A U-shaped distribution exists, and it is the opposite of a bell curve. Most of the observations lie near the two extremes of the scale, and very few fall in the middle. On a histogram or density plot it forms a literal “U”.
The lowest and the highest values are by far the most common, and the middle values are rare. Imagine the marks of a group of students: most students got either very poor or very good marks, and few got average marks.
Online product ratings often look U-shaped. Customers with an average experience rarely write a review, while those with an extreme experience feel motivated to.
On a controversial law, opinion in a polarised climate can split into two camps.
Cloud cover is measured in oktas, from 0 (clear sky) to 8 (fully overcast). In many places the sky tends to be either clear or overcast.
A U shape on the interval from 0 to 1 is most commonly modelled with the Beta distribution, \(\text{Beta}(\alpha, \beta)\), whose density is
\[ f(x) = \frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)}, \qquad 0 < x < 1. \]
When both shape parameters are below one (\(\alpha < 1\) and \(\beta < 1\)), the exponents \(\alpha - 1\) and \(\beta - 1\) are negative. The density then shoots up towards infinity at \(x = 0\) and \(x = 1\) and dips in the middle. The case \(\text{Beta}(0.5, 0.5)\) is also called the arcsine distribution.
| Shape parameters | Shape of the density |
|---|---|
| \(\alpha < 1\), \(\beta < 1\) | U-shaped |
| \(\alpha = \beta = 1\) | Flat (uniform) |
| \(\alpha > 1\), \(\beta > 1\) | Bell-shaped (one peak inside) |
| \(\alpha < 1\), \(\beta \ge 1\) (or the reverse) | J-shaped (one peak at one end) |
First we draw the density of \(\text{Beta}(0.5, 0.5)\) with dbeta(). The density is infinite at exactly 0 and 1, so we plot just inside the interval and cut the vertical axis.
library(tidyverse)
x <- seq(0.003, 0.997, length.out = 1000)
u <- tibble(x = x, density = dbeta(x, 0.5, 0.5))
ggplot(u, aes(x, density)) +
geom_line(colour = "royalblue", linewidth = 1.2) +
coord_cartesian(ylim = c(0, 4)) +
labs(title = "U-shaped Beta(0.5, 0.5) density", x = "Value", y = "Density")
Changing the parameters moves us between the shapes in the table above.
shapes <- tibble(a = c(0.5, 1, 4, 0.5), b = c(0.5, 1, 4, 3)) %>%
mutate(label = factor(sprintf("Beta(%g, %g)", a, b), levels = sprintf("Beta(%g, %g)", a, b))) %>%
crossing(x = seq(0.01, 0.99, length.out = 400)) %>%
mutate(density = dbeta(x, a, b))
ggplot(shapes, aes(x, density)) +
geom_line(colour = "royalblue", linewidth = 1) +
facet_wrap(~ label) +
coord_cartesian(ylim = c(0, 4)) +
labs(title = "From U-shaped to J-shaped to bell-shaped", x = "Value", y = "Density")
We draw random values with rbeta() and compare the histogram with the true density.
set.seed(1)
sim <- tibble(value = rbeta(10000, 0.5, 0.5))
ggplot(sim, aes(value)) +
geom_histogram(aes(y = after_stat(density)), bins = 40,
fill = "lightblue", colour = "white") +
geom_line(data = u, aes(x, density), colour = "royalblue", linewidth = 1) +
coord_cartesian(ylim = c(0, 4)) +
labs(title = "Simulated U-shaped data", x = "Value", y = "Density")
For this data the mean is about 0.5, the centre of the scale. Let’s see how many observations are actually near it.
mean(sim$value)[1] 0.5060249# share of values in the middle fifth, 0.4 to 0.6
mean(sim$value > 0.4 & sim$value < 0.6)[1] 0.1278# share of values in the outer fifths (below 0.1 or above 0.9)
mean(sim$value < 0.1 | sim$value > 0.9)[1] 0.4135A flat (uniform) distribution would put 20% of values in the middle fifth, and a bell curve far more. Here only about 13% are there, while about 41% lie in the outer tenths at the two ends. The average describes a value that few observations are close to.
When a histogram looks U-shaped, report the two groups separately, or the share in each tail, instead of a single mean and standard deviation.