## Quick Reference

If a set of numerical data has *n* elements and is arranged in increasing order

*x*_{1}≤*x*_{2}≤…≤*x** _{n}*,

the lower quartile (*Q*_{1}) may be taken to be the median of the lower half of the data, i.e. of *x*_{1}, *x*_{2},…, *x*_{½(n−1)} if *n* is odd, and the median of *x*_{1}, *x*_{2},…, *x*_{½n} if *n* is even. The upper quartile (*Q*_{3}) may be taken to be the median of the upper half of the data, i.e. of *x*_{½(n+1)}, *x*_{½(n+3)},…, *x** _{n}* if

*n*is odd, and the median of

*x*

_{½(n+2)},

*x*

_{½(n+4),}…,

*x*

*if*

_{n}*n*is even. The difference

*Q*

_{3}−

*Q*

_{1}is the interquartile range, a term introduced by Galton in 1882. An alternative term is the midspread.

As an example, consider the ordered data:

101, 103, 104, 105, 106, 107, 108, 109, 111, 111, 111, 115, 118, 121, 124, 127, 130, 156, 199.

There are nineteen observations. The tenth largest is 111, the median. Within the lower nine values, the fifth largest is 106 (=*Q*_{1}). Within the upper nine values the fifth largest is 124 (=*Q*_{3}). The inter-quartile range is 124−106=18.

When there are many observations it may be easier to read approximate values for the lower and upper quartiles from a cumulative frequency graph. These will be the values of the variable corresponding to cumulative relative frequencies of 25% and 75%, respectively.

For a continuous random variable *X*, the lower quartile of the distribution is such that P(*X*<*Q*_{1})=¼ and the upper quartile is such that P(*X*<*Q*_{3})=¾.

In his 1970 book on exploratory data analysis, Tukey referred (in the context of data) to the quartiles as hinges and he called the interquartile range the H-spread. Tukey defined a step as 1.5 × H-spread, and proposed that values one step beyond a hinge should be called inner fences and values two steps beyond a hinge should be called outer fences. Any data item beyond an outer fence would be called far out.

In the previous data the hinges are 106 and 124, thus the H-spread is 124−106=18 and the step is 1.5 × 18=27. The inner fences are at 106−27=79 and 124+27=151. The outer fences are at 79−27=52 and 151+27=178. The observation 199 is greater than 178 and is therefore far out.

See also boxplot; outlier; quantile; skewness; trimean.

**From:**
quartile
in
A Dictionary of Statistics »

*Subjects:*
Probability and Statistics.

## Related content in Oxford Index

##### Reference entries

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.