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, 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,
- First Quartile(Q1)=((n+1)/4)th Term also known as the lower quartile.
- The second quartile or the 50th percentile or the Median is given as: Second Quartile(Q2)=((n+1)/2)th Term
- 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.
- 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,
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.