Visualizing Mean, Median, Mean Deviation, and Standard Deviation of a Set of Numbers

ABSTRACT

We review the existing visualizations of the mean and the median of a given set of numbers. Then we give an alternative visualization of the mean using the empirical cumulative distribution function of the given numbers. Next, we visualize the mean deviation (MD) and the mean square deviation (MSD) of the given numbers from any arbitrary value, including the variance. In light of these new visualizations, we revisit the well-known optimal properties of the MD from the median and the MSD from the mean. We also give a more elementary explanation of why the denominator of the sample variance of a set of numbers is one less than the sample size.

KEYWORDS:

• Center
• Empirical cumulative distribution function
• Fulcrum
• Mean squared deviation
• Minimization
• Prism
• Solid of revolution
• Spread
• Variance

Acknowledgment

The authors are grateful to two anonymous referees, the associate editor, and the editor for giving us valuable suggestions, including a reference, to improve an earlier version of this article.

Notes

¹ If not, add a sufficiently large constant C to all numbers to make them all positive. Then consider the mean of the transformed values. Finally, subtract C from that calculated mean to recover the mean of the given values.

2 Even though s² is an unbiased estimator of σ², it is well known that s is not an unbiased estimator of σ, although it is less biased than would be if the denominator were n.