Deciles - Complete Guide

Deciles Overview

Deciles divide a dataset into 10 equal parts, with 9 dividing points: D₁, D₂, …, D₉. Each decile represents 10% of the data. For example, D₁ is the value below which 10% of the data falls.

Deciles for Ungrouped Data

Formula

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

$$D_i = \text{Value of } \left(\frac{i(n+1)}{10}\right)^{th} \text{ observation, } i=1,2,…,9$$

where n is the total number of observations.

Example: Ungrouped Data

The daily sales (in hundreds of dollars) for 20 days:

12, 15, 18, 22, 25, 28, 30, 32, 35, 38, 40, 42, 45, 48, 50, 52, 55, 58, 60, 65

Find D₁, D₂, D₃, and D₉.

Solution:

The data is already arranged in ascending order with n = 20.

First Decile (D₁):

$$D_1 = \text{Value of } \left(\frac{1(20+1)}{10}\right)^{th} \text{ obs.} = \text{Value of } (2.1)^{th} \text{ obs.}$$

$$= 2^{nd} + 0.1(3^{rd} - 2^{nd}) = 15 + 0.1(18 - 15) = 15.3$$

Second Decile (D₂):

$$D_2 = \text{Value of } \left(\frac{2(20+1)}{10}\right)^{th} \text{ obs.} = \text{Value of } (4.2)^{th} \text{ obs.}$$

$$= 4^{th} + 0.2(5^{th} - 4^{th}) = 22 + 0.2(25 - 22) = 22.6$$

Third Decile (D₃):

$$D_3 = \text{Value of } \left(\frac{3(20+1)}{10}\right)^{th} \text{ obs.} = \text{Value of } (6.3)^{th} \text{ obs.}$$

$$= 6^{th} + 0.3(7^{th} - 6^{th}) = 28 + 0.3(30 - 28) = 28.6$$

Ninth Decile (D₉):

$$D_9 = \text{Value of } \left(\frac{9(20+1)}{10}\right)^{th} \text{ obs.} = \text{Value of } (18.9)^{th} \text{ obs.}$$

$$= 18^{th} + 0.9(19^{th} - 18^{th}) = 58 + 0.9(60 - 58) = 59.8$$

Deciles for Grouped Data

Formula

For grouped data:

$$D_i = L + \left(\frac{\frac{i \cdot N}{10} - CF}{f}\right) \times h$$

where:

• L = Lower class boundary of the decile class
• N = Total frequency
• CF = Cumulative frequency before the decile class
• f = Frequency of the decile class
• h = Class width
• i = 1, 2, 3, …, 9

Example: Grouped Data

Find D₁, D₅, and D₉.

Solution:

Cumulative Frequencies:

N = 50

D₁ (First Decile):

Position = (1 × 50)/10 = 5

Decile class: 0-20

$$D_1 = 0 + \left(\frac{5 - 0}{6}\right) \times 20 = 16.67$$

D₅ (Fifth Decile - Median):

Position = (5 × 50)/10 = 25

Decile class: 40-60

$$D_5 = 40 + \left(\frac{25 - 16}{14}\right) \times 20 = 52.86$$

D₉ (Ninth Decile):

Position = (9 × 50)/10 = 45

Decile class: 80-100 (CF crosses 45 here)

$$D_9 = 80 + \left(\frac{45 - 42}{8}\right) \times 20 = 87.5$$

Decile Interpretation

Relationship to Other Measures

• D₁ = P₁₀ (First decile = 10th percentile)
• D₂ = P₂₀ (Second decile = 20th percentile)
• D₅ = P₅₀ = Q₂ (Fifth decile = Median = 50th percentile)
• D₇ = P₇₀ = Three-quarter point
• D₉ = P₉₀ (Ninth decile = 90th percentile)

Decile Class Width

The range from D₁ to D₉ represents 80% of the middle data (10% to 90%).

$$\text{Decile Range} = D_9 - D_1$$

This is useful for identifying the spread of the main bulk of your data while excluding extreme values.

Applications

• Income distribution analysis (e.g., top decile earners)
• Academic achievement levels (grouping students)
• Performance benchmarking
• Quality control and process monitoring
• Risk assessment and profiling