# Quartile

Statistics is an important part of mathematics and is used extensively. The concept of quarters is a fundamental one in statistics. Let us learn more about quarters and the Quartile Formula.

## Quartile Formula

Quartile, as it sounds phonetically, is a statistical term that divides the data into four quarters. It basically divides the data points into a data set in 4 quarters on the number line. One thing we need to keep in mind is that data points can be random and we have to put those numbers in line first on the number line in ascending order and then divide them into quartiles. It is basically an extended version of the median. Median divides the data into two equal parts which quartiles divide it into four parts. Once we divide the data, the four quartiles will be:

• 1st quartile also known as the lower quartile basically separates the lowest 25% of data from the highest 75%.
• 2nd quartile or the middle quartile also the same as the median it divides numbers into 2 equal parts.
• 3rd quartile or the upper quartile separates the highest 25% of data from the lowest 75%.

Quartile divides a set of observations into 4 equal parts. The first quartile is the value in the middle of the first term and the median. The median is the second quartile. The middle value between the median and the last term is the third quartile. Mathematically, they are represented as follows,

When the set of observations are arranged in ascending order the quartiles are represented as,

1. First Quartile(Q1)=((n+1)/4)th Term also known as the lower quartile.
2. The second quartile or the 50th percentile or the Median is given as:  Second Quartile(Q2)=((n+1)/2)th Term
3. The third Quartile of the 75th Percentile (Q3) is given as: Third Quartile(Q3)=(3(n+1)/4)th Term also known as the upper quartile.
4.  The interquartile range is calculated as: Upper Quartile – Lower Quartile.

## Solved Example for Quartile Formula

Question: Find the median, lower quartile, upper quartile and interquartile range of the following data set of values: 19, 21, 23, 20, 23, 27, 25, 24, 31?

Solution: Firstly arrange the values in ascending order.

Plugging in the values in the formulas above we get,

Median(Q2)=5th Term=23

Lower Quartile (Q1) = 2.5th Term = 11

Upper Quartile(Q3) = 7.5th Term = 24.5

IQR=Upper Quartile−Lower Quartile

IQR = 24.5 – 11

IQR = 13.5

Question: Find the upper quartile for the following set of numbers:
27, 19, 5, 7, 6, 9, 15, 12, 18, 2, 1.

Solution: The upper quartile formula is: Q3 = ¾(n + 1)th Term.

The formula doesn’t give you the value for the upper quartile, it gives you the place. For example, 5th place,  8th place etc.

So firstly we put your numbers in order: 1, 2, 5, 6, 7, 9, 12, 15, 18, 19, 27. Note that for very large data sets, you may want to use Excel to place your numbers in order. And then we work the formula. There are 11 numbers in the set, so:

Q3 = ¾(n + 1)th Term.

Q3 = ¾(12)th Term. = 9th Term.

In this set of numbers given, the upper quartile (18) is the 9th term or the 9th place from the left.

Example : Calculate the median, lower quartile, upper quartile, and interquartile range of the following data set of values: 20, 19, 21, 22, 23, 24, 25, 27, 26

Solution:

Arranging the values in ascending order: 19, 20, 21, 22, 23, 24, 25, 26, 27

Putting the values in the formulas above we get,

Median(Q2) = 5th Term = 23

Lower Quartile (Q1) = Mean of 2nd and 3rd term = (20 + 21)/2 = 20.5

Upper Quartile(Q3) = Mean of 7th and 8th term = (25 + 26)/2 = 25.5

IQR = Upper Quartile−Lower Quartile

IQR = 25.5 – 20.5

IQR = 5

Answer: IQR = 5

Example : What will be the upper quartile for the following set of numbers?
26, 19, 5, 7, 6, 9, 16, 12, 18, 2, 1.

Solution:

The formula for the upper quartile formula is Q3 = ¾(n + 1)th Term.

The formula instead of giving the value for the upper quartile gives us the place. For example, 8th place, 10th place, etc.

So firstly we put your numbers in ascending order: 1, 2, 5, 6, 7, 9, 12, 16, 18, 19, 26. There are a total of 11 numbers, so:

Q3 = ¾(n + 1)th Term.

Q3 = ¾(12)th Term. = 9th Term.

Solution: The upper quartile (18) is the 9th term or on the 9th place from the left.

Example : Find the 3rd quartile in the following data set: 4, 5, 8, 7, 11, 9, 9

Solution:

Let us first arrange our array in ascending order and it becomes 4, 5, 7, 8, 9, 9, 11

The median of our data is 8.

In order to find the 3rd quartile, we have to deal with the data points that are greater than the median that is 9, 9, 10.

In order to find the 3rd quartile, we have to find the median of the data points that are greater than the median that is 9, 9, 10.

Answer: Hence, the 3rd quartile of our data set is 9.