The statistics of a function

Abstract

One of the most important applications of the definite integral in a modern calculus course is the mean value of a function. Thus, if a function f is defined on an interval [a, b], then the mean, or average value, of f is given by

In this note, we will investigate the meaning of other statistics associated with a function f on an interval, including the standard deviation, the skewness, the median and the shape of the distribution of values of the function about the mean, things that typically do not appear in introductory calculus courses, though they may arise in calculus-based statistics courses. In this way, we demonstrate how one can build much stronger interrelationships and connections between different branches of mathematics, something that is very important for students to see.

Keywords:

• statistics of a function
• standard deviation of a function
• median of a function
• distribution of a function
• skewness of a function

Acknowledgement

The work described in this article was supported by the Division of Undergraduate Education of the National Science Foundation under grants DUE-0089400, DUE-0310123 and DUE-0442160. However, the views expressed do not necessarily reflect those of the Foundation.