Quartile Formula (Table of Contents)
Quartile Formula Definition
Quartile, as its name sounds, is a statistical term which divides the data into quarters or four defined intervals. 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 number 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:
 1^{st} quartile or lower quartile basically separate the lowest 25% of data from the highest 75%.
 2^{nd} quartile or middle quartile also same as median it divides numbers into 2 equal parts.
 3^{rd} quartile or the upper quartile separate the highest 25% of data from the lowest 75%.
Formula For Quartile:
Let’s say that we have a data set with N data points:
X – {X1, X2, X3……….. XN}
The formula for quartiles is given by:
Lower Quartile (Q1) = (N+1) * 1 / 4
Middle Quartile (Q2) = (N+1) * 2 / 4
Upper Quartile (Q3 )= (N+1) * 3 / 4
Interquartile Range = Q3 – Q1
What it basically means is that in a data set with N data points:
((N+1) * 1 / 4)^{th} term is the lower quartile
((N+1) * 2 / 4)^{th} term is the middle quartile
((N+1) * 3 / 4)^{th} term is the upper quartile
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Interquartile range basically distances between lower quartile and upper quartile.
Examples of Quartile Formula (With Excel Template)
Let’s take an example to understand the calculation of Quartile in a better manner.
Quartile Formula – Example #1
Let’s say we have a data sets A which contains 19 data points. Calculate the Quartile for data set A.
Data Set:
First of all, you have to arrange this ascending order i.e. from lowest to highest:
Number of data points is calculated as:
Quartile is calculated using below given formula
Lower Quartile (Q1) = (N+1) * 1 / 4
 Lower Quartile (Q1) = (19+1) * 1/4
 Lower Quartile (Q1) = 20 / 4 = 5^{th} data point
So Lower Quartile (Q1) = 29
Middle Quartile (Q2) = (N+1) * 2 / 4
 Middle Quartile (Q2) = (19+1) * 2/4
 Middle Quartile (Q2) = 40 / 4 = 10^{th} data point
So Middle Quartile (Q2) = 43
Upper Quartile (Q3)= (N+1) * 3 / 4
 Upper Quartile (Q3)= (19+1) * 3/4
 Upper Quartile (Q3)= 60 / 4 = 15^{th} data point
So Upper Quartile (Q3)= 67
Interquartile Range is calculated using the formula given below
Interquartile Range = Q3 – Q1
 Interquartile Range = 15– 5
 Interquartile Range = 10^{th} data point
So Interquartile Range = 43
If you see the data set, the median of this set is: (n+1)/2 = 20/2 = 10^{th} value i.e. 43, this is same as Q2.
Inference:
 Value 29 divides the data set in such a way that the lowest 25% are above it and the highest 75% are below it
 Value 43 divides the data set in two equal parts
 Value 67 divides the data set in such a way that the highest 25% are below it and the lowest 75% are above it
Quartile Formula – Example #2
Let’s see another example of how companies and businesses can use this tool to make an informed decision on which product to produce.
Suppose that you are a manufacturer of running shoes and a wellknown brand amongst the athletes who run a marathon, play sports, etc. You have collected the data of the shoe sizes these athletes wear so that in future you produce more of that size to meet demand.
You have collected a sample of 15 athletes from different sports. Calculate the Quartile.
Data set is given below:
Arrange the shoe size in ascending order.
Quartile is calculated using below given formula
Lower Quartile (Q1) = (N+1) * 1 / 4
 Lower Quartile (Q1)= (15+1) * 1/4
 Lower Quartile (Q1)= 16 / 4 = 4^{th} data point
So Lower Quartile (Q1)= 10
Middle Quartile (Q2) = (N+1) * 2 / 4
 Middle Quartile (Q2) = (15+1) * 2/4
 Middle Quartile (Q2)= 32 / 4 = 8^{th} data point
So Middle Quartile (Q2)= 10
Upper Quartile (Q3) = (N+1) * 3 / 4
 Upper Quartile (Q3)= (15+1)*3/4
 Upper Quartile (Q3)= 48 / 4 = 12^{th} data point
So Upper Quartile (Q3) = 11
Interquartile Range is calculated using the formula given below
Interquartile Range = Q3 – Q1
 Interquartile Range = 12 – 4
 Interquartile Range = 8^{th} data point
So Interquartile Range = 10
Explanation
To get a better understanding of quartiles, we need to understand the median in a better way. The median divides the data set in exactly two equal halves but it does not tell us anything about the spread of the data on either side. A quartile is an extended version of that and by dividing the data set in four parts, it deals with the spread of values above and below the mean. Also, there are other statistical tools which tell us about the range of the data set, the center of the data set, etc. But quartile formula helps us in understanding all these elements. Median, which is middle quartile tells us the center point and upper and lower quartiles tell us the spread.
Relevance and Uses of Quartile Formula
As discussed above, the quartile formula helps us in dividing the data into four parts very quickly and eventually makes it easy for us to understand the data in these parts. For example, a class teacher wants to award the top 25% of students with goodies and gifts and wants to give another chance to bottom 25% of students to improve their score. He can use quartiles and can divide the data. So if the quartiles are said 51, 65, 72 and a student score is say 78, he will get goodies. If another student has a score of 48, he will another chance to improve the score, quick and easy interpretation.
Recommended Articles
This has been a guide to Quartile Formula. Here we discuss the definition and how to calculate Quartile along with practical examples and downloadable excel template. You may also look at the following articles to learn more –
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