What is the Interquartile Range (IQR)?

The Interquartile Range (IQR) is a measure of statistical dispersion, equal to the difference between the third quartile (Q3, 75th percentile) and the first quartile (Q1, 25th percentile). It represents the range of the middle 50% of the data and is resistant to outliers, making it a robust measure of variability.

How is IQR calculated?

IQR is calculated in three steps:

• Step 1: Sort the data in ascending order.
• Step 2: Find Q1 (the median of the lower half) and Q3 (the median of the upper half).
• Step 3: Compute IQR = Q3 − Q1.

This calculator uses Tukey's hinges method, which includes the overall median in both halves when the dataset has an odd number of values — the standard approach for box plot construction.

What is a Box Plot (Box-and-Whisker Plot)?

A box plot is a standardized graphical representation of the five-number summary: minimum, Q1, median, Q3, and maximum. The "box" spans from Q1 to Q3 with a line at the median. The "whiskers" extend to the smallest and largest data points within 1.5 × IQR from the quartiles. Points beyond the whiskers are flagged as potential outliers.

How are outliers identified using the IQR method?

The 1.5 × IQR rule is the standard criterion:

• Lower Fence = Q1 − 1.5 × IQR
• Upper Fence = Q3 + 1.5 × IQR

Any data point below the lower fence or above the upper fence is considered a potential outlier. Some analysts also use 3 × IQR to distinguish "extreme" outliers from "mild" ones.

Why use IQR instead of standard deviation?

While standard deviation is sensitive to every data point (including outliers), the IQR is robust to extreme values. For skewed distributions or datasets with anomalies, IQR often provides a more honest picture of variability. It's widely used in finance, environmental science, quality control, and educational assessment.

What are the different methods for computing quartiles?

There are several methods, which can yield slightly different results for small datasets:

• Tukey's Hinges — used in box plots; includes median in both halves for odd n.
• Exclusive Median — excludes the median when splitting halves (used by some textbooks).
• Linear Interpolation — used by numpy, Excel's QUARTILE.EXC/INC, and other software.

This tool uses Tukey's method, which aligns with standard box plot conventions.

Can I use this calculator for large datasets?

Yes! This calculator efficiently handles datasets with hundreds or even thousands of values. Simply paste your numbers (separated by commas, spaces, tabs, or line breaks) and the tool instantly computes all statistics and generates the box plot visualization.

What does a narrow vs. wide IQR tell me?

A narrow IQR indicates that the middle 50% of the data is tightly clustered around the median — suggesting low variability. A wide IQR means the central data is more spread out. Comparing IQRs across groups helps identify differences in consistency or dispersion.