What Mean Absolute Deviation Measures
The Mean Absolute Deviation (MAD) is a simple, practical way to describe how spread out your numbers are. Instead of squaring differences (like variance and standard deviation), MAD looks at the absolute distance of each value from a chosen center and then averages those distances. Because it uses absolute values, it stays in the same units as your data, which makes it easy to interpret.
If your dataset is measured in dollars, MAD is measured in dollars. If your dataset is measured in minutes, MAD is measured in minutes. This “same-units” feature makes MAD intuitive for reporting and quick comparisons, especially when you want a “typical distance from the center” summary that non-technical readers can understand.
MAD Formula
For a list of n values, pick a center (most commonly the mean). Then compute:
The absolute value bars mean we ignore the sign of the difference. A value 5 above the center and a value 5 below the center both contribute 5 to the total absolute deviation.
MAD About the Mean vs MAD About the Median
Many textbooks define “mean absolute deviation” as deviations about the mean. In real-world work, you will also see absolute deviations computed about the median because the median is more robust when there are outliers or extreme spikes.
- MAD about the mean: Great as a smooth “average distance from the mean.”
- MAD about the median: Often preferred when you want resistance to outliers.
This calculator supports both, so you can choose the center that matches your course, reporting standard, or analysis goal.
MAD vs Median Absolute Deviation
The term “MAD” can be confusing because some fields use it to mean median absolute deviation (the median of |x − median|). That is a different statistic than the mean absolute deviation shown here. Always verify the definition:
| Statistic | Center | Aggregation | Outlier sensitivity |
|---|---|---|---|
| Mean absolute deviation (this tool) | Mean or median | Average of |x − center| | Moderate (lower than SD) |
| Median absolute deviation | Median | Median of |x − median| | High robustness |
How to Calculate MAD by Hand
You can compute MAD in a predictable sequence. The calculator mirrors these steps and shows them in the step-by-step section:
- List your values.
- Compute the center (mean or median).
- Compute each deviation x − center.
- Take absolute values |x − center|.
- Add the absolute values.
- Divide by the number of observations (or by total frequency in grouped data).
Frequency Table MAD
If your data is already summarized into a frequency table (value x with frequency f), you can compute MAD without expanding the dataset. Use weighted formulas:
This is especially useful when you have repeated values or large datasets. The frequency tab in the tool calculates weighted mean, weighted median (based on cumulative frequency), and MAD efficiently.
How to Interpret MAD
MAD tells you the “typical absolute distance” from the center. It is not a strict bound, but it is a good single-number summary of spread:
- If MAD is small relative to your values, the data is tightly clustered.
- If MAD is large, the data is widely spread out around the center.
- If you compare MAD across groups, ensure units and scale are comparable.
The tool also shows MAD as a percentage of the chosen center (when the center is not zero). This can help compare variability across similar datasets, such as comparing delivery times across weeks or comparing prices across categories.
Common Pitfalls
- Mixing up MAD definitions: Mean absolute deviation vs median absolute deviation.
- Using the wrong center: Your assignment may require “about the mean.”
- Ignoring units: MAD is always in the same units as your data.
- Comparing across different scales: Use percent-of-center cautiously and only when meaningful.
- Assuming MAD behaves like SD: They measure spread differently and won’t match numerically.
FAQ
Mean Absolute Deviation FAQs
Quick answers about formulas, mean vs median centering, frequency tables, and interpretation.
Mean absolute deviation (MAD) is the average distance of data points from a chosen center (usually the mean). It is computed as the average of |xᵢ − center| over all observations. Smaller MAD means less spread.
Both are used in practice. “MAD about the mean” is common in textbooks and aligns naturally with the mean. “MAD about the median” is more robust to outliers. This calculator supports both choices.
Mean absolute deviation averages absolute deviations. Median absolute deviation takes the median of absolute deviations (usually from the median). People sometimes use “MAD” for both, so always check the definition.
Yes. If you have values and frequencies, MAD can be computed using weighted sums: MAD = (Σ f·|x − center|) / (Σ f). This tool calculates mean/median and MAD directly without expanding the data.
For mean absolute deviation, most definitions use a simple average over the available observations (divide by n). There isn’t a universal “n−1” correction like with sample variance. This calculator uses division by n (or total frequency).
MAD is in the same units as your data. If your MAD is 3 minutes, a typical value is about 3 minutes away from the chosen center. Comparing MAD across datasets is easiest when they share the same units and scale.
MAD is easier to explain (average absolute distance) and is less sensitive to extreme outliers than standard deviation. Standard deviation is more common in statistical modeling, but MAD is often a clearer “spread” summary.
Non-numeric entries and blanks are ignored. You need at least two valid values to compute a meaningful deviation measure.