⚡ Quick IQR Facts
Robust to Outliers
Unlike standard deviation, IQR isn't distorted by extreme values
Industry Standard
Used in quality control, finance, and scientific research worldwide
1.5×IQR Rule
The gold standard for detecting potential outliers in datasets
📚 New to IQR? Jump to the step-by-step guide on this page.
Rule of five readiness • Last Updated: December 6, 2025
Interquartile Range (IQR) Calculator
Drop any sample, pick your quartile definition, and instantly see Q1, median, Q3, IQR, Tukey fences, and a live box-plot that highlights potential outliers. Ideal for exploratory data analysis, SPC charts, and reporting you can defend.
Invalid tokens are ignored automatically—use "Clean list" to strip them from the input.
Quartile method
Key quartile stats
Q1 (25th percentile)
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Median (Q2)
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Q3 (75th percentile)
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IQR (Q3 − Q1)
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Lower fence
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Upper fence
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No values analysed yet.
IQR plot & suspected outliers
Box spans Q1–Q3, median line splits the box, dotted fences show ±1.5 × IQR. Points beyond the fences are highlighted.
Outlier list
Insights
- No values analysed yet.
How to Calculate IQR: Step-by-Step Guide
Quick Method (Using This Calculator)
- Paste your data (comma, space, or newline separated)
- Choose quartile method (Tukey for classical box plots, Inclusive for Excel compatibility)
- Click "Calculate IQR" — get instant Q1, Q3, IQR, and outlier detection
Manual Calculation (Tukey Method)
- Sort data: Arrange values from smallest to largest
- Find Q2 (median): Middle value (or average of two middle values if even n)
- Find Q1: Median of lower half (excluding Q2 if odd n)
- Find Q3: Median of upper half (excluding Q2 if odd n)
- Calculate IQR: IQR = Q3 − Q1
- Optional - Outlier fences: Lower = Q1 − 1.5×IQR, Upper = Q3 + 1.5×IQR
The interquartile range may look deceptively simple—just Q3 minus Q1—but it encodes an entire philosophy of robust statistics. By focusing on the middle 50 percent of the distribution, the IQR resists the pull of extreme values that can whiplash the standard deviation. Tukey's exploratory data analysis framed the IQR as the backbone of the "box-and-whisker" summary because it highlights the core body of observations while still keeping an eye on the tails via the 1.5×IQR fences. That balance between stability and vigilance is why the IQR shows up in pharma stability studies, manufacturing SPC charts, and modern machine-learning feature diagnostics.
Excel Formulas for IQR Calculation
Calculate IQR directly in Microsoft Excel or Google Sheets using these formulas. Assume your data is in cells A1:A100.
Method 1: Using QUARTILE.INC (Excel/Google Sheets)
=QUARTILE.INC(A1:A100, 3) - QUARTILE.INC(A1:A100, 1)
This matches the "Inclusive" method in our calculator above.
Method 2: Using PERCENTILE.INC
=PERCENTILE.INC(A1:A100, 0.75) - PERCENTILE.INC(A1:A100, 0.25)
Equivalent to QUARTILE.INC, uses decimal percentiles.
Method 3: Individual Quartile Formulas
Q1 (25th percentile) - Cell B1:
=QUARTILE.INC(A1:A100, 1)
Q2 (Median) - Cell B2:
=QUARTILE.INC(A1:A100, 2)
Q3 (75th percentile) - Cell B3:
=QUARTILE.INC(A1:A100, 3)
IQR - Cell B4:
=B3-B1
Lower Fence (Outlier Detection) - Cell B5:
=B1-1.5*B4
Upper Fence (Outlier Detection) - Cell B6:
=B3+1.5*B4
📊 Excel vs Tukey Method
QUARTILE.INC uses linear interpolation (matches "Inclusive" in our calculator).
QUARTILE.EXC is closer to the Tukey method but not identical.
For large datasets (n > 50), the difference is negligible. For small datasets, results may vary slightly.
LaTeX Formulas for Academic Papers
Use these LaTeX codes to include IQR formulas in your research papers, thesis, or presentations.
Interquartile Range Formula
\text{IQR} = Q_3 - Q_1
Quartile Positions (Tukey Method)
Q_1 = \text{median}(x_1, x_2, \ldots, x_{\lfloor n/2 \rfloor})
Q_3 = \text{median}(x_{\lceil n/2 \rceil + 1}, \ldots, x_n)
Outlier Detection Fences
\text{Lower Fence} = Q_1 - 1.5 \times \text{IQR}
\text{Upper Fence} = Q_3 + 1.5 \times \text{IQR}
Percentile-Based Definition
\text{IQR} = P_{75} - P_{25}
Where P₇₅ is the 75th percentile and P₂₅ is the 25th percentile.
Box Plot Definition
\text{Outlier} \iff x < Q_1 - 1.5 \cdot \text{IQR} \text{ or } x > Q_3 + 1.5 \cdot \text{IQR}
Real-World Use Cases for IQR
🏭 Manufacturing Quality Control
Use Case: Detecting defective products in production batches
Example: Widget weights should be 100g ± tolerance. IQR identifies outliers that fall outside specification limits.
Why IQR? Robust to occasional extreme measurements (sensor errors) unlike standard deviation which gets distorted.
💰 Financial Fraud Detection
Use Case: Identifying suspicious transactions
Example: Credit card transactions analyzed daily. Amounts beyond 1.5×IQR flagged for review (e.g., $5,000 charge when typical spending is $50-$200).
Why IQR? Catches unusual activity without being fooled by occasional large legitimate purchases.
📊 Academic Research & Grading
Use Case: Analyzing test score distributions
Example: Professor wants to understand typical student performance. IQR shows middle 50% scored 68-85, with outliers at 25 and 98.
Why IQR? Not affected by the one genius or one failing student — shows true classroom performance.
🏥 Medical & Clinical Studies
Use Case: Blood pressure, cholesterol, or treatment response analysis
Example: Clinical trial measuring drug efficacy. IQR helps identify patients with abnormal responses that may need investigation.
Why IQR? Medical data often skewed (e.g., reaction times, recovery periods) — IQR works where normal distribution assumptions fail.
📈 Stock Market & Investment Analysis
Use Case: Identifying volatile trading days or price anomalies
Example: Daily stock returns analyzed using IQR. Returns beyond fences indicate unusual market events (crash/boom days).
Why IQR? Markets have fat tails (extreme events) — IQR captures typical volatility without SD inflation from crashes.
🌐 Website Analytics & Performance
Use Case: Detecting slow page load times or server issues
Example: Page load times usually 120-180ms (IQR). Outliers at 850ms+ indicate server problems or bot traffic.
Why IQR? Web performance has natural variability — IQR separates real issues from normal fluctuations.
🏡 Real Estate Price Analysis
Use Case: Determining fair market value, identifying overpriced/underpriced properties
Example: Homes in neighborhood priced $250k-$320k (IQR). $850k mansion is outlier — not comparable for valuation.
Why IQR? Real estate has outliers (luxury estates, fixers) — IQR focuses on typical market segment.
🔬 Scientific Data Cleaning
Use Case: Identifying sensor errors, data entry mistakes, or experimental anomalies
Example: Temperature readings in lab experiment. Value of -273°C flagged by IQR as impossible outlier (sensor malfunction).
Why IQR? Robust outlier detection method recommended by NIST for quality assurance protocols.
💡 When to Use IQR vs Other Measures
- Use IQR: Data is skewed, outliers present, non-normal distributions, robust summary needed
- Use Standard Deviation: Normal distribution, no outliers, inferential statistics (t-tests, ANOVA)
- Use Range: Quick checks only — highly sensitive to extreme values
Technical Details & Statistical Properties
Robustness & Breakdown Point
The IQR has a breakdown point of 25%, meaning up to 25% of the data can be arbitrarily extreme without affecting the IQR value. This makes it significantly more robust than the standard deviation (breakdown point ≈ 0%) and comparable to the median (breakdown point = 50%). This property is crucial in real-world applications where data contamination is common.
Asymptotic Efficiency
For normally distributed data, the IQR has an asymptotic efficiency of approximately 0.37 relative to the standard deviation. This means that for Gaussian data, you would need ~2.7 times more observations using IQR to achieve the same statistical power as using SD. However, this disadvantage vanishes for heavy-tailed or contaminated distributions, where IQR often outperforms SD.
Quartile Calculation Methods
There are multiple methods for calculating quartiles, leading to slight differences in IQR values:
| Method | Description | Used By |
|---|---|---|
| Tukey (T1) | Hinges method, median excluded for odd n | Classical statistics, box plots, our calculator default |
| Inclusive (R-7) | Linear interpolation, continuous CDF | Excel QUARTILE.INC, S-PLUS, our calculator option |
| Exclusive (R-6) | Linear interpolation, p ∈ (0,1) | Excel QUARTILE.EXC, Minitab |
| Moore & McCabe | Median included in both halves | Some textbooks |
Note: For large datasets (n > 100), all methods produce nearly identical results. Differences matter primarily for small samples.
Computational Complexity
Computing IQR requires sorting the dataset, giving it a time complexity of O(n log n) for comparison-based algorithms. However, the median-of-medians algorithm can find quartiles in O(n) worst-case time. For streaming data or online calculations, approximate quantile algorithms like t-digest or Q-digest provide O(log n) per update with controllable accuracy.
Relationship to Other Statistics
For Normal Distribution N(μ, σ):
- IQR ≈ 1.349 × σ (standard deviation)
- IQR ≈ 0.7413 × (90th percentile − 10th percentile)
- Q1 ≈ μ − 0.6745σ, Q3 ≈ μ + 0.6745σ
- Coefficient of Quartile Deviation: (Q3 − Q1) / (Q3 + Q1)
Sample Size Considerations
Minimum n = 4 required for meaningful IQR (otherwise Q1 = Q3). For practical use, n ≥ 20 recommended for stable quartile estimates. Standard error of IQR decreases as SE(IQR) ∝ 1/√n, so larger samples provide more precise estimates of population interquartile range.
Historical Note
The concept of quartiles dates to Francis Galton (1882), but the modern definition and widespread adoption came from John Tukey's 1977 book "Exploratory Data Analysis", which introduced the box-and-whisker plot and established the 1.5×IQR rule as a practical outlier detection standard. The National Institute of Standards and Technology (NIST) and ISO standards have since formalized IQR-based methods for quality control and uncertainty quantification.
How to Cite This Calculator
If you use this IQR calculator in your research, academic work, or professional reports, please cite it as follows:
APA Style (7th Edition)
Bhakuni, P. (2025). IQR Calculator - Free Interquartile Range Calculator with Outliers. CalcArena. https://calcarena.com/calculators/interquartile-range-calculator.html
MLA Style (9th Edition)
Bhakuni, Pawan. "IQR Calculator - Free Interquartile Range Calculator with Outliers." CalcArena, 2025, calcarena.com/calculators/interquartile-range-calculator.html.
Chicago Style (17th Edition)
Bhakuni, Pawan. "IQR Calculator - Free Interquartile Range Calculator with Outliers." CalcArena. Accessed December 6, 2025. https://calcarena.com/calculators/interquartile-range-calculator.html.
BibTeX
@misc{bhakuni2025iqrcalc,
author = {Bhakuni, Pawan},
title = {IQR Calculator - Free Interquartile Range Calculator with Outliers},
year = {2025},
howpublished = {\url{https://calcarena.com/calculators/interquartile-range-calculator.html}},
note = {Accessed: 2025-12-06}
}
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FAQ
When should I prefer Tukey over the inclusive method?
Use Tukey when following classical box-plot definitions or when sample sizes are modest. Use the inclusive method to match Excel, Google Sheets, or ISO 16269-4 reports where quartiles are interpolated.
Does every point beyond the fence get removed?
No. Fences are a screening device. Outliers might be errors, but they could also be rare yet valid events. Always review domain context before deleting data.
Can I change the 1.5 multiplier?
This UI locks the multiplier to the Tukey convention for speed. If you need custom thresholds, export the summary and replicate the logic in Python, R, or Excel with your desired constant.
Author & Reviewer
Pawan builds CalcArena’s statistical tooling with an emphasis on transparent formulas and visual diagnostics.
Reviewed by Kushal Singh, AI Research Program Manager.