What Quantity Appears on the Vertical Axis of a Normal Distribution? A Student Survey

Abstract

How accurately can final-year students majoring in statistics, physics, and finance label the vertical axis of a normal distribution, explain their label, identify units, and answer a question about the impact of horizontal-axis rescaling? Our survey finds that only 27 out of 148 students surveyed (i.e., 18.2%) could label the vertical axis of the normal distribution correctly, and of these, only five students (i.e., 3.4%) could explain their label. Performance on individual survey questions differed by degree program, as might be expected, but overall survey performance varied very little, ranging between only 8.8% and 12.5% of survey questions answered correctly across degree programs. Common misconceptions included labeling the vertical axis as “probability,” “count,” or “frequency.” To address these demonstrated gaps in statistics education, we give counterexamples to show why these labels cannot be correct; we explain why “probability density” is the correct label; and we give an intuitive explanation of probability density and its units. We also discuss the impact of horizontal-axis rescaling, and we indicate how the units of probability density change if the probability density function is for a bivariate or trivariate distribution instead of a univariate distribution. This article is intended for all levels of statistics education.

Keywords:

• cdf
• Linear density
• pdf
• pmf
• Probability density
• Rescaling

Acknowledgments

We thank Nicola Beatson, Gurmeet Bhabra, Philip Brydon, David Fletcher, Matthew Schofield, Jeffrey Smith, Jin Zhang, two anonymous referees, and the editors for comments and assistance. Any errors are our own.

Notes

1 We used a small-sample test of differences in proportions assuming independent samples. We simulated 250,000 observations of possible observed proportion pairs for each of 100 different null hypothesis values for the true population proportion. The p-value for a two-sided test was no larger than 3.1% for any of these hypothesized population proportions.

2 If we drop the statistics majors from the sample, $\text{corr}(C,L)=13.3\%$ (t-statistic 1.55) and ${\chi }^2=5.93$ with a p-value of 9.4% (but the expected count is below 5 in two cells).

3 Note that some authors referred to the probability mass function simply as a “probability function” (Spiegel 1975; Evans, Hastings, and Peacock 1994).

4 Note that some authors referred to the cumulative distribution function simply as the “distribution function” (Spiegel 1975; Evans, Hastings, and Peacock 1994). Feller (1968, p. 179) argued that the adjective “cumulative” is redundant.

5 It is possible, however, to create a generalized pdf for discrete or mixed random variables using the Dirac delta function (Chakraborty 2008), but this generalization is above the level of this article.

6 Survey Question 4 asked students to ignore the units. This is because you might argue that height has not changed if units are taken into account. For example, just as with currencies, where 1USD $\approx$ 1.40NZD (i.e., one U.S. dollar approximately equals 1.40 New Zealand dollars at the time of writing), you might argue that $0.20{\text{inch}}^{-1}=0.08{\text{cm}}^{-1}$.