Partition Values for Ungrouped (Raw) Data

Position Formulas (Raw Data)

$Q_i = \text{value of the } \left[\frac{i(n+1)}{4}\right]^{\text{th}} \text{ observation}, \quad i = 1, 2, 3$

$D_j = \text{value of the } \left[\frac{j(n+1)}{10}\right]^{\text{th}} \text{ observation}, \quad j = 1, 2, \ldots, 9$

$P_k = \text{value of the } \left[\frac{k(n+1)}{100}\right]^{\text{th}} \text{ observation}, \quad k = 1, 2, \ldots, 99$

Here $n$ is the number of observations. Notice all three use $(n+1)$ — the median position $\tfrac{n+1}{2}$ is just the special case $Q_2$, $D_5$ or $P_{50}$.

When the Position Is Not a Whole Number

If the position works out to, say, the $7.5$th observation, the partition value lies between the 7th and 8th ordered observations. Take it by linear interpolation:

$\text{value} = x_{(7)} + 0.5\,\big(x_{(8)} - x_{(7)}\big),$

i.e. the lower observation plus the fractional part times the gap to the next observation. (In general, for position $m.f$ the value is $x_{(m)} + f\,[\,x_{(m+1)} - x_{(m)}\,]$.)