Percentile, Quartile, and Interquartile Range (IQR)

This Percentile, Quartile, and IQR Calculator helps you analyze the distribution and spread of your data. It calculates percentiles (values below which a given percentage of observations fall), quartiles (values that divide data into four equal parts), and the interquartile range (IQR, a measure of statistical dispersion). For example, you can analyze test scores to find the 75th percentile, determine salary quartiles, or use the IQR to identify outliers in any numerical dataset.

Quick Calculator

Need a quick calculation? Enter your numbers below:

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Calculator

1. Load Your Data

2. Select Column & Enter Percentile

Select Numeric Column:

Enter Percentile (0-100):

Exclude Outliers

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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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