Mean Absolute Deviation (MAD) Calculator

What is Mean Absolute Deviation?

Mean Absolute Deviation (MAD) is a measure of variability in a dataset that calculates the average distance between each data point and the mean. It provides insight into the spread of the data and is less sensitive to outliers compared to standard deviation.

Formula and Its Meaning

The formula for Mean Absolute Deviation is:

\[MAD = \frac{\sum_{i=1}^{n} |x_i - \bar{x}|}{n}\]

Where:

• \(x_i\) represents each value in the dataset
• \(\bar{x}\) is the mean of the dataset
• \(n\) is the number of values in the dataset
• \(|x_i - \bar{x}|\) represents the absolute difference between each value and the mean

Calculation Steps

1. Calculate the mean of the dataset.
2. Subtract the mean from each data point and take the absolute value of the difference.
3. Sum up all these absolute differences.
4. Divide the sum by the number of data points to get the MAD.

Example Calculation

Let's calculate the MAD for the dataset: 2, 4, 4, 4, 5, 5, 7, 9

1. Calculate the mean: \(\bar{x} = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = 5\)
2. Calculate absolute deviations: |2-5|, |4-5|, |4-5|, |4-5|, |5-5|, |5-5|, |7-5|, |9-5|
3. Sum of absolute deviations: 3 + 1 + 1 + 1 + 0 + 0 + 2 + 4 = 12
4. MAD = 12 / 8 = 1.5

Visual Representation

This scatter plot represents the example dataset. The red dashed line indicates the mean (5), and the spread of points illustrates the Mean Absolute Deviation.