Step-by-Step Guide to Finding Mean Deviation for Ungrouped Data

Mean deviation provides a quantitative measure of the average amount by which terms in a data set deviate from a measure of central tendency such as the mean or median. Precise computation of mean deviation for ungrouped data requires systematic execution of calculation steps adhering to academic conventions used in statistics.

Formal Definition of Mean Deviation for Ungrouped Data

Let $x_1, x_2, x_3, \ldots, x_n$ denote $n$ distinct observations comprising an ungrouped data set. The mean deviation about a central value $a$ is defined as the arithmetic mean of the absolute differences between each observation and $a$:

$\displaystyle\text{Mean Deviation about } a = \frac{1}{n}\sum_{i=1}^n |x_i - a|$

The value of $a$ is generally taken to be the mean $\overline{x}$ or the median $M$ of the data set. Both cases require explicit calculation using the formulas for mean and median specific to ungrouped data. For a detailed exposition regarding central tendency, reference may be made to the section on the Measures of Central Tendency.

Calculation of Arithmetic Mean for Ungrouped Data

For $n$ observations $x_1, x_2, \ldots, x_n$, the arithmetic mean $\overline{x}$ is computed as follows:

$\displaystyle \overline{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{\sum_{i=1}^n x_i}{n}$

Calculation of Median for Ungrouped Data

For $n$ observations ordered in ascending order, the median $M$ is defined differently based on the parity of $n$:

If $n$ is odd,

$M = x_{\frac{n+1}{2}}$

If $n$ is even,

$M = \dfrac{ x_{\frac{n}{2}} + x_{\frac{n}{2} + 1} }{2} $

Here, $x_k$ denotes the $k$-th observation in the ordered data. For formal properties relevant to competitive exams, consult Properties of Median in Statistics.

Computation Steps for Mean Deviation about the Mean

Given the data set $x_1, x_2, \ldots, x_n$, the mean deviation about the mean $\overline{x}$ is calculated as follows:

$\displaystyle \text{Mean Deviation about } \overline{x} = \frac{1}{n}\sum_{i=1}^n |x_i - \overline{x}|$

The procedure precisely involves:

1. Compute the arithmetic mean $\overline{x}$.

2. For each $x_i$, evaluate the absolute deviation $|x_i - \overline{x}|$.

3. Sum all absolute deviations: $\sum_{i=1}^{n} |x_i - \overline{x}|$.

4. Divide this sum by $n$ to obtain the mean deviation about $\overline{x}$.

This approach can also be generalized for mean deviation about the median by replacing $\overline{x}$ with $M$ as needed. The distinction between grouped and ungrouped data can be reviewed under Difference Between Mean, Median and Mode.

Worked Example: Mean Deviation about the Mean

Given: The set of observations: $12, \; 45, \; 60, \; 66, \; 77, \; 84, \; 90$.

Step 1: Calculate the arithmetic mean $\overline{x}$.

$\overline{x} = \dfrac{12 + 45 + 60 + 66 + 77 + 84 + 90}{7}$

$\overline{x} = \dfrac{434}{7} = 62$

Step 2: Find absolute deviations from the mean:

\[ \begin{align*} |12 - 62| &= 50 \\ |45 - 62| &= 17 \\ |60 - 62| &= 2 \\ |66 - 62| &= 4 \\ |77 - 62| &= 15 \\ |84 - 62| &= 22 \\ |90 - 62| &= 28 \\ \end{align*} \]

Step 3: Sum the absolute deviations:

$50 + 17 + 2 + 4 + 15 + 22 + 28 = 138$

Step 4: Divide by $n$ to find the mean deviation:

$\displaystyle \text{Mean Deviation about } \overline{x} = \dfrac{138}{7} = 19.714$

Result: The mean deviation about the mean is $19.714$.

Worked Example: Mean Deviation about the Median

Given: The set of marks: $68,\, 75,\, 79,\, 86,\, 88,\, 91,\, 95,\, 99$

Step 1: Order the data (already ascending). $n=8$, which is even, so the median is:

$M = \frac{ x_{4} + x_{5} }{2 } = \frac{86 + 88}{2} = 87$

Step 2: Obtain all absolute deviations from the median:

\[ \begin{align*} |68 - 87| &= 19 \\ |75 - 87| &= 12 \\ |79 - 87| &= 8 \\ |86 - 87| &= 1 \\ |88 - 87| &= 1 \\ |91 - 87| &= 4 \\ |95 - 87| &= 8 \\ |99 - 87| &= 12 \\ \end{align*} \]

Step 3: Sum the absolute deviations: $19 + 12 + 8 + 1 + 1 + 4 + 8 + 12 = 65$

Step 4: Divide by $n$ to find the mean deviation:

$\text{Mean Deviation about } M = \dfrac{65}{8} = 8.125$

Result: The mean deviation about the median is $8.125$.

Interpretation and Mathematical Notes

Mean deviation quantifies dispersion and is always a non-negative value, with lower values indicating greater concentration about the chosen central tendency. As only absolute differences are used, the measure remains unaffected by the direction of deviation. This attribute distinguishes it from variance and standard deviation, which involve squared deviations; a detailed comparative discussion with variance may be found in the Statistics and Probability section.

For advanced considerations between computation methods for grouped and ungrouped data, refer to How To Find Mean Deviation.