Understanding Mean Deviation and Central Tendency

The question asks about the specific average (measure of central tendency) from which the mean deviation is minimized. The mean deviation is a measure of dispersion that calculates the average of the absolute differences between each data point and a chosen average (like mean, median, or mode).

Mathematically, the Mean Deviation from an average $A$ for a dataset $x_1, x_2, \dots, x_n$ is given by:

\begin{equation*} MD_A = \frac{\sum_{i=1}^{n} |x_i - A|}{n} \end{equation*}

The problem states that we want to find the value of $A$ that makes this quantity $MD_A$ as small as possible.

Minimizing Mean Deviation: The Key Property

A fundamental property in statistics is that the sum of the absolute deviations of a set of observations is minimum when the deviations are taken from the median. Since the mean deviation is simply the sum of absolute deviations divided by the number of observations ($n$), minimizing the sum of absolute deviations also minimizes the mean deviation.

Therefore, the mean deviation from an average $A$ is minimum when $A$ is the median of the dataset.

Analyzing the Options

Let's consider the given options:

• Harmonic Mean: The harmonic mean is typically used for averaging rates or ratios. While it's a measure of central tendency, the sum of absolute deviations from the harmonic mean is not guaranteed to be minimum.
• Arithmetic Mean: The arithmetic mean (average) is the most common measure. A key property of the arithmetic mean is that the sum of the squared deviations ($\sum (x_i - \bar{x})^2$) is minimum when $\bar{x}$ is the arithmetic mean. However, the sum of absolute deviations ($\sum |x_i - A|$) is minimized by the median.
• Mode: The mode is the value that appears most frequently in a dataset. While useful for identifying the most typical value, it does not possess the property of minimizing the sum of absolute deviations.
• Median: The median is the middle value of a dataset when it is ordered. The sum of the absolute deviations from the median is indeed the minimum among all possible values of $A$. Consequently, the mean deviation is minimum when calculated from the median.

Why the Median Minimizes Mean Deviation

Consider a sorted dataset $x_1 \le x_2 \le \dots \le x_n$. The problem is to find $A$ that minimizes $\sum_{i=1}^{n} |x_i - A|$. Intuitively, to minimize the sum of distances from $A$, $A$ should be located somewhere in the "middle" of the data. The median serves as this optimal point for minimizing the sum of absolute differences.

For example, if you have just two points, $x_1$ and $x_2$, the sum of absolute deviations from $A$ is $|x_1 - A| + |x_2 - A|$. This sum is minimized for any $A$ between $x_1$ and $x_2$, including the median (which is $(x_1+x_2)/2$). For a larger dataset, the median balances the deviations from points below it and points above it in a way that minimizes the total sum of absolute distances.

Conclusion

Based on the property that the sum of absolute deviations is minimized when taken from the median, the mean deviation from the average $A$ is minimum if $A$ represents the median.

Revision Table: Key Concepts

Additional Information: Measures of Central Tendency

Measures of central tendency provide a single value that attempts to describe a set of data by identifying the central position within that set of data. The most common measures are the mean, median, and mode.

• Arithmetic Mean ($\bar{x}$): Calculated by summing all values and dividing by the number of values. It's sensitive to extreme values.
• Median: The value exactly in the middle of a dataset when it is ordered from least to greatest. If there is an even number of observations, the median is the average of the two middle values. It's robust to outliers.
• Mode: The value that occurs most frequently in a dataset. A dataset can have one mode (unimodal), multiple modes (multimodal), or no mode if all values appear with the same frequency.
• Harmonic Mean (HM): The reciprocal of the arithmetic mean of the reciprocals of the data points. It's often used when dealing with rates or ratios, such as average speed. HM = $\frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$.

The choice of which measure of central tendency to use depends on the nature of the data and the specific question being asked. For minimizing the mean deviation, the median is the correct choice.