Quartile Calculator

The quartile calculator is really helpful when you want to find the five-number summary for the Box-and-Whisker plots. This statistics calculator will calculate the first quartile (Q1), second quartile (Q2) or the median, third quartile (Q3), minimum value, and maximum value of the given data set. Further, it calculates the interquartile range and the range too.

You just need to type or copy and paste the data and click the “calculate” button. Make sure to separate each number with a comma or a space.

Quartiles

The quartiles are one of the measures of position. They help to describe the position of some value in relation to other values in a data set.

Quartiles are used to divide an increasing array of data (data is arranged in ascending order) into four equal sections. Each of these sections contains an equal number of items. We can calculate three quartiles for a data set.

• First quartile (Q1 or the lower quartile)
• Second quartile (Q2 or the median)
• Third quartile (Q3 or the upper quartile)

The first quartile (Q1) is the data value that separates the bottom 25% and top 75% of the data which is arranged in ascending order. So, the first quartile has 25% of the items lower than it and 75% of items greater than it. This is equal to the 25th percentile of the data set.

The second quartile (Q2) is the data value that separates the bottom 50% and top 50% of the data which is arranged in ascending order. So, the second quartile has 50% of the items lower than it and 50% of items greater than it. The second quartile is exactly equal to the median as well as the 50th percentile of the data set.

The third quartile (Q3) is the data value that separates the bottom 75% and top 25% of the data which is arranged in ascending order. So, the third quartile has 75% of the items lower than it and 25% of items greater than it. This is equal to the 75th percentile of the data set.

Quartiles Calculation

You can follow the below steps to find the quartiles:

• Arrange the data in ascending order.
• Find the median of the data values. This is the second quartile.
• Find the median of the data values which are below the second quartile. This is the first quartile.
• Find the median of the data values which are above the second quartile. This is the third quartile.

Example 1

The following data set is representing the starting salary of newly graduated accountants in a college. Find the median (Q2), lower quartile (Q1), and upper quartile (Q3) for the starting salaries. Interpret your results.

$55,000, $60,000, $52,000, $45,000, $74,000, $75,000, $48,000, $58,000, $72,000, $66,000, $45,000, $50,000, $54,000, $65,000, $71,000

Solution

First, we will arrange the data in increasing order.

$45,000, $45,000, $48,000, $50,000, $52,000, $54,000, $55,000, $58,000, $60,000, $65,000, $66,000, $71,000, $72,000, $74,000, $75,000

Then, we will find the location of the second quartile or the median.

$$Second\ quartile(Q2)=\left(\frac{N+1}{2}\right)^{th}item=\left(\frac{15+1}{2}\right)^{th}item=8^{th}item=58,000$$

Next, find the median of the data values below the Q2 to find the Q1.

$45,000, $45,000, $48,000, $50,000, $52,000, $54,000, $55,000

First quartile (Q1) = $50,000

Next, find the median of the data values above Q2 to find the Q3.

$60,000, $65,000, $66,000, $71,000, $72,000, $74,000, $75,000

Third quartile (Q3) = $71,000

You can interpret the above quartiles as follows.

25% of newly graduated accountants earn less than $50,000, and 25% earn more than $71,000. 50% of newly graduated accountants earn more than $58,000, while the other 50% earn less than that.

You can see that from the above example, for an odd number of data, the quartiles will be original data values. However, with an even number of data, the quartiles will not correspond to the initial values. Let’s modify the above example to learn this.

Example 2

Assume that you missed including one salary data to the data in Example 1. The salary that you missed is $95,000. Find the revised median (Q2), lower quartile (Q1), and upper quartile (Q3) for the starting salaries.

Solution

First, we will arrange the data in increasing order.

$45,000, $45,000, $48,000, $50,000, $52,000, $54,000, $55,000, $58,000, $60,000, $65,000, $66,000, $71,000, $72,000, $74,000, $75,000, $95,000

Then, we will find the location of the quartiles.

$$Second\ quartile(Q2)=\left(\frac{N+1}{2}\right)^{th}item=\left(\frac{16+1}{2}\right)^{th}item=8.5^{th}item$$

$$Second\ quartile(Q2)=\frac{8^{th}item+9^{th}item}{2}=\frac{58,000+60,000}{2}=59,000$$

Now, divide the data set at the median into two groups. Find the median of the data values below the Q2 to find the Q1.

$45,000, $45,000, $48,000, $50,000, $52,000, $54,000, $55,000, $58,000

First quartile (Q1)=($50,000 + $52,000)/2 = $51,000

Next, find the median of the data values above the Q2 to find the Q3.

$60,000, $65,000, $66,000, $71,000, $72,000, $74,000, $75,000, $95,000

Third quartile (Q3) = ($71,000 + $72,000)/2 = $71,500

Interquartile range

The difference between the upper quartile (Q3) and the lower quartile (Q1) is known as the interquartile range.

• Interquartile range (IQR) = Upper quartile - Lower quartile
• Interquartile range (IQR) = Third quartile - First quartile
• Interquartile range (IQR) = Q3- Q1

The interquartile range eliminates the lowest 25% of items and the highest 25% of items of the data array. In other words, the interquartile range focuses on the spread of the middle 50% of the data array. As the interquartile range eliminates the items below the lower quartile and the items above the upper quartile, the interquartile range is free of the extreme values or the outliers of the data set. This eliminates the major drawback of the range calculation.

Example 3

Find the interquartile range for Example 1.

Solution

We have already found the quartiles for the data range:

• First quartile (Q1) = $50,000
• Second quartile (Q2) = $58,000
• Third quartile (Q3) = $71,000

Let’s apply the above data to the interquartile formula.

Interquartile range (IQR) = Third quartile (Q3)- First quartile (Q1) = $71,000 - $50,000 = $21,000

Example 4

Find the interquartile range for Example 2.

Solution

We have already found the quartiles for the data range:

• First quartile (Q1) = $51,000
• Second quartile (Q2) = $59,000
• Third quartile (Q3) = $71,500

Let’s apply the above data to the interquartile range formula.

Interquartile range (IQR) = Third quartile (Q3) - First quartile (Q1) = $71,500 - $51,000 = $20,500

Minimum and maximum values

The minimum value of a data set means the lowest value of the data set. When you arrange a data set in increasing order, it is the first value of your data set.

The maximum value of a data set means the highest value of the data set. When you arrange a data set in increasing order, it is the last value of your data set.

The minimum value and the maximum value help to understand the total spread of the data set. The range which is the basic measure of dispersion is based on the minimum value and the maximum value of the data set.

Example 5

Find the minimum and maximum values of the data set of the starting salary of newly graduated accountants of Example 1.

Solution

We have already arranged the data set in ascending order as below.

$45,000, $45,000, $48,000, $50,000, $52,000, $54,000, $55,000, $58,000, $60,000, $65,000, $66,000, $71,000, $72,000, $74,000, $75,000

The minimum salary is the first salary data in the above array. Therefore,

The minimum starting salary of newly graduated accountants = $45,000

The maximum salary is the last salary data in the above array. Therefore,

The maximum starting salary of newly graduated accountants = $75,000

Example 6

Find the minimum and maximum values of the data set of the starting salary of newly graduated accountants of Example 2.

Solution

We have already arranged the data set in ascending order as below.

$45,000, $45,000, $48,000, $50,000, $52,000, $54,000, $55,000, $58,000, $60,000, $65,000, $66,000, $71,000, $72,000, $74,000, $75,000, $95,000

The minimum salary is the first salary data in the above array. Therefore,

The minimum starting salary of newly graduated accountants = $45,000

The maximum salary is the last salary data in the above array. Therefore,

The maximum starting salary of newly graduated accountants = $95,000

Range of a set

The range in statistics is the most basic measure of the dispersion of a data set. It is calculated as the difference between the largest (maximum) value and the smallest (minimum) value of the data set.

The range of a set = Maximum value - Minimum value

The range of a set = Largest value - Smallest value

The range is the total distance or the total spread between the extreme values of the data set. It is a rough measure of dispersion.

The range is depending only on two extreme items of the data set. If the extreme values contain any outliers, the range is easily distorted and biased.

As the range is not based on all the data of the data set, the range is not considered a good measure of dispersion.

Example 7

Find the range of the data set of the starting salary of newly graduated accountants of Example 1.

Solution

Previously we have found the minimum value and the maximum value of the data set.

The minimum starting salary of newly graduated accountants = $45,000

The maximum starting salary of newly graduated accountants = $75,000

Now we will apply the above values to the range formula.

The range of a set = Maximum value - Minimum value = $75,000 - $45,000 = $30,000

Example 8

Find the range of the data set of the starting salary of newly graduated accountants of Example 2.

Solution

Previously we have found the minimum value and the maximum value of the data set.

The minimum starting salary of newly graduated accountants = $45,000

The maximum starting salary of newly graduated accountants = $95,000

Now we will apply the above values to the range formula.

The range of a set = Maximum value - Minimum value = $95,000 - $45,000 = $50,000

Quartile Calculations Applications in Real World

The quartile computations are useful when we wish to eliminate the data set's extreme values and examine its distribution. The list below shows the several fields that use quartiles to make decisions.

Human resources - The quartiles of salaries are determined before establishing the salary range of employees in a company. It helps in the elimination of extremely low salaries, such as trainee salaries, and extremely high salaries resulting from the experience and excellent talents of employees.

Finance - When planning monthly spending, quartiles are calculated to get an idea of how the expenses were spread out in the past. It helps in avoiding over- and under-budgeting.

This helps provide data on the range of production capabilities that are not distorted by power outages, strikes, days of out-of-stock materials, and so on.

Marketing - When marketers analyze their competitors' price ranges, they identify the quartiles for the competitor prices. They can then omit the pricing of low-quality and highly branded products during the analysis.