Mean Deviation Coefficient Explained

The question asks for the name given to the quantity obtained when the mean deviation is divided by the average value used in its calculation.

Understanding Mean Deviation

Mean deviation measures the average absolute difference between each data point in a set and a central point (like the mean or median). It gives us an idea of how spread out the numbers are.

The formula for Mean Deviation calculated from the mean ($\bar{x}$) is:

$ \text{MD}_{\bar{x}} = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}| $

Where:

• $x_i$ represents each individual data point.
• $\bar{x}$ is the mean of the data set.
• $n$ is the total number of data points.
• $|x_i - \bar{x}|$ is the absolute difference between the data point and the mean.

Similarly, Mean Deviation calculated from the median ($M$) is:

$ \text{MD}_{M} = \frac{1}{n} \sum_{i=1}^{n} |x_i - M| $

Deriving the Coefficient of Mean Deviation

To understand the variability relative to the central value, we often use a relative measure. This is achieved by dividing the measure of dispersion (like mean deviation) by the central value itself.

When the mean deviation (calculated either from the mean or the median) is divided by the specific average value (mean or median) that was used to compute it, the resulting value is called the coefficient of mean deviation.

The formulas are:

• Coefficient of Mean Deviation from the Mean:

  $ \text{CMD}_{\bar{x}} = \frac{\text{MD}_{\bar{x}}}{\bar{x}} $

• Coefficient of Mean Deviation from the Median:

  $ \text{CMD}_{M} = \frac{\text{MD}_{M}}{M} $

This coefficient is a unitless measure, which makes it useful for comparing the dispersion of two or more datasets that might have different scales or units.

Therefore, when mean deviation is divided by the average used, the resulting quantity is the coefficient of mean deviation.