Median Absolute Deviation Calculator

Q: What is the difference between Mean Absolute Deviation and Median Absolute Deviation?

Mean Absolute Deviation uses the mean as the centre: Σ|xi−x̄|/n. Median Absolute Deviation uses the median as the centre and then takes the median of deviations. The median-based version is more robust because it is resistant to both outliers in the centre measure and in the deviation step.

Enter any list of numbers to get the Median Absolute Deviation, normalized MAD, and a per-value deviation table with outlier detection.

📖 What is Median Absolute Deviation (MAD)?

Median Absolute Deviation (MAD) is a robust measure of statistical dispersion — a way to quantify how spread out the values in a dataset are. Unlike standard deviation, which squares deviations and is therefore heavily influenced by outliers, MAD uses the median in two places: first as the measure of central tendency, then as the way to summarise the deviations. The result is a spread measure with a 50% breakdown point, meaning up to half the data can be extreme without corrupting the MAD.

The computation has three steps: (1) find the median of the dataset, (2) compute the absolute deviation of each value from that median, (3) find the median of those absolute deviations. The final value is MAD. Because both steps use the median rather than the mean, no single outlier (or even a small fraction of outliers) can pull MAD to an unreasonable extreme the way a single extreme value can inflate standard deviation.

MAD was first described by Carl Friedrich Gauss in 1816 and was later popularised in modern robust statistics by Peter Rousseeuw and Christophe Croux (1993) as a reliable alternative to SD. Its primary use cases are: outlier detection (the 3.5×MAD rule), robust regression diagnostics, financial risk measurement on skewed return distributions, quality control on production lines with occasional defects, and any domain where a few extreme measurements are expected but should not dominate the spread estimate.

For normally distributed data, MAD and SD are related by the constant 1.4826: SD ≈ 1.4826 × MAD. This means you can multiply MAD by 1.4826 to get a consistent, unbiased estimate of the population standard deviation σ under normality. This “normalized MAD” (shown in the results above) lets you directly compare MAD-based spread to SD-based spread on the same scale.

This calculator also shows the MAD/SD ratio for your dataset. For perfectly normal data this ratio should be approximately 0.6745. A much lower ratio (MAD ≪ SD) indicates that outliers are inflating SD but not MAD — a clear sign of a heavy-tailed or contaminated distribution. This single diagnostic number can alert you to the presence of outliers before you even look at the deviation table.

📐 Formula

MAD = median(|x_i − median(x)|)

Step-by-step:

Sort the dataset and find the median M
Compute the absolute deviation for each value: d_i = |x_i − M|
Find the median of {d_i} — that is MAD

Normalized MAD (consistent estimator of σ under normality):

σ̂ = 1.4826 × MAD

Outlier threshold (Leys et al., 2013):

Outlier if |x_i − M| > 3.5 × MAD

Worked example: Dataset = {2, 3, 4, 5, 100}. Step 1: Sorted = {2, 3, 4, 5, 100}, median M = 4. Step 2: Absolute deviations = {|2−4|, |3−4|, |4−4|, |5−4|, |100−4|} = {2, 1, 0, 1, 96}. Step 3: Sorted deviations = {0, 1, 1, 2, 96}, median = 1. So MAD = 1. Compare: SD ≈ 43.3. The outlier (100) inflated SD enormously but barely affected MAD.

✍ How to Use This Calculator

Enter your data — type or paste numbers into the text area, separated by commas or spaces. Negative values and decimals are fully supported. At least 3 values are required.
Click Calculate — MAD, normalized MAD, median, mean, SD, count, and the MAD/SD ratio all appear instantly.
Check the deviation table — each value is listed in sorted order with its absolute deviation from the median. Values flagged with ⚠ exceed the 3.5×MAD outlier threshold.
Interpret the MAD/SD ratio — for normal data this should be around 0.6745. A much lower ratio means outliers are inflating SD but not MAD.

💡 Example Calculations

Example 1 — Simple clean dataset

Dataset: {2, 3, 4, 5, 6} — five evenly spaced values

Median M = 4. Absolute deviations: {2, 1, 0, 1, 2}

Sorted deviations: {0, 1, 1, 2, 2}. Median of these = 1.

MAD = 1. Normalized MAD = 1.4826. SD = 1.581. MAD/SD = 0.632 (close to 0.6745 for normal data).

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Example 2 — Dataset with an extreme outlier

Dataset: {1, 2, 3, 4, 100} — four typical values and one extreme outlier

Median M = 3. Deviations: {2, 1, 0, 1, 97}. Sorted: {0, 1, 1, 2, 97}.

MAD = 1 (unchanged by the 97-unit outlier). SD ≈ 43.3 (completely distorted).

The value 100 is flagged as an outlier (deviation=97 > 3.5×1=3.5). MAD/SD ≈ 0.023 — the extreme low ratio confirms outlier contamination.

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Example 3 — Skewed income data

Annual salaries ($): {28000, 32000, 35000, 38000, 41000, 150000}

Median = (35000+38000)/2 = 36500. Deviations: {8500, 4500, 1500, 1500, 4500, 113500}

Sorted deviations: {1500, 1500, 4500, 4500, 8500, 113500}. MAD = (4500+4500)/2 = 4500.

SD ≈ 46,800. MAD of $4,500 is far more representative of typical spread than SD of $46,800.

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Example 4 — Even number of values with repeats

Dataset: {5, 5, 5, 6, 7, 8} — six values, three identical

Sorted: {5, 5, 5, 6, 7, 8}. Median M = (5+6)/2 = 5.5

Deviations: {0.5, 0.5, 0.5, 0.5, 1.5, 2.5}. Sorted: {0.5, 0.5, 0.5, 0.5, 1.5, 2.5}.

MAD = (0.5+0.5)/2 = 0.5. Normalized MAD = 0.741. SD ≈ 1.152. MAD/SD ≈ 0.434.

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❓ Frequently Asked Questions

What is Median Absolute Deviation (MAD)?+

MAD is the median of the absolute deviations from the dataset's median: MAD = median(|x_i − median(x)|). It measures spread and is much more robust to outliers than standard deviation because using the median twice makes it highly resistant to extreme values.

What is the formula for Median Absolute Deviation?+

Step 1: Find the median M of your dataset. Step 2: Compute |x_i − M| for each value. Step 3: Find the median of those absolute deviations. That value is MAD.

How is MAD different from standard deviation?+

Standard deviation squares deviations before averaging, which amplifies outliers. MAD uses the median of absolute deviations, so a single extreme value cannot distort it. MAD is preferred for skewed distributions and datasets with outliers. For the dataset {1, 2, 3, 4, 100}: MAD = 1, SD ≈ 43.3.

What is the normalized MAD and why is it used?+

For normally distributed data, multiply MAD by 1.4826 to get an estimate of σ (population SD). This constant (1/Φ⁻¹(0.75) ≈ 1.4826) makes MAD a consistent estimator of SD under normality, allowing direct comparison with standard deviation.

What is a good MAD value?+

There is no universal threshold — MAD is relative to the scale of your data. A MAD of 5 on data ranging 0–10 indicates high spread; the same MAD on data ranging 0–1000 indicates low spread. Compare MAD to the median for scale-relative context.

How is MAD used for outlier detection?+

A common rule: any point where |x − median| > 3.5 × MAD is a potential outlier (Leys et al., 2013). This is the MAD-based equivalent of the Z-score rule (|Z| > 3.5). This calculator flags these points in the deviation table automatically.

What is the difference between Mean Absolute Deviation and Median Absolute Deviation?+

Mean Absolute Deviation uses the mean as the centre: Σ|x_i−x̄|/n. Median Absolute Deviation uses the median as centre and then takes the median of deviations. The median-based version is more robust because it is resistant to outliers in both the centre measure and the deviation step.

Can MAD be zero?+

Yes. If more than half the values in the dataset are identical, the median of |x_i − M| will be zero. This happens with heavily repeated values (e.g., ordinal ratings). Zero MAD signals no spread around the median in that dataset and makes outlier detection unreliable.

Is MAD resistant to outliers?+

Yes. MAD has a breakdown point of 50%, meaning up to half the data can be contaminated by extreme values before MAD becomes unreliable. By contrast, the mean and SD have a breakdown point of 0% — a single extreme value can arbitrarily distort them.

When should I use MAD instead of standard deviation?+

Use MAD when: your data has outliers or is heavily skewed; you are working with small samples where one extreme value matters a lot; your data is ordinal; or you need a robust measure for anomaly detection. Standard deviation is preferred for symmetric, well-behaved distributions where its mathematical properties are important.

Median Absolute Deviation Calculator

📖 What is Median Absolute Deviation (MAD)?

📐 Formula

✍ How to Use This Calculator

💡 Example Calculations

Example 1 — Simple clean dataset

Example 2 — Dataset with an extreme outlier

Example 3 — Skewed income data

Example 4 — Even number of values with repeats

❓ Frequently Asked Questions

🔗 Related Calculators

📌 Quick Tips