Understanding the Average Deviation Formula

Average deviation, commonly referred to as mean absolute deviation, provides a measure of dispersion in a data set by quantifying the average absolute difference between each data value and a chosen central tendency (mean, median, or mode). This concept is fundamental to statistical analysis and is frequently encountered in mathematics, physics, and chemistry.

Mathematical Structure of Average Deviation Formula

Consider a data set consisting of $n$ observations, represented as $x_1, x_2, x_3,\ldots, x_n$. The arithmetic mean (average) of these observations is defined as

$\displaystyle \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$

The average deviation about the mean is defined as the arithmetic mean of the absolute values of the deviations of each observation from the mean. The formal expression is

$\displaystyle \text{Average deviation about mean} = \frac{1}{n} \sum_{i=1}^{n} \left| x_i - \bar{x} \right|$

If the average deviation is to be calculated with respect to the median $M$ or mode $Z$, the formula is similarly written as

$\displaystyle \text{Average deviation about median} = \frac{1}{n}\sum_{i=1}^{n} \left| x_i - M \right|$

$\displaystyle \text{Average deviation about mode} = \frac{1}{n}\sum_{i=1}^{n} \left| x_i - Z \right|$

The presence of the absolute value in each formula ensures that all deviations are treated as positive, thereby eliminating cancellation of negative and positive differences. This property distinguishes average deviation from variance or standard deviation. 

Explicit Derivation of Average Deviation Formula

Let $x_1, x_2, \ldots, x_n$ be $n$ data values. The steps for deriving the average deviation about the mean are:

First, compute the arithmetic mean:

$\displaystyle \bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n}$

For each observation $x_i$, find its deviation from the mean:

$\displaystyle d_i = x_i - \bar{x}$

Each deviation $d_i$ can be positive, negative, or zero; to measure total dispersion, take the absolute value:

$\displaystyle |d_i| = |x_i - \bar{x}|$

Sum all absolute deviations:

$\displaystyle S = |x_1 - \bar{x}| + |x_2 - \bar{x}| + \ldots + |x_n - \bar{x}| = \sum_{i=1}^n |x_i - \bar{x}|$

Divide $S$ by the number of observations $n$ to obtain the average deviation:

$\displaystyle \text{Average deviation about the mean} = \frac{1}{n}\sum_{i=1}^{n} |x_i - \bar{x}|$

Formulation for Grouped Data

For data provided in a frequency distribution, let $x_i$ denote the mid-value of the $i$-th class interval, $f_i$ denote the corresponding frequency, and $N = \sum_{i=1}^k f_i$ the total frequency. The average deviation about the mean for grouped data is calculated as follows:

$\displaystyle \text{Average deviation about mean} = \frac{1}{N} \sum_{i=1}^{k} f_i |x_i - \bar{x}|$

Where $\bar{x} = \frac{1}{N} \sum_{i=1}^{k} f_i x_i$ is the mean for grouped data. This formula can also be applied analogously for the median or mode as the measure of central tendency. A stepwise computation of each component is essential for accuracy in competitive exams, particularly when dealing with class boundaries and frequencies.

Calculation of Average Deviation: Worked Examples

Given: Data set: $12, 14, 16, 18, 20, 22$

Step 1: Calculate the arithmetic mean.

$\displaystyle \bar{x} = \frac{12 + 14 + 16 + 18 + 20 + 22}{6} = \frac{102}{6} = 17$

Step 2: Compute each absolute deviation:

$|12 - 17| = 5$
$|14 - 17| = 3$
$|16 - 17| = 1$
$|18 - 17| = 1$
$|20 - 17| = 3$
$|22 - 17| = 5$

Step 3: Find the sum of absolute deviations.

$5 + 3 + 1 + 1 + 3 + 5 = 18$

Step 4: Divide by the total number of observations.

$\text{Average deviation} = \dfrac{18}{6} = 3$

Result: The average deviation of the data set is $3$.

Given: Data set: $33, 44, 55, 66, 77, 88, 99$

Step 1: Calculate the arithmetic mean.

$\bar{x} = \frac{33 + 44 + 55 + 66 + 77 + 88 + 99}{7} = \frac{462}{7} = 66$

Step 2: Compute each absolute deviation:

$|33 - 66| = 33$
$|44 - 66| = 22$
$|55 - 66| = 11$
$|66 - 66| = 0$
$|77 - 66| = 11$
$|88 - 66| = 22$
$|99 - 66| = 33$

Step 3: Sum of absolute deviations:

$33 + 22 + 11 + 0 + 11 + 22 + 33 = 132$

Step 4: Divide by the total number of observations.

$\text{Average deviation} = \dfrac{132}{7} \approx 18.857$

Result: The average deviation of the data set is approximately $18.857$.

Interpretation and Properties of Average Deviation

Average deviation is used to measure the degree of dispersion or variability in a data set. Unlike standard deviation, average deviation does not involve squaring the deviations; instead, it only deals with absolute values. This makes it less sensitive to extreme values (outliers) but easier to interpret in terms of the original data units. 

The value of average deviation is always non-negative, and its minimum value is zero, achieved only when all the data points coincide with the central value. The units of average deviation are the same as those of the original observations, making it directly comparable to the data values.

For broader coverage of statistical concepts and their interrelations, consult Statistics and Probability Concepts.