Mean Absolute Deviation (MAD) Formula

The Formula

\text{MAD} = \frac{\sum |x - \bar{x}|}{n}

When to use: Find how far each number is from the mean (ignoring +/-), then average those distances. It tells you: on average, how far is a typical value from the center?

Quick Example

Data: 2, 4, 6, 8. Mean = 5. Distances from mean: 3, 1, 1, 3. MAD = \frac{3+1+1+3}{4} = 2

Notation

MAD stands for Mean Absolute Deviation. |x_i - \bar{x}| is the absolute deviation of the ith data point from the mean \bar{x}.

What This Formula Means

The Mean Absolute Deviation (MAD) is the average of the absolute distances between each data point and the mean of the dataset. It measures how spread out data values are from the center, with larger MAD values indicating more variability.

Find how far each number is from the mean (ignoring +/-), then average those distances. It tells you: on average, how far is a typical value from the center?

Formal View

For a dataset \{x_1, x_2, \ldots, x_n\} with mean \bar{x}, the Mean Absolute Deviation is MAD = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|.

Worked Examples

Example 1

easy
Find the mean absolute deviation (MAD) of the data set: 4, 6, 8, 10, 12.

Solution

  1. 1
    Step 1: Find the mean: \frac{4+6+8+10+12}{5} = \frac{40}{5} = 8.
  2. 2
    Step 2: Find the absolute deviations from the mean: |4-8|=4, |6-8|=2, |8-8|=0, |10-8|=2, |12-8|=4.
  3. 3
    Step 3: Find the mean of these absolute deviations: MAD = \frac{4+2+0+2+4}{5} = \frac{12}{5} = 2.4.

Answer

MAD = 2.4.
The mean absolute deviation measures how far data values are from the mean on average. A smaller MAD indicates data clustered closely around the mean, while a larger MAD indicates more spread. MAD uses absolute values to avoid positive and negative deviations cancelling out.

Example 2

medium
Two data sets: A = {10, 10, 10, 10, 10} and B = {2, 6, 10, 14, 18}. Both have a mean of 10. Calculate the MAD for each and explain what it tells you.

Common Mistakes

  • Forgetting absolute value
  • Dividing by wrong number
  • Confusing with standard deviation

Why This Formula Matters

MAD is an intuitive and accessible measure of data spread used in weather forecasting, quality control, and classroom statistics. It serves as a conceptual stepping stone to understanding the more widely used standard deviation.

Frequently Asked Questions

What is the Mean Absolute Deviation (MAD) formula?

The Mean Absolute Deviation (MAD) is the average of the absolute distances between each data point and the mean of the dataset. It measures how spread out data values are from the center, with larger MAD values indicating more variability.

How do you use the Mean Absolute Deviation (MAD) formula?

Find how far each number is from the mean (ignoring +/-), then average those distances. It tells you: on average, how far is a typical value from the center?

What do the symbols mean in the Mean Absolute Deviation (MAD) formula?

MAD stands for Mean Absolute Deviation. |x_i - \bar{x}| is the absolute deviation of the ith data point from the mean \bar{x}.

Why is the Mean Absolute Deviation (MAD) formula important in Statistics?

MAD is an intuitive and accessible measure of data spread used in weather forecasting, quality control, and classroom statistics. It serves as a conceptual stepping stone to understanding the more widely used standard deviation.

What do students get wrong about Mean Absolute Deviation (MAD)?

Students forget to take absolute values before averaging, which causes deviations to cancel and gives zero — making all data sets look identical.

What should I learn before the Mean Absolute Deviation (MAD) formula?

Before studying the Mean Absolute Deviation (MAD) formula, you should understand: mean fair share.

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