Percentiles and Quartiles

Overview

Percentiles and quartiles describe the position of a value within a dataset, telling us what percentage of data falls below that value.

Definitions

Term	Definition	Notation
Percentile	Value below which a given % falls	$P_k$
Quartile	Divides data into 4 equal parts	$Q_1, Q_2, Q_3$
Decile	Divides data into 10 equal parts	$D_1, \ldots, D_9$

Percentiles

Definition

$$\text{Percentile rank of } x = \frac{\text{Number of values below } x}{\text{Total values}} \times 100$$

Key Percentiles

Percentile	Also Known As	Meaning
$P_{25}$	$Q_1$, First quartile	25% below
$P_{50}$	$Q_2$, Median	50% below
$P_{75}$	$Q_3$, Third quartile	75% below
$P_{10}$	$D_1$, First decile	10% below
$P_{90}$	$D_9$, Ninth decile	90% below

Percentile Calculation

Locator Formula

For the $p$-th percentile:

$$L = \frac{p}{100}(n + 1)$$

Interpolation

If $L = k + d$ (where $k$ is integer, $d$ is decimal):

$$P_p = x_k + d(x_{k+1} - x_k)$$

Example

Data (sorted): 12, 15, 18, 22, 25, 28, 32, 35 ($n = 8$)

Find $P_{30}$:

$$L = \frac{30}{100}(8 + 1) = 2.7$$ $$P_{30} = x_2 + 0.7(x_3 - x_2) = 15 + 0.7(18 - 15) = 15 + 2.1 = 17.1$$

Quartiles

Definitions

Quartile	Percentile	Description
$Q_1$	$P_{25}$	First quartile
$Q_2$	$P_{50}$	Second quartile (median)
$Q_3$	$P_{75}$	Third quartile

Method 1: Median of Halves

1. Find the median ($Q_2$)
2. $Q_1$ = median of values below $Q_2$
3. $Q_3$ = median of values above $Q_2$

Method 2: Position Formula

$$Q_1 \text{ position} = \frac{n+1}{4}$$ $$Q_2 \text{ position} = \frac{2(n+1)}{4} = \frac{n+1}{2}$$ $$Q_3 \text{ position} = \frac{3(n+1)}{4}$$

Calculating Quartiles Example

Data: 3, 7, 8, 12, 13, 14, 18, 21, 23, 27 ($n = 10$)

$$Q_2 = \text{median} = \frac{13 + 14}{2} = 13.5$$

Lower half: 3, 7, 8, 12, 13

$$Q_1 = 8$$

Upper half: 14, 18, 21, 23, 27

$$Q_3 = 21$$

Z-Score and Percentiles

For normal distributions:

Z-Score	Percentile
-3	0.13
-2	2.28
-1	15.87
0	50
1	84.13
2	97.72
3	99.87

Converting z-score to percentile for normal distribution:

$$\text{Percentile} = \Phi(z) \times 100$$

Where $\Phi(z)$ is the standard normal CDF.

Applications

Use Case	Percentile/Quartile
Standardized tests	SAT, GRE percentile ranks
Growth charts	Height/weight percentiles
Income distribution	Household income quartiles
Academic rankings	Class rank percentile

Interpreting Percentiles

"A score at the 90th percentile" means:

• 90% of scores are at or below this value
• 10% of scores are above this value
• NOT the same as "scoring 90%"

Comparing Methods

Different software may use different methods:

Method	Description
Inclusive	Uses all data for each half
Exclusive	Excludes median from halves
Linear interpolation	Interpolates between values

Always specify the method when precision matters!