## 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.