STATISTICSDescriptive StatisticsStatistics Calculator
📊

Median Absolute Deviation Calculator

Free MAD calculator. Median absolute deviation, normalized MAD, modified Z-score, outlier detection.

Run CalculatorExplore data analysis and statistical calculations

Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

MAD
STATISTICSDescriptive Statistics

Median Absolute Deviation — Robust Spread & Modified Z-Scores

MAD(median), normalized MAD, modified Z-scores for outlier detection. Iglewicz-Hoaglin criterion.

Mean MAD →IQR →

Real-World Scenarios — Click to Load

Data Input

mad_results.sh
CALCULATED
$ mad --n=10
Median
10.1500
MAD
0.0500
Normalized MAD
0.0741
Mean
10.1400
SD
0.0966
Share:
Median Absolute Deviation
MAD = 0.0500
0.050
Normalized: 0.074Outliers: 0
numbervibe.com/calculators/statistics/median-absolute-deviation-calculator

Data with MAD Band (median ± MAD) & Outliers

Red = outliers (|Modified Z| > 3.5). Green = median band.

Outlier Identification

Red bars = flagged outliers (Iglewicz-Hoaglin: |Mᵢ| > 3.5).

Modified Z-Scores (red = outlier)

Calculation Breakdown

COMPUTATION
Median of data
10.1500
ext{Robust} ext{center}
COMPUTATION
Absolute deviations
|xᵢ − median|
[0.14999999999999858, 0.05000000000000071, 0.049999999999998934, 0.15000000000000213, 0.05000000000000071, 0.049999999999998934, ...]
RESULT
MAD
0.0500
ext{median} ext{of} ext{deviations}
SCALE
Normalized MAD
0.0741
ext{MAD} imes 1.4826 approx \text{sigma} ext{for} ext{normal}
OUTLIERS
Modified Z
Mᵢ = 0.6745(xᵢ−median)/MAD
ext{Outlier} ext{if} |Mᵢ| > 3.5
OUTLIERS
Outliers detected
None
ext{Iglewicz}- ext{Hoaglin} ext{criterion}

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • • MAD = median(|xᵢ − median(x)|) — robust measure of spread, unaffected by outliers
  • • Normalized MAD = MAD × 1.4826 estimates σ for normal data
  • • Modified Z-score: Mᵢ = 0.6745(xᵢ − median)/MAD. Outlier if |Mᵢ| > 3.5 (Iglewicz-Hoaglin)
  • • MAD band: median ± MAD. Data points outside the band are relatively far from center
  • • For normal data: MAD × 1.4826 ≈ SD. MAD is more robust than SD to contamination

Did You Know?

📊MAD was popularized by Hampel (1974) and is the default scale estimator in robust statistics.Source: Hampel et al.
📐The constant 1.4826 makes MAD comparable to σ: for N(μ,σ²), E[MAD] ≈ σ/1.4826.Source: Normal theory
🔍Modified Z-scores use 0.6745 and 3.5 to approximate standard Z-scores for normal data.Source: Iglewicz-Hoaglin
📈MAD has 50% breakdown point — up to half the data can be outliers before MAD breaks.Source: Robust stats
🧪MAD is used in R (mad()), Python (scipy.stats.median_abs_deviation), and many robust methods.Source: Software
📉For heavy-tailed or contaminated data, MAD is preferred over standard deviation.Source: Best practice

How It Works

1. Find Median of Data

Compute the median of your dataset. This is the robust center.

2. Compute Absolute Deviations

For each point: |xᵢ − median|. These are the distances from the center.

3. MAD = Median of Deviations

MAD is the median of those absolute deviations. It measures typical distance from the median.

4. Normalized MAD & Modified Z

MAD × 1.4826 ≈ σ for normal data. Mᵢ = 0.6745(xᵢ−median)/MAD. |Mᵢ| > 3.5 flags an outlier.

Expert Tips

When to Use MAD

Contaminated data, heavy tails, or when SD is distorted by outliers. MAD is robust.

3.5 Threshold

Iglewicz-Hoaglin: 3.5 is conservative. For stricter detection, use 2.5 or 2.

Frequently Asked Questions

What is MAD?

MAD = median(|xᵢ − median(x)|). It measures typical distance from the median. Robust to outliers.

Why 1.4826?

For normal data, MAD ≈ σ/1.4826. So MAD×1.4826 ≈ σ, making normalized MAD comparable to SD.

What is the modified Z-score?

Mᵢ = 0.6745(xᵢ−median)/MAD. For normal data, |Mᵢ| ≈ |Z|. |Mᵢ| > 3.5 flags outliers.

MAD vs standard deviation?

MAD is robust; outliers barely affect it. SD is sensitive to outliers. Use MAD for contaminated data.

Can MAD be zero?

Yes, if more than half the points equal the median (e.g., many ties). Modified Z becomes undefined.

MAD by the Numbers

1.4826
MAD→σ Factor
0.6745
Modified Z Factor
3.5
Outlier Threshold
50%
Breakdown Point

Note: When MAD = 0 (many ties at median), modified Z-scores are undefined. Consider using IQR-based outlier detection instead.

👈 START HERE
⬅️Jump in and explore the concept!
AI

Related Calculators

Outlier Calculator

Detect outliers using IQR (Tukey), Z-score, Modified Z-score (MAD), Grubbs' test, and Dixon's Q test. Highlights which points are outliers.

Statistics

Mean Absolute Deviation Calculator

Compute MAD = Σ|xᵢ - x̄|/n from data. Compare MAD about the mean vs MAD about the median. See relationship to standard deviation.

Statistics

Dispersion Calculator

Compute ALL measures of dispersion: Range, IQR, Variance (population & sample), Standard Deviation, MAD, CV, Relative Range, Quartile Deviation. Side-by-side...

Statistics

Frequency Polygon Calculator

Create frequency polygons from grouped data. Plots midpoints vs frequencies. Supports overlaying multiple datasets for comparison. Raw data or frequency...

Statistics

Median Calculator

Step-by-step median with sorting, odd/even handling, weighted median, and running median. Robust measure of center.

Statistics

Percentile Rank Calculator

Calculates the percentile rank of a specific value within a dataset. What percent of values fall below a given score? Batch mode, reverse mode, CDF chart.

Statistics