One of the four divisions of observations which have been grouped into four equal-sized sets based on their statistical rank. The quartile including the top statistically ranked members is called the first quartile and denoted Q_1. The other quartiles are similarly denoted Q_2, Q_3, and Q_4. For N data points with N of the form 4n+5 (for n=0, 1, ...), the hinges are identical to the first and third quartiles.

The following table summarizes a number of common methods for computing the position of the first and third quartiles from a sample size n (P. Stikker, pers. comm., Jan. 24, 2005). In the table, [x] denotes the nearest integer function.

method1st quartile1st quartile3rd quartile3rd quartile
n oddn evenn oddn even
Tukey (Hoaglin et al. 1983)(n+3)/4(n+2)/4(3n+1)/4(3n+2)/4
Moore and McCabe (2002)(n+1)/4(n+2)/4(3n+3)/4(3n+2)/4
Mendenhall and Sincich (1995)[(n+1)/4][(n+1)/4][(3n+3)/4][(3n+3)/4]
Freund and Perles (1987)(n+3)/4(n+3)/4(3n+1)/4(3n+1)/4

See also

Hinge, Interquartile Range, Percentile, Quantile, Quartile Deviation, Quartile Variation Coefficient

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Freund, J. and Perles, B. "A New Look at Quartiles of Ungrouped Data." American Stat. 41, 200-203, 1987.Hoaglin, D.; Mosteller, F.; and Tukey, J. (Ed.). Understanding Robust and Exploratory Data Analysis. New York: Wiley, pp. 39, 54, 62, 223, 1983.Kenney, J. F. and Keeping, E. S. "Quartiles." §3.3 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 35-37, 1962.Mendenhall, W. and Sincich, T. L. Statistics for Engineering and the Sciences, 4th ed. Prentice-Hall, 1995.Moore, D. S. and McCabe, G. P. Introduction to the Practice of Statistics, 4th ed. New York: W. H. Freeman, 2002.Whittaker, E. T. and Robinson, G. The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 184-186, 1967.

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Cite this as:

Weisstein, Eric W. "Quartile." From MathWorld--A Wolfram Web Resource.

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