statistics

Median Absolute Deviation Calculator

The median absolute deviation (MAD) quantifies how spread out data points are around the median, making it invaluable for analysing skewed or non-normal distributions. Unlike standard deviation, which can be distorted by outliers, MAD provides a stable measure of variability. Data analysts, researchers, and quality-control specialists rely on it to assess consistency when traditional assumptions of normality don't hold.

Last updated: April 30, 2026

Creators Wojciech Sas, PhD

Reviewers Anna Szczepanek and Davide Borchia

749 people find this calculator helpful

What Is Median Absolute Deviation?

Median absolute deviation measures the typical distance between each data point and the median of the dataset. It belongs to a family of robust statistics—tools designed to resist the influence of extreme values.

In contrast, standard deviation treats all deviations equally and can be pulled heavily by outliers. If you're working with real-world measurements that contain occasional errors or unusual observations, MAD offers a more trustworthy picture of variation.

The term "absolute" is crucial: it means we ignore whether a point lies above or below the median. We care only about how far away it is. The "median" step at the end ensures that even the deviations themselves are analysed robustly, creating a double shield against outliers.

Median Absolute Deviation Formula

To find MAD, you work through five straightforward steps. Begin with your raw dataset, sort it, and find the middle value (the median). Then subtract that median from every point, take absolute values to eliminate negative signs, sort the resulting deviations, and finally extract the median of those deviations.

MAD = median(|X_i − m|)

where:

m = median of the original dataset
X_i = each individual data point

MAD — The median absolute deviation—the final robust measure of spread
m — The median (middle value) of your dataset
X<sub>i</sub> — Each individual observation in your dataset

Step-by-Step Worked Example

Suppose you recorded sprint times (in seconds) for six runners: 12, 16, 12, 11, 14, 15.

Step 1: Sort the data. Ordered times: [11, 12, 12, 14, 15, 16].

Step 2: Find the median. With six values, the median is the average of the 3rd and 4th: m = (12 + 14) ÷ 2 = 13 seconds.

Step 3: Subtract the median from each value.
11 − 13 = −2
12 − 13 = −1
12 − 13 = −1
14 − 13 = 1
15 − 13 = 2
16 − 13 = 3

Step 4: Take absolute values. [2, 1, 1, 1, 2, 3].

Step 5: Find the median of the deviations. Sorted: [1, 1, 1, 2, 2, 3]. The median of these six values is (1 + 2) ÷ 2 = 1.5 seconds.

Common Pitfalls and Practical Considerations

When computing or interpreting median absolute deviation, watch for these frequent mistakes:

Forgetting to sort before finding the median — The median is meaningless without ordering. Always arrange your dataset in ascending or descending order first, then locate the centre value (or average of the two central values if your dataset has an even count).
Neglecting the absolute value step — If you skip taking absolute values of the deviations, negative and positive distances will partially cancel out, giving a misleading result. Always convert every deviation to its positive magnitude before proceeding.
Confusing MAD with mean absolute deviation — Mean absolute deviation (average of absolute deviations) is different from median absolute deviation (median of absolute deviations). Both are robust, but MAD is more resistant to outliers because it uses the median rather than the mean in the final step.
Over-relying on a single statistic — MAD describes spread but says nothing about location, skewness, or the shape of your distribution. Pair it with complementary measures like the median itself, quartiles, or a histogram for a complete picture.

When to Use Median Absolute Deviation

Choose MAD when your data are non-normally distributed, contain outliers, or come from a process prone to occasional extreme events. Environmental monitoring, financial data analysis, and medical measurements often fit this profile.

For example, if you measure pollutant levels in a river and one storm causes a spike, MAD will remain stable while standard deviation balloons. Similarly, in clinical trials, if one patient has an unusual response, MAD isolates the typical variation experienced by the rest.

You can report MAD as is, or multiply it by 1.4826 (a scaling constant) to make it comparable to standard deviation in normally distributed data. This adjustment allows seamless comparison with legacy analyses or industry benchmarks.

Frequently Asked Questions

Why is median absolute deviation better than standard deviation for outliers?

Standard deviation squares the deviations before averaging, which dramatically amplifies the influence of distant points. A single extreme value can triple or quadruple the standard deviation. MAD uses the median instead of the mean, which is inherently resistant. Even if you add a wild outlier, the median of the deviations barely shifts because the median depends only on rank, not magnitude.

How do I interpret a median absolute deviation of 2.5?

It means that the typical data point is roughly 2.5 units away from the median. If your dataset is test scores with a median of 75 and a MAD of 2.5, most students cluster within about 2.5 points of that centre. The exact spread depends on your data's shape, but this figure gives an intuitive sense of variability around the middle.

Can I use median absolute deviation with datasets smaller than 5 values?

Technically yes, but the result becomes less meaningful. With very few observations, a single value disproportionately affects the median of the deviations. The real strength of MAD emerges with 10+ observations, where the robustness property kicks in properly.

What is the relationship between MAD and the interquartile range?

Both are robust measures of spread that avoid outlier distortion. The interquartile range (IQR) captures the middle 50% of data directly. MAD uses the full dataset but calculates variation around the median. They often produce similar insights, though MAD weights all observations whereas IQR ignores the tails entirely.

Should I scale the median absolute deviation for statistical tests?

If you need to compare your MAD to standard deviation or use it in a parametric framework, multiply by 1.4826. This constant assumes normally distributed data and makes MAD directly comparable to σ. For descriptive purposes alone, you don't need to scale—report it as your computation produces it.

How does MAD differ from mean absolute deviation?

Mean absolute deviation averages the deviations; median absolute deviation finds their median. Because averaging is sensitive to extreme values while the median is not, MAD provides one additional layer of outlier resistance compared to mean absolute deviation. For clean datasets they're similar, but for messy real-world data, MAD is the safer choice.

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