# Average Deviation Formula

The average deviation is a part of several indices of variability that is used by statisticians to characterize the dispersion among the measures in a given population. The average deviation of a set of scores is calculated by computing the mean and then the specific distance between each score and that mean without regard to whether the score is above or below the mean. It is also called an average absolute deviation. Below mentioned is the formula to calculate the average deviation.

$$\begin{array}{l}\text { Average Deviation }=\frac{1}{n}\sum_{i=1}^{n}|x-\bar{x}|\end{array}$$
Where,
xÂ represents the observation.
$$\begin{array}{l}\bar{x}\end{array}$$
Â represents the mean.
nÂ represents the number of observations.

### Solved Examples

QuestionÂ 1: Calculate the average deviation for the given data: 4, 6, 8, 10, 12, 14.
Solution:

Given:
n =Â 6
First lets find the mean by using the formula,
$$\begin{array}{l}\overline{x} = \frac{4+6+8+10+12+14}{6}\end{array}$$

$$\begin{array}{l}\overline{x} = 9\end{array}$$

$$\begin{array}{l}Average \: Deviation= \frac{\left | 4-9 \right |+\left | 6-9 \right |+\left | 8-9 \right |+\left | 10-9 \right |+\left | 12-9 \right |+\left | 14-9 \right |}{6}\end{array}$$

$$\begin{array}{l}Average \: Deviation=\frac{5+ 3+ 1+ 1+ 3+ 5}{6}\end{array}$$

$$\begin{array}{l}Average \: Deviation=3\end{array}$$

Question 2: Find the average deviation for the following set of observations.

11, 6, 6, 12, 12, 7,Â  7, 9

Solution:

Given set of observations:

11, 6, 6, 12, 12, 7,Â  7, 9

Sum of observations = 11 + 6 + 6 + 12 + 12 + 7 + 7 + 9 = 70

Number of observations = n = 8

Mean = Sum of observations/Total number of observations

$$\begin{array}{l}\bar{x}=\frac{70}{8}=8.75\\ Average\ deviation = \frac{\left | 11-8.75 \right | +\left | 6-8.75 \right |+ \left | 6-8.75 \right |+\left | 12-8.75 \right |+\left | 12-8.75 \right |+\left | 7-8.75 \right |+\left | 7-8.75 \right |+\left | 9-8.75 \right |}{8}\end{array}$$

$$\begin{array}{l}=\frac{2.25+ 2.75+ 2.75+ 3.25+ 3.25+ 1.75+1.75+ 0.25}{8}\\=\frac{18}{8}\\=2.25\end{array}$$