Mean Absolute Deviation (MAD) — the clean definition

Mean Absolute Deviation is the average of the absolute distances from a chosen center. The “absolute” part matters: instead of letting negative and positive deviations cancel each other out, we take the magnitude of the distance.

For a dataset with n values \(x_1, x_2, ..., x_n\) and a chosen center \(c\), the Mean Absolute Deviation is:

\( \text{MAD} = \frac{1}{n} \sum_{i=1}^{n} |x_i - c| \)

• Classic MAD: \(c = \bar{x}\) (the mean), so \(\text{MAD} = \frac{1}{n}\sum |x_i - \bar{x}|\).
• Robust option: \(c = \text{median}(x)\), so you measure distances from the median instead.

If you used \((x_i - c)\) without absolute value, the positive and negative deviations would sum to zero when \(c\) is the mean — which tells you nothing about spread. Absolute values turn “direction” into “distance,” which is exactly what spread is about.

• MAD: uses absolute distances; straightforward, less sensitive to extreme squaring effects.
• Standard deviation: uses squared distances; penalizes large deviations more heavily.
• Practical takeaway: MAD is often easier to explain and interpret quickly.

How to interpret your MAD number

MAD is measured in the same units as your data. If your numbers are test scores, MAD is “points.” If your numbers are minutes, MAD is “minutes.” That’s why it feels intuitive: it tells you the typical distance from the center, in real-world units.

• Small MAD: most values cluster near the center (consistent results).
• Moderate MAD: some variation; values spread out but not wildly.
• Large MAD: high variability; values frequently sit far from the center.

If you want a quick “how big is big?” check, compare MAD to your data’s range: \(\text{Relative spread} \approx \frac{\text{MAD}}{\max(x)-\min(x)}\). This calculator shows a small meter using that ratio (it’s informal, but useful).

If your dataset includes an outlier, the mean can shift toward that extreme value. That shift can increase the absolute distances for many points and inflate mean-based MAD. Median-based MAD tends to stay more stable in the presence of outliers, which is why some teachers and textbooks prefer it.

Example 1: MAD around the mean (classic)

Suppose you have the dataset: 10, 12, 9, 15, 14. We’ll compute the mean, then the absolute deviations from the mean, then average them.

Add the values: \(10 + 12 + 9 + 15 + 14 = 60\). There are \(n=5\) values, so: \(\bar{x} = 60/5 = 12\).

• |10 − 12| = 2
• |12 − 12| = 0
• |9 − 12| = 3
• |15 − 12| = 3
• |14 − 12| = 2

Sum of absolute deviations = \(2 + 0 + 3 + 3 + 2 = 10\). MAD = \(10/5 = 2\).

Interpretation: Your values are typically about 2 units away from the mean. If these were quiz scores, it means a “typical” score sits roughly 2 points from the average.

Example 2: Outlier impact (mean vs median)

Consider: 10, 12, 9, 15, 100. That last value is a big outlier. Watch what happens if you measure deviations from the mean versus the median.

Mean = \((10+12+9+15+100)/5 = 146/5 = 29.2\). Absolute deviations: 19.2, 17.2, 20.2, 14.2, 70.8. Average = \((141.6)/5 = 28.32\). So mean MAD ≈ 28.32.

Median of (9,10,12,15,100) is 12. Absolute deviations from 12: 2, 0, 3, 88, ? (and ? ) → specifically: |10−12|=2, |12−12|=0, |9−12|=3, |15−12|=3, |100−12|=88. Average = \((96)/5 = 19.2\). So median MAD = 19.2.

Takeaway: Both versions increase because an outlier increases spread, but the median-based MAD often stays more stable because the median center does not shift as dramatically as the mean.

What this calculator does (step-by-step, behind the scenes)

This page runs entirely in your browser. When you click Calculate MAD, it follows a simple pipeline:

We split your input using commas, spaces, and line breaks. Any empty pieces are ignored. Each token is converted to a number. If any token is not a valid number, you’ll see an error that points you back to the input field.

If you choose Mean, we compute \(\bar{x}\) by summing values and dividing by \(n\). If you choose Median, we sort the list and pick the middle value (or average the two middle values for an even-sized dataset).

For each value \(x_i\), we compute the distance to the center: \(|x_i - c|\). Distances are always nonnegative.

Finally, we add the distances and divide by \(n\) to get MAD. The result is rounded based on your chosen decimal setting.

• We show the center value (mean or median), the sample size \(n\), and the min/max/range.
• We generate a short step list (first 12 rows) so you can screenshot proof for homework.
• We create a share-ready summary so you can paste it into a chat or post.

Note: There are multiple definitions floating around online (especially “MAD” in robust statistics, which can refer to median absolute deviation with a scaling constant). This calculator explicitly uses mean of absolute deviations from your chosen center (mean or median) — exactly as shown in the formula section above.

Make your MAD more meaningful

MAD becomes more useful when you pair it with context. Here are quick tips to get more value out of the number.

• If your data are dollars, MAD is dollars. If your data are centimeters, MAD is centimeters.
• Always state the unit when you report MAD: “MAD = 2.4 points,” not just “MAD = 2.4”.

• Two classes can have the same average score, but different MAD — meaning different consistency.
• In business, two products can have the same average delivery time, but different MAD — meaning one is more reliable.

• If one value is wildly different (data entry error? special case?), try median center to see sensitivity.
• If you’re analyzing performance and want to penalize big misses, consider standard deviation too.

A nice one-liner: “On average, values are about MAD units away from the center.” It’s clean, intuitive, and easy to remember.

Frequently Asked Questions

• What is Mean Absolute Deviation (MAD) in simple words?

  MAD is the average distance between each value and a chosen center (usually the mean). It tells you, in the same units as your data, how far values typically are from the “middle.”

• Is MAD the same as “median absolute deviation” used in robust statistics?

  Not always. Some sources use “MAD” to mean median absolute deviation (with optional scaling constants). This calculator uses the mean of absolute deviations from your chosen center (mean or median), as shown in the formula.

• Why divide by n and not (n − 1)?

  MAD is typically defined as an average of absolute deviations, so dividing by n is standard. The “n − 1” adjustment is common for variance/standard deviation when estimating population variance from a sample.

• Can MAD be negative?

  No. Because it’s built from absolute values, MAD is always zero or positive. MAD equals zero only when all values are identical.

• When should I use median-based MAD?

  Use it when outliers might distort the mean, or when your data distribution is skewed. Median-based MAD gives a more “typical” spread around a center that doesn’t shift as much.

• What’s a “good” MAD?

  There’s no universal “good” number because it depends on your unit and context. A MAD of 2 minutes might be excellent for one process and terrible for another. Compare MAD to your goals, your range, or other groups.

• Can I use this for homework?

  Yes — and the step list is designed to be screenshot-friendly. Just make sure your assignment defines MAD the same way (mean vs median center). If your class uses a special definition, follow your instructor.

How to report MAD (copy/paste format)

If you’re writing a lab report, stats homework, or a quick business summary, use a consistent template:

• Dataset size: n = …
• Center: mean/median = …
• MAD: …
• Interpretation: “Values are typically about … units away from the center.”

Bonus: If you’re comparing two groups, report both MAD values side-by-side and note which group is more consistent.