Mean Deviation: Coefficient of Mean Deviation, Merits, and Demerits

Last Updated : 06 Nov, 2023

Range, Interquartile range, and Quartile deviation all have the same defect; i.e., they are determined by considering only two values of a series: either the extreme values (as in range) or the values of the quartiles (as in quartile deviation). This approach of analysing dispersion by determining the location of limits is also known as the Limits Approach because it does not consider all the items in the series. Therefore, it is always preferable to have a measure of dispersion that accounts for all of a series’ observations and is calculated in relation to a central value. If the variations of items are derived from an average, then such a measure of dispersion could throw light on the creation of the series and the dispersal of items around a central value. One such measure of dispersion is Mean Deviation.

What is Mean Deviation?

The arithmetic average of the deviations of various items from a measure of central tendency (mean, median, or mode) is known as the Mean Deviation of a series. Other names for Mean Deviation are the First Moment of Dispersion and Average Deviation.

Mean deviation is calculated by using all of the items in the series. It can theoretically be calculated by taking deviations from any of the three averages. However, in reality, either the mean or the median is used to determine the mean deviation. Mode is usually not considered because its value is uncertain and provides incorrect results. Since the sum of deviations from the median is less than the sum of deviations from the mean, the former is considered better than the latter.

Note: The sign (+ or -) of deviations is ignored while calculating deviations from the selected average, assuming all deviations are positive.

Coefficient of Mean Deviation

Mean Deviation is an absolute measure of dispersion. In order to convert it into a relative measure, it is divided by the average from which it has been calculated. It is known as the Coefficient of Mean Deviation.

Coefficient of Mean Deviation from Mean ( $MD_{\bar{X}}$ ) = $\frac{MD_{\bar{X}}}{\bar{X}}$

Coefficient of Mean Deviation from Median ( $MD_{M}$ ) = $\frac{MD_{M}}{M}$

The mean deviation can be calculated using the mean, median and mode. The formula for calculating are:

(I) Mean Deviation from the Mean

To calculate the mean, first add up all of the observations and then, divide the total by the number of observations. The following are formulas for mean deviation about the mean in different statistical series:

In case of Individual Series:

$Mean~Deviation~from~Mean~(MD_{\bar{X})}=\frac{\sum{|X-\bar{X}|}}{N}=\frac{\sum{|D|}}{N}$

Where,

$Mean=\frac{Sum~of~all~observations}{N}$

In case of Discrete Series:

$Mean~Deviation~from~Mean~(MD_{\bar{X})}=\frac{\sum{f|X-\bar{X}|}}{N}=\frac{\sum{f|D|}}{N}$

Where,

$Mean=\Sigma\frac{fX}{f}$

In case of Continuous Series:

$Mean~Deviation~from~Mean~(MD_{\bar{X})}=\frac{\sum{f|m-\bar{X}|}}{N}=\frac{\sum{f|D|}}{N}$

Where,

$\bar{X}=A+\frac{\sum{fd'}}{\sum{f}}\times{C}$

X̄ = Arithmetic Mean; A = Assumed Mean; d = m – A; i.e., deviations of mid-points from assumed mean; d’= Step Deviations (deviations divided by common factor); Σfd’ = Sum of product of step deviations (d’) with their respective frequencies (f); C = Common Factor; Σf =Total number of items.

(II) Mean Deviation from the Median

The value that divides the data into its upper and lower half is called the median. The number in the middle of an arranged set of values, either ascending or descending, is called the median. The following are formulas for mean deviation about the median: