Percentiles - Complete Guide

Percentiles Overview

Percentiles divide data into 100 equal parts. The i-th percentile is the value below which i% of the observations fall. For example, the 25th percentile (P₂₅) is the value below which 25% of the data falls.

Percentiles for Ungrouped Data

Formula

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

$$P_i = \text{Value of } \left(\frac{i(n+1)}{100}\right)^{th} \text{ observation, } i=1,2,3,…,99$$

where n is the total number of observations.

Example 1: Finding Specific Percentiles

The test score of a sample of 20 students:

20, 30, 21, 29, 10, 17, 18, 15, 27, 25, 16, 15, 19, 22, 13, 17, 14, 18, 12, 9

Find P₁₀, P₂₀, and P₈₀.

Solution:

Arrange the data in ascending order:

9, 10, 12, 13, 14, 15, 15, 16, 17, 17, 18, 18, 19, 20, 21, 22, 25, 27, 29, 30

Tenth Percentile (P₁₀):

$$P_{10} = \text{Value of } \left(\frac{10(20+1)}{100}\right)^{th} \text{ obs.} = \text{Value of } (2.1)^{th} \text{ obs.}$$

$$= 2^{nd} \text{ obs.} + 0.1(3^{rd} \text{ obs.} - 2^{nd} \text{ obs.})$$

$$= 10 + 0.1(12 - 10) = 10.2$$

Thus, 10% of students scored 10.2 or below.

Twentieth Percentile (P₂₀):

$$P_{20} = \text{Value of } \left(\frac{20(20+1)}{100}\right)^{th} \text{ obs.} = \text{Value of } (4.2)^{th} \text{ obs.}$$

$$= 4^{th} \text{ obs.} + 0.2(5^{th} \text{ obs.} - 4^{th} \text{ obs.})$$

$$= 13 + 0.2(14 - 13) = 13.2$$

Thus, 20% of students scored 13.2 or below.

Eightieth Percentile (P₈₀):

$$P_{80} = \text{Value of } \left(\frac{80(20+1)}{100}\right)^{th} \text{ obs.} = \text{Value of } (16.8)^{th} \text{ obs.}$$

$$= 16^{th} \text{ obs.} + 0.8(17^{th} \text{ obs.} - 16^{th} \text{ obs.})$$

$$= 22 + 0.8(25 - 22) = 24.4$$

Thus, 80% of students scored 24.4 or below.

Example 2: Comparing Data Points

The hourly wage rates (in dollars) of 25 employees:

20, 28, 30, 18, 27, 19, 22, 21, 24, 25, 18, 25, 20, 27, 24, 20, 23, 32, 20, 35, 22, 26, 25, 28, 31

Find: a) Upper wage for lowest 15% (P₁₅), b) Upper wage for lowest 45% (P₄₅), c) Lower wage for upper 25% (P₇₅)

Solution:

Arrange in ascending order:

18, 18, 19, 20, 20, 20, 20, 21, 22, 22, 23, 24, 24, 25, 25, 25, 26, 27, 27, 28, 28, 30, 31, 32, 35

a) P₁₅:

$$P_{15} = \text{Value of } \left(\frac{15(25+1)}{100}\right)^{th} \text{ obs.} = \text{Value of } (3.9)^{th} \text{ obs.}$$

$$= 3^{rd} + 0.9(4^{th} - 3^{rd}) = 19 + 0.9(20 - 19) = 19.9 \text{ dollars}$$

b) P₄₅:

$$P_{45} = \text{Value of } \left(\frac{45(25+1)}{100}\right)^{th} \text{ obs.} = \text{Value of } (11.7)^{th} \text{ obs.}$$

$$= 11^{th} + 0.7(12^{th} - 11^{th}) = 23 + 0.7(24 - 23) = 23.7 \text{ dollars}$$

c) P₇₅:

$$P_{75} = \text{Value of } \left(\frac{75(25+1)}{100}\right)^{th} \text{ obs.} = \text{Value of } (19.5)^{th} \text{ obs.}$$

$$= 19^{th} + 0.5(20^{th} - 19^{th}) = 27 + 0.5(28 - 27) = 27.5 \text{ dollars}$$

Percentiles for Grouped Data

Formula

For grouped data:

$$P_i = L + \left(\frac{\frac{i \cdot N}{100} - CF}{f}\right) \times h$$

where:

• L = Lower class boundary of the percentile class
• N = Total frequency
• CF = Cumulative frequency before the percentile class
• f = Frequency of the percentile class
• h = Class width
• i = Percentile number (1-99)

Example: Grouped Data

Find P₂₅, P₅₀, and P₇₅.

Solution:

Cumulative Frequencies:

N = 50

P₂₅ (25th Percentile):

Position = (25 × 50)/100 = 12.5

Percentile class: 20-30

$$P_{25} = 20 + \left(\frac{12.5 - 12}{14}\right) \times 10 = 20.36$$

P₅₀ (50th Percentile - Median):

Position = (50 × 50)/100 = 25

Percentile class: 20-30

$$P_{50} = 20 + \left(\frac{25 - 12}{14}\right) \times 10 = 29.29$$

P₇₅ (75th Percentile):

Position = (75 × 50)/100 = 37.5

Percentile class: 30-40

$$P_{75} = 30 + \left(\frac{37.5 - 26}{16}\right) \times 10 = 37.19$$

Common Percentile Values

• P₁: First percentile
• P₁₀: Tenth percentile (lower decile)
• P₂₅: Twenty-fifth percentile (first quartile)
• P₅₀: Fiftieth percentile (median)
• P₇₅: Seventy-fifth percentile (third quartile)
• P₉₀: Ninetieth percentile (upper decile)
• P₉₉: Ninety-ninth percentile

Applications of Percentiles

• Test Scoring: Interpreting standardized test results
• Health Metrics: Growth charts and BMI percentiles
• Academic Performance: Class rankings
• Quality Control: Setting acceptance standards
• Income Distribution: Analyzing wealth inequality
• Performance Analysis: Benchmarking against competition