Mean Deviation for Ungrouped Data

Mean Deviation for Ungrouped Data

When we have $n$ observations $x_1, x_2, x_3, \ldots, x_n$, the mean deviation measures the average distance of each observation from a chosen central value. The central value we use is either the mean ($\bar{x}$) or the median ($M$). The process follows four clear steps.

The Four-Step Procedure

Step 1: Choose the central value. Decide whether you are finding the mean deviation about the mean or about the median. Call this central value $a$.

Step 2: Find the deviations. For each observation $x_i$, compute $x_i - a$. These deviations can be positive, negative, or zero.

Step 3: Take absolute values. Drop any negative signs to get $|x_i - a|$. This gives the distance of each observation from $a$, regardless of direction.

Step 4: Find the mean of these absolute deviations. Add up all the absolute deviations and divide by $n$.

When the central value is the mean $\bar{x}$, we write:

When the central value is the median $M$, we write:

Worked Example 1: Mean Deviation About the Mean

Find the mean deviation about the mean for the data: 6, 7, 10, 12, 13, 4, 8, 12.

Step 1: Compute the mean.

Step 2: Find the deviations $x_i - \bar{x}$.

Step 3: Take absolute values $|x_i - \bar{x}|$.

Step 4: Compute the mean deviation.

Worked Example 2: Larger Data Set

Find the mean deviation about the mean for: 12, 3, 18, 17, 4, 9, 17, 19, 20, 15, 8, 17, 2, 3, 16, 11, 3, 1, 0, 5.

Step 1: Compute the mean.

Step 2 and 3: Find the absolute deviations $|x_i - \bar{x}|$.

Step 4: Sum the absolute deviations and divide.

Worked Example 3: Mean Deviation About the Median

Find the mean deviation about the median for: 3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21.

Step 1: Find the median. First arrange the data in ascending order.

There are 11 observations (odd). The median is the $\left(\frac{11+1}{2}\right)^{\text{th}} = 6^{\text{th}}$ observation.

$M = 9$ …