Median Absolute Deviation (MAD) Calculator

Paste a dataset to compute its median absolute deviation, MAD = median(|xᵢ − median|) — a robust, outlier-resistant measure of spread, plus the median and the 1.4826-scaled estimate of the standard deviation.

Quick Facts

Formula
MAD = median(|xᵢ − median(x)|)
Robust to outliers; multiply by 1.4826 for a normal-consistent estimate of σ.

Your Results

Calculated
Median absolute deviation (MAD)
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median(|xᵢ − median|)
Median of the data
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Center used for deviations
Scaled MAD (× 1.4826)
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Estimate of σ for normal data
Count
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Number of values

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Enter your data values and press Calculate.

What the median absolute deviation measures

The median absolute deviation (MAD) is a robust measure of statistical dispersion — it tells you how spread out a set of numbers is, using medians instead of means so that outliers barely affect the result. It answers the question: "typically, how far is a value from the center of my data?"

The formula

For a dataset x₁, x₂, …, xₙ, the MAD is computed in two steps:

  • Step 1 — find the median of the data. Call it m = median(x).
  • Step 2 — take the median of the absolute deviations from that center: MAD = median(|xᵢ − m|).

Worked example with the dataset 1, 2, 4, 6, 8: the median is 4. The absolute deviations are |1−4|=3, |2−4|=2, |4−4|=0, |6−4|=2, |8−4|=4, i.e. {0, 2, 2, 3, 4}. The median of those deviations is 2, so the MAD is 2.

The 1.4826 scale factor

The raw MAD is smaller than the standard deviation, so it is often multiplied by a constant to make the two comparable. The standard constant is k ≈ 1.4826, which equals 1 ⁄ Φ⁻¹(0.75), the reciprocal of the 75th-percentile point of the standard normal distribution. For data drawn from a normal distribution, 1.4826 × MAD is an unbiased, consistent estimator of the standard deviation σ. If your MAD is 2, the scaled estimate of σ is 2 × 1.4826 ≈ 2.965.

Why use MAD instead of standard deviation?

The standard deviation squares every deviation from the mean, so a single extreme value can dominate it. The MAD is built entirely from medians, giving it a breakdown point of 50% — up to half the data can be corrupted before the statistic becomes meaningless. That makes MAD the tool of choice for skewed data (incomes, home prices, response times) and for automated outlier detection.

Detecting outliers with MAD

A common robust rule flags a point as an outlier when its modified z-score exceeds about 3.5. The modified z-score is defined as z = 0.6745 × (xᵢ − median) ⁄ MAD, where 0.6745 ≈ 1 ⁄ 1.4826. Values with |z| > 3.5 are treated as outliers. This is far more reliable than a mean-and-standard-deviation rule when the data already contain extreme values.

Frequently Asked Questions

How do you calculate the median absolute deviation?
Find the median of the data, subtract it from each value and take the absolute value, then take the median of those absolute deviations: MAD = median(|xᵢ − median(x)|). For 1, 2, 4, 6, 8 the median is 4, the absolute deviations are {0, 2, 2, 3, 4}, and their median — the MAD — is 2.
Why use MAD instead of the standard deviation?
MAD is robust to outliers. Because it uses medians rather than means and squared terms, a single extreme value changes it very little (its breakdown point is 50%). The standard deviation squares each deviation, so one large outlier can inflate it dramatically. Prefer MAD when data are skewed or contain outliers.
What does the 1.4826 factor do?
Multiplying MAD by k ≈ 1.4826 (which is 1/Φ⁻¹(0.75)) makes it a consistent estimator of the standard deviation σ for normally distributed data. So for a roughly normal sample, 1.4826 × MAD ≈ standard deviation. The raw, unscaled MAD is what you use if you just want the median distance from the center.
Is the median absolute deviation the same as the mean absolute deviation?
No — they share the initials but differ. The mean absolute deviation averages the absolute deviations, usually about the mean, and is still influenced by outliers. The median absolute deviation takes the median of the absolute deviations about the median, making it robust. This calculator computes the median version.

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