Octiles for ungrouped data
Octiles are the values of arranged data which divide whole data into eight equal parts. They are 7 in numbers namely $O_1,O_2, \cdots, O_7$. Here $O_1$ is first octile, $O_2$ is second octile, $O_3$ is third octile and so on.
The formula for $i^{th}$ octile is
$O_i =$ Value of $\bigg(\dfrac{i(n+1)}{8}\bigg)^{th}$ observation, $i=1,2,3,\cdots, 7$
where,
- $n$ is the total number of observations.
Example
Diastolic blood pressure (in mmHg) of a sample of 18 patients admitted to the hospitals are as follows:
65, 76, 64, 73, 74, 80, 71, 68, 66,
81, 79, 75, 70, 62, 83, 63, 77, 78.
Find the value of $O_4$ and $O_6$.
Solution
The formula for $i^{th}$ octile is
$O_i =$ Value of $\bigg(\dfrac{i(n+1)}{8}\bigg)^{th}$ observation, $i=1,2,3,\cdots, 7$
where $n$ is the total number of observations.
Arrange the data in ascending order
62, 63, 64, 65, 66, 68, 70, 71, 73,
74, 75, 76, 77, 78, 79, 80, 81, 83
Fourth octile $O_4$
The fourth octile $O_4$ can be computed as follows:
$$ \begin{aligned} O_{4} &=\text{Value of }\bigg(\dfrac{4(n+1)}{8}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{4(18+1)}{8}\bigg)^{th} \text{ obs.}\\ &= \text{Value of }\big(9.5\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(9\big)^{th} \text{ obs.}+0.5 \big(\text{Value of } \big(10\big)^{th}\text{ obs.}-\text{Value of }\big(9\big)^{th} \text{ obs.}\big)\\ &=73+0.5\big(74 -73\big)\\ &=73.5 \text{ mmHg}. \end{aligned} $$
Thus, $50$ % of patients had diastolic blood pressure less than or equal to $73.5$ mmHg.
Sixth octile $O_6$
The sixth octile $O_6$ can be computed as follows:
$$ \begin{aligned} O_{6} &=\text{Value of }\bigg(\dfrac{6(n+1)}{8}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{6(18+1)}{8}\bigg)^{th} \text{ obs.}\\ &= \text{Value of }\big(14.25\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(14\big)^{th} \text{ obs.}+0.25 \big(\text{Value of } \big(15\big)^{th}\text{ obs.}-\text{Value of }\big(14\big)^{th} \text{ obs.}\big)\\ &=78+0.25\big(79 -78\big)\\ &=78.25 \text{ mmHg}. \end{aligned} $$
Thus, $75$ % of the patients had diastolic blood pressure less than or equal to $78.25$ mmHg.
