Octiles - Complete Guide

Octiles Overview

Octiles divide a dataset into 8 equal parts, with 7 dividing points: O₁, O₂, …, O₇. Each octile represents 12.5% of the data. Octiles are less commonly used than quartiles or deciles but are useful for more granular data division.

Octiles for Ungrouped Data

Formula

The formula for the i-th octile in ungrouped data is:

$$O_i = \text{Value of } \left(\frac{i(n+1)}{8}\right)^{th} \text{ observation, } i=1,2,…,7$$

where n is the total number of observations.

Example: Ungrouped Data

Test scores of 32 students (arranged in ascending order):

25, 28, 30, 32, 35, 38, 40, 42, 45, 48, 50, 52, 55, 58, 60, 62, 65, 68, 70, 72, 75, 78, 80, 82, 85, 88, 90, 92, 95, 98, 100, 102

Find O₁, O₂, O₄, and O₇.

Solution:

n = 32

First Octile (O₁):

$$O_1 = \text{Value of } \left(\frac{1(32+1)}{8}\right)^{th} \text{ obs.} = \text{Value of } (4.125)^{th} \text{ obs.}$$

$$= 4^{th} + 0.125(5^{th} - 4^{th}) = 32 + 0.125(35 - 32) = 32.375$$

Second Octile (O₂):

$$O_2 = \text{Value of } \left(\frac{2(32+1)}{8}\right)^{th} \text{ obs.} = \text{Value of } (8.25)^{th} \text{ obs.}$$

$$= 8^{th} + 0.25(9^{th} - 8^{th}) = 42 + 0.25(45 - 42) = 42.75$$

Fourth Octile (O₄ - Median):

$$O_4 = \text{Value of } \left(\frac{4(32+1)}{8}\right)^{th} \text{ obs.} = \text{Value of } (16.5)^{th} \text{ obs.}$$

$$= 16^{th} + 0.5(17^{th} - 16^{th}) = 62 + 0.5(65 - 62) = 63.5$$

Seventh Octile (O₇):

$$O_7 = \text{Value of } \left(\frac{7(32+1)}{8}\right)^{th} \text{ obs.} = \text{Value of } (28.875)^{th} \text{ obs.}$$

$$= 28^{th} + 0.875(29^{th} - 28^{th}) = 92 + 0.875(95 - 92) = 94.625$$

Octiles for Grouped Data

Formula

For grouped data:

$$O_i = L + \left(\frac{\frac{i \cdot N}{8} - CF}{f}\right) \times h$$

where:

• L = Lower class boundary of the octile class
• N = Total frequency
• CF = Cumulative frequency before the octile class
• f = Frequency of the octile class
• h = Class width
• i = 1, 2, 3, …, 7

Example: Grouped Data

Find O₁, O₄, and O₇.

Solution:

Cumulative Frequencies:

N = 55

O₁ (First Octile):

Position = (1 × 55)/8 = 6.875

Octile class: 10-20

$$O_1 = 10 + \left(\frac{6.875 - 5}{8}\right) \times 10 = 12.34$$

O₄ (Fourth Octile - Median):

Position = (4 × 55)/8 = 27.5

Octile class: 30-40

$$O_4 = 30 + \left(\frac{27.5 - 25}{14}\right) \times 10 = 31.79$$

O₇ (Seventh Octile):

Position = (7 × 55)/8 = 48.125

Octile class: 40-50

$$O_7 = 40 + \left(\frac{48.125 - 39}{10}\right) \times 10 = 49.125$$

Octile Interpretation

Octiles vs Other Position Measures

When to Use Octiles

• Detailed data distribution analysis
• When you need 8-part segmentation (e.g., weekly data analysis)
• Academic or research contexts requiring finer granularity
• Performance evaluation with 8 performance tiers
• Quality control with 8-level acceptance criteria

Relationship to Other Measures

• O₁ = P₁₂.₅ (First octile = 12.5th percentile)
• O₂ = P₂₅ = Q₁ (Second octile = 25th percentile = First quartile)
• O₄ = P₅₀ = Q₂ (Fourth octile = Median)
• O₆ = P₇₅ = Q₃ (Sixth octile = 75th percentile = Third quartile)