Sum of Absolute Deviations from Central Tendency Measures

This question asks us to identify the measure of central tendency from which the sum of the absolute deviations of the data points is the smallest. In statistics, "ignoring signs" means we are considering the absolute value of the deviations.

Understanding Deviations from the Median

A fundamental property related to the Median is that it uniquely minimizes the sum of the absolute deviations for a given dataset. This means if you calculate the difference between each data point and the Median, take the absolute value of each difference, and then sum them up, you will get the smallest possible sum compared to using any other single value from the dataset or outside of it.

For a dataset denoted as {$x_1, x_2, ..., x_n$}, the value $M$ that minimizes the sum $S = \sum_{i=1}^{n} |x_i - M|$ is the Median of the dataset.

Comparing with Other Measures

Let's look at why the other options are not correct:

• Arithmetic Mean: The sum of deviations of items from the arithmetic mean ($\bar{x}$) always equals zero. That is, $\sum_{i=1}^{n} (x_i - \bar{x}) = 0$. However, the sum of the *absolute* deviations from the arithmetic mean, $\sum_{i=1}^{n} |x_i - \bar{x}|$, is generally larger than the sum of absolute deviations from the median. The arithmetic mean is known for minimizing the sum of *squared* deviations, $\sum_{i=1}^{n} (x_i - \bar{x})^2$.
• Harmonic Mean: The harmonic mean is typically used for averaging rates or ratios. It does not possess the property of minimizing the sum of absolute deviations.
• Mode: The mode is the most frequently occurring value in a dataset. Similar to the harmonic mean, the sum of absolute deviations from the mode is not generally minimized.

Conclusion

Based on the properties of measures of central tendency, the sum of the absolute deviations (i.e., deviations ignoring signs) of the items from the Median is the least.