IQR Calculator

Compute quartiles (Q1, Q2, Q3), IQR, lower/upper fences, and identify outliers from any numeric dataset. Interactive box plot visualizes the five-number summary. Designed for statisticians, students, and data analysts — all calculations performed locally in your browser.

Accepts integers or decimals. Use commas, spaces, or line breaks.
Try examples:
? Symmetric (1-10)
? With outlier (100)
? Odd count (11 values)
? Decimal values
⚠️ Multiple outliers
Privacy-first: All data processing happens inside your browser. No dataset ever leaves your device.

What is the Interquartile Range (IQR)?

The Interquartile Range (IQR) is a measure of statistical dispersion, representing the middle 50% of a dataset. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1): IQR = Q3 – Q1. Unlike range, IQR is resistant to outliers, making it a robust tool for identifying extreme values and understanding data spread.

Methodological note: The IQR is fundamental in exploratory data analysis (EDA) and forms the backbone of the box plot (Tukey, 1977). It is widely used in fields like finance (volatility trimming), biomedical research (outlier removal), and quality control. By focusing on the central half of observations, the IQR provides a clearer picture of typical variation without distortion by anomalies.

Alternative outlier rules: While the 1.5×IQR rule is standard, some analysts use a 3×IQR threshold to flag only “extreme” outliers. The choice depends on domain and data distribution.

How the Calculator Works

Our calculator implements the standard Tukey method (also known as the "hinge" method):

  1. Sort the dataset in ascending order.
  2. Median (Q2): If odd number of observations, median is the middle value. If even, average of the two central values.
  3. Q1 (first quartile): Median of the lower half of the data (excluding overall median if N is odd).
  4. Q3 (third quartile): Median of the upper half (excluding overall median if N is odd).
  5. IQR = Q3 - Q1
  6. Lower fence = Q1 - 1.5 × IQR, Upper fence = Q3 + 1.5 × IQR. Any data point outside the fences is considered a mild outlier (Tukey’s rule).

This method is used by statistical software (R, Python's numpy, SAS) and is recommended for general exploratory analysis. The box plot visualizes the five-number summary: minimum (within fences), Q1, median, Q3, maximum (within fences), plus outliers as individual points.

Real‑World Applications of IQR

  • Finance and Investing: Removing extreme returns (outliers) to estimate typical volatility and risk metrics.
  • Healthcare: Detecting anomalous patient measurements (blood pressure, lab results) based on IQR thresholds.
  • Education: Analyzing test scores to identify students who need intervention or to detect grading anomalies.
  • Quality Control: Monitoring manufacturing processes; values beyond 1.5×IQR signal potential defects.
  • Environmental Science: Identifying pollution outliers in air/water quality datasets.
Case Study: Salary Analysis in Tech Industry

An HR analyst examines annual salaries (in $k): 62, 68, 72, 74, 78, 82, 85, 88, 92, 150. The IQR = Q3 (88) – Q1 (72) = 16. Lower fence = 72 - 1.5*16 = 48, Upper fence = 88 + 24 = 112. The value 150 lies beyond 112 → classified as outlier (likely executive salary). Using IQR avoids misleading mean and allows fair baseline compensation analysis. The box plot reveals the skewed distribution instantly.

Limitations and Considerations

While IQR is robust, it discards 50% of the data. For symmetric distributions, it pairs well with median. For multimodal data, consider additional visualization. Also, the "1.5*IQR" rule identifies potential outliers, but domain knowledge should guide final decisions.

Frequently Asked Questions

Standard deviation measures spread around the mean and is sensitive to outliers. IQR measures spread around the median and is resistant to extreme values, making it more reliable for skewed distributions.

Our calculator uses the Tukey method (excludes median from halves for odd N). This is the most common approach in introductory statistics and software like TI calculators, R's fivenum(), and Excel's QUARTILE.EXC. Results may differ slightly from QUARTILE.INC, but for large datasets differences are negligible.

Outliers may represent measurement errors, rare events, or genuine extreme observations. They warrant further investigation but should not be automatically discarded.

Absolutely. The algorithm is O(n log n) due to sorting. Your browser can handle tens of thousands of numbers effortlessly.

John Tukey empirically derived this multiplier for Gaussian-like distributions; it flags approximately 0.7% of observations as outliers. For stricter detection (e.g., in finance or extreme value analysis), a 3×IQR rule is sometimes applied to identify only “far outliers”. The choice depends on the field and the analyst’s tolerance for false positives.
References & methodology sources: Tukey, J.W. (1977). Exploratory Data Analysis. Pearson; NIST/SEMATECH e-Handbook of Statistical Methods; Moore, D.S., McCabe, G.P. Introduction to the Practice of Statistics. This tool implements publicly documented statistical formulas and does not rely on proprietary algorithms. All calculations are performed locally for transparency.
Last updated: June 2026. No user data is collected or transmitted.