Interquartile Range (IQR) Calculator

Central Tendency and Dispersion Measures

Enter a comma-separated list of numbers to calculate the Interquartile Range (IQR), quartiles, and identify any outliers.

Practical Examples

See how the IQR calculator works with different datasets.

Basic Even Numbered Dataset

Simple Dataset

A simple dataset with an even number of values to find the IQR.

Data: 2, 4, 4, 5, 6, 7, 8, 9

Dataset with an Odd Number of Values

Odd Numbered Dataset

An example demonstrating how quartiles are calculated for a dataset with an odd count.

Data: 10, 20, 30, 40, 50, 60, 70

Identifying Outliers

Dataset with Outliers

This dataset includes clear outliers to show how the calculator identifies them.

Data: 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49, 78, 108

Test Scores Analysis

Real-World Data

Analyzing a set of student test scores to find the spread of the middle 50%.

Data: 88, 92, 80, 78, 95, 84, 76, 90, 81, 85, 93

Other Titles
Understanding the Interquartile Range (IQR): A Comprehensive Guide
Dive deep into the concepts behind the IQR, its calculation, application, and importance in statistical analysis.

What is the Interquartile Range (IQR)?

  • Defining Statistical Dispersion
  • The Role of Quartiles
  • Why IQR is a Robust Measure
The Interquartile Range (IQR) is a measure of statistical dispersion, or in simpler terms, how spread out a set of data is. It represents the range of the middle 50% of the data points. Unlike the standard range (maximum value minus minimum value), the IQR is not affected by extreme values, known as outliers. This makes it a more robust and reliable measure for understanding the variability in most datasets.
The Role of Quartiles

To understand the IQR, you first need to understand quartiles. Quartiles divide a rank-ordered data set into four equal parts. The three points that create these four parts are:

  • Q1 (First Quartile or Lower Quartile): The 25th percentile. 25% of the data falls below this value.
  • Q2 (Second Quartile or Median): The 50th percentile. This is the median of the entire dataset.
  • Q3 (Third Quartile or Upper Quartile): The 75th percentile. 75% of the data falls below this value. The IQR is simply the difference between the third and first quartiles: IQR = Q3 - Q1.

Conceptual Example

  • Imagine a list of student scores: the IQR tells you the score range for the middle half of the class, ignoring the very top and very bottom scores.
  • In real estate, the IQR of house prices in a neighborhood can give a better idea of typical prices than the average, which might be skewed by a few mansions or very cheap properties.

Step-by-Step Guide to Using the IQR Calculator

  • Inputting Your Data
  • Interpreting the Results
  • Using the Outlier Detection Feature
1. Inputting Your Data
Enter your data into the input field. The numbers should be separated by commas. The calculator is designed to handle integers, decimals, and negative numbers. Any non-numeric text will be automatically ignored during the calculation.
2. Interpreting the Results

After clicking 'Calculate', you will see several key metrics:

  • Sorted Data: Your numbers, ordered from least to greatest.
  • Q1, Q2 (Median), Q3: The calculated quartiles.
  • IQR: The result of Q3 - Q1.
  • Outlier Bounds: The calculator uses the standard formula (Q1 - 1.5IQR and Q3 + 1.5IQR) to determine the thresholds beyond which a data point is considered an outlier.
  • Outliers: Any data points from your set that fall outside these bounds.

Practical Walkthrough

  • Data Input: 10, 2, 5, 8, 12, 15, 7
  • Calculation: The calculator first sorts the data: 2, 5, 7, 8, 10, 12, 15. It finds Q1=5, Median=8, Q3=12. The IQR is 12 - 5 = 7.
  • Result: You'll see these values clearly displayed, along with the outlier bounds and a confirmation that no outliers were found.

Real-World Applications of the IQR

  • Data Science and Analytics
  • Finance and Economics
  • Scientific Research and Quality Control
Data Science and Analytics
The IQR is fundamental in exploratory data analysis and is the core component of box plots (box-and-whisker diagrams). Box plots visually represent the five-number summary (minimum, Q1, median, Q3, maximum) and help in quickly identifying the data's spread and potential outliers.
Finance and Economics
Analysts use the IQR to understand the volatility of stock prices, income distribution in a population, or housing market fluctuations. Because it resists outliers, it provides a stable measure of where the bulk of the data lies, filtering out extreme, and often misleading, high or low values.

Application Scenarios

  • A retail company might analyze the IQR of daily sales figures to understand typical performance and set realistic sales targets, ignoring anomalous days like major holidays or technical outages.
  • A medical researcher might use the IQR to report the range of blood pressure readings in a clinical trial, providing a clear picture of the central tendency without distortion from a few subjects with extremely high or low readings.

Common Misconceptions and Correct Methods

  • IQR vs. Range vs. Standard Deviation
  • Inclusive vs. Exclusive Quartile Methods
  • Handling Even and Odd Datasets
IQR vs. Range vs. Standard Deviation
A common mistake is to use the simple range (max - min) for data spread, which is highly sensitive to outliers. Standard deviation is another popular measure, but it assumes a normal distribution and can also be affected by extreme values. The IQR is non-parametric and does not assume any particular distribution, making it more versatile.
Quartile Calculation Methods
There are several methods for calculating quartiles. This calculator uses the widely accepted Tukey method (inclusive method), where the median is included in the calculation of Q1 and Q3 if the total number of data points is odd. Some methods exclude the median, which can lead to slightly different results. Our approach is standard in most introductory statistics and data analysis platforms.

Methodology Clarification

  • For the dataset (1, 2, 3, 4, 5, 6, 7), the median (Q2) is 4. The lower half is (1, 2, 3). The median of this lower half is 2 (Q1). The upper half is (5, 6, 7). The median of this upper half is 6 (Q3).
  • For the dataset (1, 2, 3, 4, 5, 6), the median (Q2) is the average of 3 and 4, which is 3.5. The lower half is (1, 2, 3). The median of this is 2 (Q1). The upper half is (4, 5, 6). The median of this is 5 (Q3).

Mathematical Derivation and Formulas

  • The Five-Number Summary
  • The IQR Formula
  • The Outlier Rule
The IQR Formula
The core formula is straightforward: IQR = Q3 - Q1. The complexity lies in accurately finding Q1 and Q3.
Finding Quartiles
  1. Order the data from smallest to largest.
  2. Find the median (Q2) of the entire dataset.
  3. Identify the lower half of the data (all values to the left of the median, including the median itself if the dataset count is odd, depending on the method).
  4. Find the median of the lower half. This is Q1.
  5. Identify the upper half of the data (all values to the right of the median).
  6. Find the median of the upper half. This is Q3.
The 1.5 x IQR Rule for Outliers

To identify outliers, we establish a 'fence' around the central data. Any values that fall outside this fence are considered outliers.

  • Lower Fence = Q1 - (1.5 * IQR)
  • Upper Fence = Q3 + (1.5 * IQR)

Formula Application

  • Data: {2, 3, 5, 6, 8, 10, 12}. Q1 = 3, Q3 = 10. IQR = 10 - 3 = 7.
  • Outlier Check: Lower Fence = 3 - (1.5 * 7) = -7.5. Upper Fence = 10 + (1.5 * 7) = 20.5. Any data point less than -7.5 or greater than 20.5 is an outlier.