Mean Absolute Deviation Calculator
Enter your numbers, choose a central point (mean, median, or a custom value), and get the mean absolute deviation instantly. A step-by-step panel shows every individual deviation and how they average to the final MAD. Use this calculator for statistics homework, quality-control checks, data science, or any time you need a simple, robust measure of spread.
Formula
Worked example
Dataset: 4, 7, 13, 2, 1, 9, 5, 8. Mean = (4+7+13+2+1+9+5+8)/8 = 49/8 = 6.125. Absolute deviations: |4-6.125|=2.125, |7-6.125|=0.875, |13-6.125|=6.875, |2-6.125|=4.125, |1-6.125|=5.125, |9-6.125|=2.875, |5-6.125|=1.125, |8-6.125|=1.875. Sum = 25. MAD = 25/8 = 3.125.
What is mean absolute deviation?
Mean absolute deviation (MAD) is a measure of statistical dispersion: it tells you, on average, how far each value in a dataset is from a chosen central point. The central point is usually the arithmetic mean, but you can also use the median or any other reference value. Because it uses absolute values rather than squared differences, MAD is expressed in the same units as the original data and is far easier to interpret at a glance than variance. A MAD of 3 for a dataset of weights measured in kilograms means the typical data point sits about 3 kg away from the center.
How to calculate mean absolute deviation - the formula
The formula is MAD = (1/n) * sum of |xi - m| for every value xi, where m is the central point (usually the mean) and n is the number of values. The steps are: (1) find the mean of your dataset; (2) subtract the mean from each value and take the absolute value of the result; (3) sum all those absolute differences; (4) divide by n. For example, the dataset 4, 7, 13, 2, 1, 9, 5, 8 has a mean of 6.125. The absolute deviations are 2.125, 0.875, 6.875, 4.125, 5.125, 2.875, 1.125, and 1.875, which sum to 25. Dividing by 8 gives MAD = 3.125.
MAD vs. standard deviation: when to use each
Standard deviation also measures spread, but it squares the deviations before averaging, which means a single large outlier inflates it dramatically. MAD, using absolute values instead, is far less sensitive to extreme values. This makes MAD the preferred dispersion measure when your data contains outliers, when you need a result in the original units of measurement without any squaring, or when communicating results to a non-technical audience. Standard deviation is preferred when working with normal distributions or when your statistical test formally requires it, because it has convenient mathematical properties in those settings. In most everyday data analysis, both measures are useful together.
Choosing the right central point
The default central point for MAD is the arithmetic mean, which minimises the sum of squared deviations. However, you can compute MAD around the median instead: median absolute deviation (also called MAD in some literature) is even more robust to outliers because the median itself is not pulled by extremes. Use the mean-centered MAD when you want a measure consistent with variance and standard deviation. Use the median-centered MAD for heavy-tailed data, data with outliers you cannot remove, or data from distributions that are not symmetric. The custom central point option lets you measure dispersion around any reference: for instance, a regulatory target, a historical baseline, or a model prediction.
Applications of mean absolute deviation
MAD is used across many fields. In finance, it measures the average deviation of returns from a benchmark, giving a more outlier-resistant view of risk than standard deviation. In quality control, relative MAD (MAD divided by the mean, expressed as a percentage) serves as a coefficient of dispersion to compare variability across processes with different scales. In forecasting, MAD is a common accuracy metric: it tells you how many units off your predictions are on average. In education, teachers use it to show students how spread out test scores are in a concrete, easy-to-understand way. Epidemiologists use it when reporting dispersion of lab measurements that contain occasional erroneous readings.
Relative MAD interpretation guide
| Relative MAD | Interpretation | Typical context |
|---|---|---|
| < 5% | Very low dispersion | Precision manufacturing, tight tolerances |
| 5% - 10% | Low dispersion | Quality-controlled processes, repeatable experiments |
| 10% - 20% | Moderate dispersion | Survey data, biological measurements |
| 20% - 30% | Moderately high dispersion | Financial returns, economic data |
| 30% - 50% | High dispersion | Highly variable phenomena, heterogeneous populations |
| > 50% | Very high dispersion | Heavy-tailed distributions, data with outliers |
Relative MAD (MAD divided by the mean, expressed as a percentage) gives a scale-free measure of dispersion. These bands are commonly used in quality control and applied statistics.
Frequently asked questions
What does a MAD of zero mean?
A MAD of exactly zero means every value in the dataset is identical: every absolute deviation from the center is zero. There is no spread at all. This can happen with real data (for example, if all measurements were rounded to the same value) or it may indicate a data entry error.
Is mean absolute deviation the same as median absolute deviation?
No, though both are abbreviated MAD. Mean absolute deviation averages the absolute deviations using the arithmetic mean. Median absolute deviation takes the median of the absolute deviations (rather than the mean). The median-based version is even more robust to outliers. This calculator computes mean absolute deviation; if you choose the median as the central point, you are still averaging the deviations with the mean, not the median.
How does MAD compare to variance and standard deviation?
All three measure spread, but they weigh deviations differently. Variance averages squared deviations, so large outliers count disproportionately (a point twice as far contributes four times as much). Standard deviation is the square root of variance, returning to the original units but still inflated by outliers. MAD uses absolute values, giving every deviation linear weight, so it is far more resistant to extreme values. For a perfectly normal distribution, MAD is approximately 0.8 times the standard deviation.
Can I use MAD with a small dataset?
Yes, but interpret the result cautiously. With only 2 or 3 data points, the MAD is a sample estimate and can vary widely from the true population spread. As a rule of thumb, at least 5 to 10 values are needed for a reasonably stable estimate, and the more values you have, the more reliable the MAD becomes.
What is relative MAD and why is it useful?
Relative MAD expresses MAD as a percentage of the mean, making it a scale-free measure of dispersion sometimes called the coefficient of dispersion or mean absolute percentage deviation. It is useful when comparing the variability of datasets that are measured in different units or have very different magnitudes. For example, comparing the spread of house prices in different cities is clearer in percentage terms than in absolute dollar amounts.
Why does this calculator show a step-by-step breakdown?
Because MAD is a taught concept in schools and universities, and seeing every intermediate value helps students check their own manual calculations. The step panel shows the mean (or median), each individual absolute deviation, the sum of those deviations, and the final division, so you can follow the exact arithmetic from start to finish.