Percentile, Quartile, and Interquartile Range (IQR)

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Percentiles, Quartiles, Percentile Ranks, and IQR

Definition

A percentile indicates the value below which a given percentage of observations fall in a dataset.Quartiles divide data into four equal parts, and a percentile rank represents the percentage of scores that fall below a particular value. The Interquartile Range (IQR) is the range between the 25th and 75th percentiles.

Key Formulas

Percentile (Linear Interpolation):

\begin{align*} \text{Rank} &= \frac{\text{Percentile Rank} * (\text{Sample Size} - 1)}{100} + 1 \\ \text{Percentile} &= x_{\lfloor \text{Rank} \rfloor} + (\text{Rank} - \lfloor \text{Rank} \rfloor) * (x_{\lceil \text{Rank} \rceil} - x_{\lfloor \text{Rank} \rfloor}) \end{align*}

Percentile Rank:

\text{PR} = \frac{CF - (0.5 \times F)}{N} \times 100

CF = cumulative frequency up to score
F = frequency of score
N = total number of scores

Quartiles:

\begin{align*} Q_1 &= P_{25} \text{ (First Quartile)} \\ Q_2 &= P_{50} \text{ (Median)} \\ Q_3 &= P_{75} \text{ (Third Quartile)} \\ IQR &= Q_3 - Q_1 \text{ (Interquartile Range)} \end{align*}

Examples

Percentile Calculation:

For dataset: $[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]$

\begin{align*} \text{25th Percentile:} \\ \text{Rank} &= \frac{25 * (10 - 1)}{100} + 1 = 3.25 \\ P_{25} &= 3 + (3.25 - 3) * (4 - 3) = 3.25 \end{align*}

Percentile Rank Example:

For value 6 in dataset $[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]$ :

PR = \frac{6 - 0.5}{10} \times 100 = 55\%

Interpretation Guide

Basic Concepts

25th percentile (Q1): Lower quarter of data
50th percentile (Q2): Median, typical value
75th percentile (Q3): Upper quarter of data

Key Insights

IQR (Q3 - Q1) contains middle 50% of data
Equal quartile spacing suggests symmetry
Larger IQR indicates more variability

Outlier Detection

Potential outliers fall outside this range: $\left[ Q1 - 1.5 \times IQR \text{, } Q3 + 1.5 \times IQR \right]$

Applications & Uses

Common Applications

• Educational assessment scores
• Growth charts in healthcare
• Financial performance metrics
• Quality control measures

Distribution Analysis

• Data spread assessment
• Outlier identification
• Relative performance evaluation
• Box plot visualization

Limitations & Considerations

Different calculation methods may yield slightly different results
Small sample sizes can affect reliability
Outliers can significantly impact percentile ranks

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