Interquartile Range (IQR) Calculator

Calculate IQR, quartiles, and outlier fences from a dataset. IQR = Q3 - Q1 measures the spread of the middle 50% of data. Identify potential outliers using the 1.5×IQR rule.

📊 Statistics📐 IQR = Q3 - Q1🔢 Math
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Formula & Reference

VariableSymbolFormulaUnits
Interquartile Range (IQR) Calculator—IQR = Q3 - Q1same as data units

Step-by-Step Examples

Example 1
Exam Scores

Dataset: 55,60,65,70,75,80,85,90,95,100.

  • n=10 sorted values
  • Q1 = 65 (25th percentile)
  • Q2 = 77.5 (median)
  • Q3 = 90 (75th percentile)
  • IQR = 90 - 65 = 25
  • Fences: 65-37.5=27.5 to 90+37.5=127.5: no outliers
✓ IQR = 25, no outliers
Example 2
Outlier Detection

Income data: 30,35,40,45,50,55,60,200 (thousands).

  • Q1 = 36.25, Q3 = 56.25, IQR = 20
  • Lower fence = 36.25 - 30 = 6.25
  • Upper fence = 56.25 + 30 = 86.25
  • 200 > 86.25: outlier detected
✓ 200 is an outlier
Example 3
Box Plot Summary

Data: 10,12,15,18,20,25,30. Summarize for box plot.

  • Min=10, Q1=12, Median=18, Q3=25, Max=30
  • IQR=13, Fences: -7.5 to 44.5
  • No outliers — all within fences
  • Box from 12 to 25, whiskers to 10 and 30
✓ Box plot: [12|18|25], whiskers 10-30

Real-World Applications

📊
Exploratory Data Analysis
IQR is the primary measure of spread in box plots — standard in EDA.
💊
Medical Research
Reference ranges in lab tests often use percentile ranges (2.5th-97.5th) based on IQR logic.
💼
Business Analytics
Outlier detection in sales data, transaction amounts, customer behavior.
🌍
Environmental Science
Detect anomalous pollution readings, extreme weather events using IQR fences.

Common Mistakes to Avoid

⚠️
Treating IQR outliers as errors

Values beyond 1.5×IQR are potential outliers worth examining, but they may be real and important data. Always investigate before removing.

⚠️
Using IQR for small datasets

With n < 10, quartile estimates are unreliable. IQR-based outlier detection works best with n ≥ 20.

⚠️
Not sorting data first

Quartile calculation requires sorted data. Always sort before finding Q1 and Q3 positions.

Frequently Asked Questions

What is IQR? ▾
The difference between the third quartile (Q3) and first quartile (Q1). Contains the middle 50% of data. Robust measure of spread: unaffected by outliers or extreme values.
How does the 1.5×IQR rule work? ▾
Lower fence = Q1 - 1.5×IQR. Upper fence = Q3 + 1.5×IQR. Any data point beyond these fences is flagged as a potential outlier. This rule was proposed by John Tukey.
Why is IQR better than range for spread? ▾
Range = max - min and is heavily influenced by one extreme value. IQR ignores the top and bottom 25% of data, making it much more robust to outliers.
What is the 3×IQR rule? ▾
For severe/extreme outliers: values beyond Q1-3×IQR or Q3+3×IQR. The 1.5×IQR rule catches mild outliers; 3×IQR catches extreme ones. Both are commonly used.
How to calculate quartiles with different methods? ▾
There are multiple conventions (R uses 7 methods!). The most common: Method 1 (textbook): median of lower/upper halves. Method 2 (linear interpolation): used by Excel and many software packages.

Related Math Calculators

📊 Percentile📊 Variance📊 Standard Deviation📊 Statistics📊 Normal Distribution

Formula Explorer connections

Interpretation: This formula summarizes data, models uncertainty or supports inference about a population or random process. Assumption: The sampling design and distribution assumptions must match the data. Independence, sample size, outliers and measurement quality can materially affect interpretation.

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