Sample quantiles

01A sample quantile locates a position in ordered observations#

A sample quantile locates a position in ordered observations. The answer depends on the convention. This lesson uses nearest rank: for $0<p\leq1$, sort, multiply $p$ by the count $n$, and round upward to a whole position.

For $1,3,6,9,12$ and $p=0.6$, the rank is $0.6\times5=3$. Read the third value, $6$.

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02Round upward even when the decimal part is small: 2.1 gives rank 3#

Round upward even when the decimal part is small: $2.1$ gives rank $3$. An exact rank such as $2$ stays $2$. At $p=1$, this rule selects the largest observation.

Counts include repeats. If $2$ occurs twice and $9$ three times, the ordered list is $2,2,9,9,9$. With $p=0.4$, rank $2$ gives value $2$.

03Different conventions can give different answers#

Different conventions can give different answers. At $p=0.5$ for $2,6,10,14$, nearest rank selects $6$. The usual even-size median averages $6$ and $10$ to get $8$.