# What Quantity Appears on the Vertical Axis of a Normal Distribution?

11 Pages Posted: 27 Aug 2018 Last revised: 17 Apr 2019

See all articles by J. Scott Chaput

## J. Scott Chaput

University of Otago - Department of Accountancy and Finance

## Timothy Falcon Crack

University of Otago - Department of Accountancy and Finance

## Olena Onishchenko

University of Otago

Date Written: April 16, 2019

### Abstract

What quantity appears on the vertical axis of a normal distribution, or any univariate probability density function? When we ask senior students this seemingly simple question, we almost always get an incorrect answer (the most common incorrect answer is "it is probability"). Some students can give an incomplete answer (e.g., "it is the height of the density function"), but many respond with an embarrassed silence. Students who do give the correct answer ("it is probability density") are unable to explain their answer. That is, they have no intuition for what probability density is. We aim to fill the gap in statistics education revealed by these student responses. So, we give two counterexamples to demonstrate why the most common incorrect answer that we have heard cannot be correct. We then explain why probability density is the correct answer, and we give an intuitive explanation of what exactly is meant by the term "probability density." We indicate how the answer changes subtly if the probability density function is for a bivariate or trivariate distribution instead of a univariate distribution. We also discuss the impact of rescaling the x-axis. This paper is intended for all levels of statistical education.

Keywords: probability density; linear denisty; pdf; pmf; cdf; resclaing

JEL Classification: A20

Suggested Citation

Chaput, J. Scott and Crack, Timothy Falcon and Onishchenko, Olena, What Quantity Appears on the Vertical Axis of a Normal Distribution? (April 16, 2019). Available at SSRN: https://ssrn.com/abstract=3229568 or http://dx.doi.org/10.2139/ssrn.3229568

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