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Mean Absolute Deviation Calculator

See how spread out your numbers are: the average distance of each value from the middle, with a table showing every distance.

Level: Grades 6–7

  • Common Core standard 6.SP.B.5.cGiving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
  • Common Core standard 7.SP.B.3Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.
Printable worksheet

Mean Absolute Deviation Calculator

Inputs

Separate with commas, spaces, or new lines. Don't use thousands separators.

Measure

Result

Mean absolute deviation

3.75

The mean absolute deviation of 8 values is 3.75.

Mean
9
Median
7.5
Count (n)
8
Mean absolute deviation around the median
3.5
Median absolute deviation
2.5
Median absolute deviation × 1.4826 (robust SD estimate)
3.7065

See the step-by-step solution ↓

Absolute deviation of each value
Absolute deviation of each value
#Value (x)|x − mean||x − median|
1364.5
2631.5
3631.5
4720.5
5810.5
61123.5
71567.5
81678.5

Step-by-step solution

  1. Find the mean (x̄): add the 8 values and divide by how many there are

    x̄=3+6+6+7+8+11+15+168=9

  2. Find how far each value is from the mean, ignoring whether it is above or below, and add the distances

    |3 − 9|+|6 − 9|+…+|16 − 9|=6+3+3+2+1+2+6+7=30

  3. Divide the total distance by the number of values to get the average distance

    MAD=308=3.75

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find out how spread out a list of numbers is, on average. The mean absolute deviation (MAD) measures spread in the most direct way. It's the average distance between each value and the mean (the ordinary average). Every distance counts as positive, so values above and below the mean don't cancel out. A MAD of 3.75 means the values sit, on average, 3.75 units away from their mean.

Paste a list of numbers, and the calculator gives the MAD with the mean, median (middle value) and count. A table shows every value's distance from the mean and from the median. Two related measures are one click away. One is the mean absolute deviation around the median. The other is the median absolute deviation. It's a sturdy measure that barely moves when the data has one extreme value far from the rest (an outlier).

Where it's used

Why this math matters outside the classroom:

  • Comparing classes. A teacher sees two classes with a mean score of 80, but MADs of 4 and 12 show the second class's scores are far more spread out.
  • Delivery times. A courier whose arrival times have a MAD of 3 minutes is more dependable than one with the same average and a MAD of 15 minutes.
  • Sports consistency. A basketball coach compares two players who both average 15 points, choosing the one whose game totals vary by a MAD of 2 points.
  • Forecast accuracy. A store measures its sales forecasts by the mean absolute error, the average distance between predicted and actual daily sales.
  • Robust lab checks. A lab flags a reading as suspect when it lies more than 3 scaled median absolute deviations from the median of the batch.

Grade level and standards: Grades 6–7; Common Core State Standards for Mathematics 6.SP.B.5.c, 7.SP.B.3. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Enter or paste your data, separated by commas, spaces or new lines.
  2. Choose the measure. Most classes and textbooks mean the first option, the average distance from the mean.
  3. Read the main result. Then see the mean, the median and the other two measures listed below it.
  4. Open the steps and the table to see each value's distance (its absolute deviation) and how they add up.

What the inputs mean

Data set
Up to 10,000 numbers. Negatives and decimals are fine. A single value has a MAD of 0.
Measure
The median absolute deviation (often also shortened to MAD) is a different measure. It's the median of the distances from the median. Check which one your course or software means.

Formula and method

mean x̄ = (x1 + x2 + ... + xn) ÷ n
MAD = (|x1 − x̄| + |x2 − x̄| + ... + |xn − x̄|) ÷ n
MAD around the median = Σ |xi − median| ÷ n
median absolute deviation = median of |xi − median|
robust SD estimate ≈ 1.4826 × median absolute deviation

In the formulas, x1, x2 and so on are your values, and n is how many there are. x̄ (read "x-bar") is the mean. The bars | | mean "distance, ignoring the sign" (the absolute value). Σ means "add them all up". SD is short for standard deviation, another common measure of spread.

Subtracting the mean from each value gives a deviation. It's positive above the mean and negative below it. Those deviations, with their signs, always add up to zero, so they can't measure spread. Dropping the sign turns each one into a plain distance. Averaging the distances gives the mean absolute deviation.

Measuring the distances from the median instead of the mean gives the smallest possible average distance for that data set. That's why some fields prefer it.

The median absolute deviation swaps both averages for medians. Medians ignore how extreme the largest values are, so one wild outlier hardly changes it. For data that follows a bell curve (a normal distribution), multiplying it by about 1.4826 gives an estimate of the standard deviation.

Worked example

Eight students read 3, 6, 6, 7, 8, 11, 15 and 16 books over the summer.

  1. Mean: (3 + 6 + 6 + 7 + 8 + 11 + 15 + 16) ÷ 8 = 72 ÷ 8 = 9 books.
  2. Distances from 9: 6, 3, 3, 2, 1, 2, 6 and 7, which add up to 30.
  3. Mean absolute deviation: 30 ÷ 8 = 3.75 books.
  4. Around the median of 7.5 the distances add up to 28, so that version is 28 ÷ 8 = 3.5.
  5. The sorted distances from the median are 0.5, 0.5, 1.5, 1.5, 3.5, 4.5, 7.5 and 8.5, so the median absolute deviation is (1.5 + 3.5) ÷ 2 = 2.5.

Result: The mean absolute deviation is 3.75 books. A typical student read about 3.75 books more or fewer than the class average of 9.

Understanding the result

MAD is in the same units as the data, so it reads naturally: 'on average, values are 3.75 books from the mean'. A MAD of 0 means every value is the same.

When you compare two groups, the one with the larger MAD is more spread out. You can also divide the gap between two group means by a MAD. That shows the gap in units of typical spread. It's a common way to judge how much two groups overlap.

For bell-shaped data, the mean absolute deviation is about 0.8 times the standard deviation. If the MAD is much smaller than that, a few large outliers are probably pushing up the standard deviation.

Assumptions and limitations

  • Every value counts equally; there are no weights or frequencies.
  • The data set is treated as complete. The sample standard deviation divides by n − 1 to adjust for using a sample. The mean absolute deviation has no such adjustment.
  • The mean absolute deviation still uses the mean, so a single extreme value pulls it up. The median absolute deviation is the sturdier choice.
  • The 1.4826 scaling assumes roughly bell-shaped data. It only gives an estimate of the standard deviation.
  • The table lists the first 100 values. Longer lists are still used in full.

Frequently asked questions

How do you calculate mean absolute deviation?

Find the mean. Find how far each value is from it, ignoring the sign. Add those distances, and divide by the number of values. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the distances add up to 12. So the MAD is 12 ÷ 8 = 1.5.

What is the difference between MAD and standard deviation?

Both measure spread around the mean. MAD averages the plain distances. Standard deviation squares them, averages, and takes the square root. Squaring gives large distances more weight. So the standard deviation is always at least as large as the MAD, and it reacts more to outliers.

Is mean absolute deviation the same as median absolute deviation?

No, although both are shortened to MAD. The mean absolute deviation averages distances from the mean. The median absolute deviation takes the median of distances from the median. For the books example, they're 3.75 and 2.5.

What does a large mean absolute deviation mean?

It means the values are widely scattered around the mean. Whether it counts as large depends on the scale of the data, so judge it next to the mean. A MAD of 5 is large for quiz scores out of 10, but small for scores out of 500.

How this calculator is checked

Method: Standard definitions of the mean absolute deviation about the mean and about the median, and of the median absolute deviation with the 1/Φ⁻¹(0.75) ≈ 1.4826 consistency constant for normal data. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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