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A quartile is a type of quantile which divides the number of data points into four more or less equal parts, or quarters. The first quartile (Q₁) is defined as the middle number between the smallest number and the median of the data set. It is also known as the lower quartile or the 25th empirical quartile and it marks where 25% of the data is below or to the left of it (if data is ordered on a timeline from smallest to largest). The second quartile (Q₂) is the median of the data and 50% of the data lies below this point. The third quartile (Q₃) is the middle value between the median and the highest value of the data set. It is also known as the upper quartile or the 75th empirical quartile and 75% of the data lies below this point. ^[1] Due to the fact that the data needs to be ordered from smallest to largest in order to compute quartiles, quartiles are a form of Order statistic.

Along with the minimum and the maximum of the data, which are also quartiles, the three quartiles described above provide a five-number summary of the data. ^[1] This summary is important in statistics because it provides information about both the center and the spread of the data. Knowing the lower and upper quartile provides information on how big the spread is and if the dataset is skewed toward one side. Since quartiles divide the number of data points evenly, the range is not the same between quartiles (ie. Q₃-Q₂ ≠ Q₂-Q₁). While the maximum and minimum also show the spread of the data, the upper and lower quartiles can provide more detailed information on the location of specific data points, the presence of outliers in the data, and the difference in spread between the middle 50% of the data and the outer data points. ^[2]

For a visual representation of quartiles: https://www.mathsisfun.com/data/quartiles.html

Computer software for Quartiles[edit]

Excel:

The Excel function QUARTILE(array, quart) provides the desired quartile value for a given array of data. In the Quartile function, array is the dataset of numbers that is being analyzed and quart is any of the following 5 values depending on which quartile is being calculated. ^[3]


Quart	Output QUARTILE Value
0	Minimum value
1	Lower Quartile (25th percentile)
2	Median
3	Upper Quartile (75th percentile)
4	Maximum value

MATLAB:

In order to calculate quartiles in Matlab, the function quantile(A,p) can be used. Where A is the vector of data being analyzed and p is the percentage that relates to the quartiles as stated below.


p	Output QUARTILE Value
0	Minimum value
0.25	Lower Quartile (25th percentile)
0.5	Median
0.75	Upper Quartile (75th percentile)
1	Maximum value

^ ^a ^b A modern introduction to probability and statistics : understanding why and how. Dekking, Michel, 1946-. London: Springer. 2005. pp. 234–238. ISBN 978-1-85233-896-1. OCLC 262680588.{{cite book}}: CS1 maint: others (link)
^ Knoch, Jessica (2018-02-23). "How are Quartiles Used in Statistics?". Magoosh Statistics Blog. Retrieved 2019-12-10.
^ "How to use the Excel QUARTILE function | Exceljet". exceljet.net. Retrieved 2019-12-10.

[:0-1] A modern introduction to probability and statistics : understanding why and how. Dekking, Michel, 1946-. London: Springer. 2005. pp. 234–238. ISBN 978-1-85233-896-1. OCLC 262680588.{{cite book}}: CS1 maint: others (link)

[2] Knoch, Jessica (2018-02-23). "How are Quartiles Used in Statistics?". Magoosh Statistics Blog. Retrieved 2019-12-10.

[3] "How to use the Excel QUARTILE function | Exceljet". exceljet.net. Retrieved 2019-12-10.

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