Calculate MAD, scaled MAD, mean absolute deviation, and MAD-based outlier detection. Compare robust vs classical spread measures with visual breakdown.
The median absolute deviation (MAD) calculator is the ultimate robust measure of statistical dispersion. MAD calculates the median of the absolute deviations from the data's median — making it resistant to up to 50% outliers, the highest breakdown point of any common spread estimator.
This tool computes the raw MAD, the scaled MAD (multiplied by 1.4826 to estimate standard deviation for normal data), the mean absolute deviation, and performs MAD-based outlier detection. A comparison chart shows how MAD, scaled MAD, standard deviation, and IQR relate for your specific data.
Enter your data, choose a center point and scaling option, and get a complete robust variability analysis with deviation tables and step-by-step calculations. 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.
MAD is the most robust measure of spread in common use, with a 50% breakdown point that standard deviation and range cannot match. This calculator provides MAD alongside scaled MAD, mean absolute deviation, and SD, with a visual comparison that instantly reveals how much outliers are inflating classical measures.
The deviation table and outlier detection give you actionable results — not just a number, but identification of which values are driving the spread in your data.
MAD = median(|xᵢ − median(x)|). Scaled MAD = 1.4826 × MAD (≈ σ for normal data). Mean Absolute Deviation = Σ|xᵢ − center| / n. Outlier threshold: |xᵢ − median| > 3 × scaled MAD.
Result: MAD = 2, Scaled MAD = 2.9652
Median = 25. Absolute deviations: |22−25|=3, |25−25|=0, |27−25|=2, ... Sorted |deviations|: 0,1,1,2,2,3,3,65. MAD = median = 2. Scaled MAD = 2 × 1.4826 = 2.9652. The value 90 has |90−25|/2.97 = 21.9 MADs from center — a clear outlier.
Peter Huber popularized MAD as a robust scale estimator in the 1960s and 70s. It forms the foundation of many robust statistical methods: M-estimators use MAD for initial scale estimates, robust regression uses MAD to identify influential outliers, and anomaly detection systems use MAD-based thresholds because they resist "masking" (where a cluster of outliers makes classical methods fail to flag any of them).
In signal processing, MAD is used to estimate the noise level in a signal. The "universal threshold" for wavelet denoising is λ = MAD × √(2 ln n) / 0.6745, where MAD is computed from the finest-scale wavelet coefficients. This approach is robust because signal components don't corrupt the noise estimate.
MAD requires sorting (O(n log n)) or can use a selection algorithm (O(n)). For streaming data, approximate MAD can be maintained with P² quantile estimators or binned approaches. In distributed systems, MAD is harder to compute than mean/SD because median is not decomposable — but approximate distributed MAD algorithms exist.
MAD is calculated in two steps: (1) find the median of your data, (2) compute the absolute deviation of each value from the median, then find the median of those deviations. It measures "typical" distance from the center using the most robust possible approach — both the center (median) and the spread (median of deviations) resist outliers.
The factor 1.4826 (= 1/Φ⁻¹(3/4) where Φ is the normal CDF) makes the scaled MAD a consistent estimator of the standard deviation when data is normally distributed. Without scaling, MAD is about 67.45% of the SD for normal data. The scaling lets you directly compare MAD to SD.
The breakdown point is the fraction of data that can be arbitrary (outliers) before a statistic becomes unreliable. MAD has a 50% breakdown point — the highest possible. Standard deviation has a 0% breakdown point — even a single extreme outlier can make it arbitrarily large. IQR has a 25% breakdown point.
MAD uses the median of absolute deviations (from the median), while mean absolute deviation uses the mean of absolute deviations (from the mean or median). MAD is much more robust because it uses the median twice. Mean absolute deviation is affected by every data point and has no outlier resistance.
Use MAD when (1) data may contain outliers, (2) the distribution is skewed, (3) you need a robust preprocessing step before formal analysis, or (4) you're comparing groups where some may have contaminated data. MAD is especially common in signal processing, anomaly detection, and robust regression.
Compute the modified z-score: zᵢ = |xᵢ − median| / (1.4826 × MAD). Values with |zᵢ| > 3 are typically flagged as outliers. This is more reliable than classical z-scores (using mean and SD) because both the center and scale estimates are robust to the very outliers you're trying to detect.