Mathwords logoMathwords

Average Absolute Deviation — Definition, Formula & Examples

Average absolute deviation is the average distance between each data value and the mean of the data set. It tells you, on average, how spread out the values are.

The average absolute deviation (AAD) of a data set is the arithmetic mean of the absolute differences between each observation and the data set's mean. Unlike variance, it does not square the deviations, keeping the result in the same units as the original data.

Key Formula

AAD=1n∑i=1n∣xi−xˉ∣\text{AAD} = \frac{1}{n}\sum_{i=1}^{n} |x_i - \bar{x}|
Where:
  • nn = Number of data values
  • xix_i = Each individual data value
  • xˉ\bar{x} = Mean of the data set
  • ∣  ∣|\;| = Absolute value (makes every deviation positive)

How It Works

First, calculate the mean of your data set. Then subtract the mean from each data value and take the absolute value of each difference—this removes negative signs so that deviations above and below the mean don't cancel out. Finally, average those absolute deviations. A small AAD means the data points cluster tightly around the mean, while a large AAD indicates greater spread.

Worked Example

Problem: Find the average absolute deviation of the data set: 2, 4, 6, 8, 10.
Find the mean: Add all values and divide by 5.
xˉ=2+4+6+8+105=6\bar{x} = \frac{2+4+6+8+10}{5} = 6
Find each absolute deviation: Subtract the mean from each value and take the absolute value.
∣2−6∣=4,  ∣4−6∣=2,  ∣6−6∣=0,  ∣8−6∣=2,  ∣10−6∣=4|2-6|=4,\;|4-6|=2,\;|6-6|=0,\;|8-6|=2,\;|10-6|=4
Average the absolute deviations: Sum the absolute deviations and divide by 5.
AAD=4+2+0+2+45=125=2.4\text{AAD} = \frac{4+2+0+2+4}{5} = \frac{12}{5} = 2.4
Answer: The average absolute deviation is 2.4. On average, each value is 2.4 units away from the mean.

Why It Matters

Average absolute deviation is a straightforward way to describe data spread before you learn standard deviation. It appears in AP Statistics preparation and data-science contexts where an intuitive, easy-to-explain measure of variability is preferred over squared-deviation methods.

Common Mistakes

Mistake: Forgetting to take absolute values, causing positive and negative deviations to cancel out.
Correction: Always apply absolute value to each deviation before averaging. Without it, the sum of deviations from the mean is always zero.