Calculate IQR, semi-IQR, quartile coefficient of dispersion, Tukey fences, and outlier detection. Includes box plot, quartile segments, and IQR vs SD comparison.
The interquartile range (IQR) calculator measures the spread of the middle 50% of your data — from Q1 (25th percentile) to Q3 (75th percentile). The IQR is the most robust measure of variability because it ignores outliers entirely, making it far more reliable than range or standard deviation for skewed or contaminated data.
This tool computes IQR along with semi-IQR, quartile coefficient of dispersion, Tukey fences for outlier detection, and a visual box plot. It also compares IQR to standard deviation, helping you assess whether your data follows a normal distribution (where IQR ≈ 1.35 × SD).
Enter your data, choose a quartile method, set the fence multiplier, and get a complete variability analysis centered on the interquartile range. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results. Validate that outputs match your chosen standards.
The IQR is the gold standard for robust variability measurement. This calculator provides the IQR alongside semi-IQR, quartile coefficient, Tukey fences, and a visual box plot — giving you the complete robust descriptive statistics toolkit in one place.
Whether you're screening for outliers, reporting skewed data, or comparing the spread of datasets with different scales, the IQR calculator delivers accurate, outlier-resistant results.
IQR = Q3 − Q1. Semi-IQR = IQR / 2. QCD = (Q3 − Q1) / (Q3 + Q1) × 100%. Lower fence = Q1 − k × IQR. Upper fence = Q3 + k × IQR. For normal data: IQR ≈ 1.35 × σ.
Result: IQR = 15, Q1 = 73, Q3 = 88
Sorted data: 68,70,72,74,76,81,84,85,88,90,92,95. Q1 = median(68,70,72,74,76,81) = 73, Q3 = median(84,85,88,90,92,95) = 89. IQR = 89 − 73 = 16. Fences: 49–113. No outliers detected.
John Tukey introduced the box plot in 1970 specifically around the IQR as its core measurement. The box spans Q1 to Q3 (the IQR), the median line divides it, and whiskers extend to the farthest non-outlier values. Everything beyond the fences (Q1 − 1.5×IQR and Q3 + 1.5×IQR) is plotted individually as an outlier. This elegant design makes the box plot the standard tool for distribution comparison.
The IQR is used to normalize robust test statistics. The IQR-based measure of kurtosis, for example, replaces the fourth moment with quartile-based metrics that are less sensitive to extreme values. Some robust regression methods use IQR to scale residuals for outlier weights.
For a normal distribution, IQR = 1.35σ. For a uniform distribution on [a,b], IQR = (b−a)/2. For an exponential distribution with rate λ, IQR = ln(3)/λ. Comparing the observed IQR/SD ratio to 1.35 provides a quick normality check.
The IQR measures the spread of the middle 50% of data — from the 25th to the 75th percentile. It tells you how concentrated or dispersed the central bulk of your data is, completely ignoring the most extreme values. A small IQR means data is clustered tightly around the median.
Range (max − min) uses only two extreme values, making it extremely sensitive to outliers. A single outlier can double the range while barely affecting IQR. IQR uses the middle 50%, so it's robust — it stays stable even when data has unusual extremes.
Tukey's method defines outliers as values beyond Q1 − 1.5×IQR or Q3 + 1.5×IQR. These "fences" work for any distribution shape. Mild outliers fall between 1.5× and 3× IQR from Q1/Q3; extreme outliers fall beyond 3× IQR. This method is used in box plots worldwide.
QCD = (Q3 − Q1) / (Q3 + Q1) is a dimensionless measure of relative variability based on quartiles. Like the coefficient of variation (CV), it allows comparing spread across datasets with different scales. Unlike CV, it's robust to outliers.
Use IQR when data is skewed, contains outliers, or isn't normally distributed. Income data, housing prices, medical lab values, and environmental measurements often require IQR-based reporting. Use SD when data is approximately normal and outliers have been verified as legitimate.
The semi-IQR (also called quartile deviation) is simply IQR / 2. It's used in older statistical texts as a measure of typical deviation from the median. For normal data, semi-IQR ≈ 0.6745 × σ (the probable error).