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The Mean Absolute Deviation (MAD) Calculator measures the average distance between each data point and the mean of the dataset. MAD is a robust and intuitive measure of variability — it tells you, on average, how far values deviate from the center. Unlike standard deviation, which squares deviations (giving more weight to outliers), MAD uses absolute values, producing a measure that is easier to interpret and more resistant to extreme observations.
MAD is widely used in quality control, forecasting accuracy assessment, financial risk analysis, and data science. In many practical situations, MAD provides a clearer picture of typical variability than standard deviation, especially when communicating with non-technical audiences. A MAD of 5 units means that, on average, each observation is about 5 units away from the mean — a straightforward interpretation that standard deviation lacks. This calculator supports up to 10 data values and computes both the MAD and the dataset mean.
The Mean Absolute Deviation from the mean is calculated as:
$$\text{MAD} = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$$
where \(\bar{x}\) is the arithmetic mean of the dataset and \(|\cdot|\) denotes absolute value. The steps are:
For a normal distribution, MAD is related to standard deviation by a constant factor:
$$\text{MAD} \approx 0.7979 \times \sigma$$
This means MAD is about 80% of the standard deviation for normally distributed data. Some analysts define MAD around the median instead of the mean, which provides even greater robustness to outliers. This calculator uses the mean-based definition, which is more common in introductory statistics.
A MAD of 5.0 means each observation deviates from the mean by 5 units on average. Smaller MAD indicates data is clustered tightly around the mean (low variability), while larger MAD indicates widely spread data (high variability). MAD = 0 only when all values are identical.
MAD is particularly useful for: detecting whether a single standard deviation figure is being inflated by a few outliers, providing forecast accuracy metrics (Mean Absolute Error in forecasting is essentially MAD of prediction errors), and communicating variability to non-statisticians in plain terms.
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Results
Six daily temperatures with mean 23.83. Deviations: |18-23.83|=5.83, |20-23.83|=3.83, |22-23.83|=1.83, |25-23.83|=1.17, |28-23.83|=4.17, |30-23.83|=6.17. MAD = 23/6 = 3.83 degrees.
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Results
Eight measurements with mean 10.0125. MAD = 0.156, indicating very tight quality control — parts deviate by only 0.156 units from the target on average.
Both measure spread, but standard deviation squares deviations (amplifying outlier influence) while MAD uses absolute values (giving outliers proportional weight). MAD is more robust and easier to interpret; SD is preferred for mathematical convenience and when normality is assumed.
For most datasets, yes. For a normal distribution, MAD is about 0.7979 times the standard deviation. However, for certain highly non-normal distributions, the relationship can differ.
Yes. The Median Absolute Deviation (also abbreviated MAD) uses the median as the center point. This version is even more robust to outliers and is commonly used in robust statistics. This calculator uses the mean-based definition.
In forecasting, Mean Absolute Deviation (or Mean Absolute Error) measures the average magnitude of forecast errors. It is preferred over RMSE when outlier errors should not disproportionately affect the accuracy metric.
There is no universal threshold — it depends on context. Compare MAD to the mean using the coefficient of variation analog (MAD/mean * 100%). A small ratio indicates consistent data; a large ratio indicates high variability relative to the average.
MAD is proportional to the L1 norm of deviations from the mean. The L1 norm sums absolute values, while the L2 norm (used in standard deviation) sums squared values. L1-based measures like MAD are central to robust regression and compressed sensing.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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