}}{2} = \frac{7 - 3}{2} = \frac{4}{2} = 2$

Quartiles and Interquartile Range - Key takeaways

A quartile is a type of quantile that divides an ordered data set into four quarters.
The interquartile range is the difference between the upper quartile and the lower quartile value.
The third quartile accounts for data under 75%.
The formula for inter-quartile range is $I Q R = Q_{3} - Q_{1}$ .
The formula for quartile deviation is $Q u a r t i l e d e v i a t i o n = \frac{Q_{3} - Q_{1}}{2}$

Frequently Asked Questions about Quartiles and Interquartile Range

IQR = Q3-Q1

where Q3 = Third quartile

Q1 = First quartile

The equation (Q3-Q1)/2 is used for calculating the semi-interquartile range.

Quartiles are the values that divide a set into quarters. The interquartile range is the difference between the upper quartile and the lower quartile values.

Plot a box plot by finding the five-number summary; the median, the lower and upper quartiles, and the lower and upper extremes of the data set.

The interquartile range, which tells us how far apart the first and third quartiles are, indicates how spread out the middle 50% of our set of data is.

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Flashcards in Quartiles and Interquartile Range13 Start learning

A type of quantile that divides an ordered data set into four parts (quarters) is known as

Quartile

What is inter-quartile range?

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

What is a type of quantile that divides an ordered data set into four parts?

Quartile

The interquartile range is the difference between what two values?

Third quartile and first quartile

What are measures of spread that give quite a vivid display of the dispersion of the data?

The quartiles and interquartile ranges

Which of these account for data under 25%?

First quartile

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Quartiles and Interquartile Range

What is a quartile?

The lower quartile

The second quartile

The third quartile

Importance of quartiles in statistics

Interquartile range and quartile deviation

Interquartile range formula

Quartiles and interquartile range calculation

Plotting interquartile ranges

Quartile deviation

Quartiles and Interquartile Range - Key takeaways

Frequently Asked Questions about Quartiles and Interquartile Range

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Entdecke Lernmaterial in der StudySmarter-App

Quartiles and Interquartile Range

What is a quartile?

The lower quartile

The second quartile

The third quartile

Importance of quartiles in statistics

Interquartile range and quartile deviation

Interquartile range formula

Quartiles and interquartile range calculation

Plotting interquartile ranges

Quartile deviation

Quartiles and Interquartile Range - Key takeaways

Frequently Asked Questions about Quartiles and Interquartile Range

What is the Interquartile range formula?

How to calculate semi-interquartile range?

What are Quartile and Inter-quartile ranges?

How to plot interquartile range?

What is the significance of the Inter-quartile range?

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Learn with 13 Quartiles and Interquartile Range flashcards in the free StudySmarter app

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