What Is an IQR Calculator?

An IQR calculator computes the Interquartile Range — the spread of the middle 50% of a dataset — along with the full five-number summary, Tukey outlier fences, and identified outliers. Pre-filled with {7, 15, 36, 39, 40, 41}, the IQR is 25 (Q3=40 − Q1=15). The five-number summary chart visualizes Min, Q1, Median, Q3, and Max.

IQR Formula

IQR = Q3 − Q1. Q1 = 25th percentile (lower quartile), Q3 = 75th percentile (upper quartile). This calculator uses linear interpolation to compute quartiles, consistent with Excel's QUARTILE.INC function. The IQR measures variability in the central half of the data, making it robust against outliers that inflate the standard deviation.

Outlier Detection with Tukey Fences

Mild outliers: below Q1 − 1.5×IQR, or above Q3 + 1.5×IQR. Extreme outliers: below Q1 − 3×IQR, or above Q3 + 3×IQR. In a normal distribution, Tukey fences capture 99.3% of data within 1.5×IQR of Q1/Q3. Only 0.7% of data in a normal distribution would be flagged as mild outliers. Extreme outlier fences capture 99.9999% — anything beyond is almost certainly a measurement error or legitimate anomaly.

Box Plots and the IQR

A box plot visually encodes the five-number summary. The box spans Q1 to Q3 (width = IQR). The line inside = median. Whiskers extend to the most extreme values within 1.5×IQR of Q1/Q3. Points beyond whiskers are individual outlier markers. Side-by-side box plots are the most efficient way to compare distributions across multiple groups simultaneously.

IQR in Robust Statistics

Robust statistics resist the influence of outliers. The IQR has a breakdown point of 25% — up to 25% of data can be extreme outliers without affecting the IQR. Standard deviation has a 0% breakdown point — even one outlier can distort it drastically. For real-world data with measurement errors, contamination, or heavy-tailed distributions, IQR-based methods (median, MAD, IQR) are preferred over mean-based methods.

Applications

Finance: VaR calculations often exclude outlier days using IQR methods. Healthcare: clinical reference ranges typically exclude the top and bottom 2.5% (similar spirit to IQR). Quality control: IQR-based control charts for non-normal processes. Machine learning: feature scaling using IQR (RobustScaler in scikit-learn). Data cleaning: automatically flag potential data entry errors using outlier fences.