Densities

The Hills are Shadows, said Tennyson

Quant Variables

Qual Variables

Density Plots

Ridge Plots

Author

Arvind V.

Published

June 22, 2024

Modified

June 19, 2025

Abstract

Quant and Qual Variable Graphs and their Siblings

WebR Status

Installing package 7 out of 8: ggridges

Slides and Tutorials

R (Static Viz)

Radiant Tutorial

Datasets

“Never let the future disturb you. You will meet it, if you have to, with the same weapons of reason which today arm you against the present.”

— Marcus Aurelius

Setting up R Packages

library(tidyverse)
library(mosaic)
library(ggformula)

# install.packages("remotes")
# library(remotes)
# remotes::install_github("wilkelab/ggridges")
library(ggridges)
library(skimr)

library(palmerpenguins) # Our new favourite dataset

Plot Theme

Show the Code

# https://stackoverflow.com/questions/74491138/ggplot-custom-fonts-not-working-in-quarto

# Chunk options
knitr::opts_chunk$set(
  fig.width = 7,
  fig.asp = 0.618, # Golden Ratio
  # out.width = "80%",
  fig.align = "center"
)
### Ggplot Theme
### https://rpubs.com/mclaire19/ggplot2-custom-themes

theme_custom <- function() {
  font <- "Roboto Condensed" # assign font family up front

  theme_classic(base_size = 14) %+replace% # replace elements we want to change

    theme(
      panel.grid.minor = element_blank(), # strip minor gridlines
      text = element_text(family = font),
      # text elements
      plot.title = element_text( # title
        family = font, # set font family
        # size = 20,               #set font size
        face = "bold", # bold typeface
        hjust = 0, # left align
        # vjust = 2                #raise slightly
        margin = margin(0, 0, 10, 0)
      ), plot.title.position = "plot",
      plot.subtitle = element_text( # subtitle
        family = font, # font family
        # size = 14,               #font size
        hjust = 0,
        margin = margin(2, 0, 5, 0)
      ),
      plot.caption = element_text( # caption
        family = font, # font family
        size = 8, # font size
        hjust = 1
      ), # right align

      axis.title = element_text( # axis titles
        family = font, # font family
        size = 10 # font size
      ),
      axis.text = element_text( # axis text
        family = font, # axis family
        size = 8
      ) # font size
    )
}

# Set graph theme
theme_set(new = theme_custom())
#

What graphs will we see today?

Variable #1	Variable #2	Chart Names	Chart Shape
Quant	None	Density plot, Ridge Density Plot

What kind of Data Variables will we choose?

No	Pronoun	Answer	Variable/Scale	Example	What Operations?
1	How Many / Much / Heavy? Few? Seldom? Often? When?	Quantities, with Scale and a Zero Value.Differences and Ratios /Products are meaningful.	Quantitative/Ratio	Length,Height,Temperature in Kelvin,Activity,Dose Amount,Reaction Rate,Flow Rate,Concentration,Pulse,Survival Rate	Correlation

Inspiration

April is the cruelest month, said T.S Eliot. But December in Nebraska must be tough.

What is a “Density Plot”?

As we saw earlier, Histograms are best to show the distribution of raw Quantitative data, by displaying the number of values that fall within defined ranges, often called buckets or bins.

Sometimes it is useful to consider a chart where the bucket width shrinks to zero!

You might imagine a density chart as a histogram where the buckets are infinitesimally small, i.e. zero width. Think of the frequency density as a differentiation (as in calculus) of the histogram. By taking the smallest of steps $\sim 0$ , we get a measure of the slope of distribution. This may seem counter-intuitive, but densities have their uses in spotting the ranges in the data where there are more frequent values. In this, they serve a similar purpose as do histograms, but may offer insights not readily apparent with histograms, especially with default bucket widths. The chunkiness that we see in the histograms is removed and gives us a smooth curve showing in which range the data are more frequent.

Case Study-1: `penguins` dataset

We will first look at at a dataset that is directly available in R, the penguins dataset. Data were collected and made available by Dr. Kristen Gorman and the Palmer Station, Antarctica LTER, a member of the Long Term Ecological Research Network.

Examine the Data

As per our Workflow, we will look at the data using all the three methods we have seen.

R
web-r

glimpse(penguins)

Rows: 344
Columns: 8
$ species           <fct> Adelie, Adelie, Adelie, Adelie, Adelie, Adelie, Adel…
$ island            <fct> Torgersen, Torgersen, Torgersen, Torgersen, Torgerse…
$ bill_length_mm    <dbl> 39.1, 39.5, 40.3, NA, 36.7, 39.3, 38.9, 39.2, 34.1, …
$ bill_depth_mm     <dbl> 18.7, 17.4, 18.0, NA, 19.3, 20.6, 17.8, 19.6, 18.1, …
$ flipper_length_mm <int> 181, 186, 195, NA, 193, 190, 181, 195, 193, 190, 186…
$ body_mass_g       <int> 3750, 3800, 3250, NA, 3450, 3650, 3625, 4675, 3475, …
$ sex               <fct> male, female, female, NA, female, male, female, male…
$ year              <int> 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007…

skim(penguins)

Data summary
Name	penguins
Number of rows	344
Number of columns	8
_______________________
Column type frequency:
factor	3
numeric	5
________________________
Group variables	None

Variable type: factor

skim_variable	n_missing	complete_rate	ordered	n_unique	top_counts
species	0	1.00	FALSE	3	Ade: 152, Gen: 124, Chi: 68
island	0	1.00	FALSE	3	Bis: 168, Dre: 124, Tor: 52
sex	11	0.97	FALSE	2	mal: 168, fem: 165

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
bill_length_mm	2	0.99	43.92	5.46	32.1	39.23	44.45	48.5	59.6	▃▇▇▆▁
bill_depth_mm	2	0.99	17.15	1.97	13.1	15.60	17.30	18.7	21.5	▅▅▇▇▂
flipper_length_mm	2	0.99	200.92	14.06	172.0	190.00	197.00	213.0	231.0	▂▇▃▅▂
body_mass_g	2	0.99	4201.75	801.95	2700.0	3550.00	4050.00	4750.0	6300.0	▃▇▆▃▂
year	0	1.00	2008.03	0.82	2007.0	2007.00	2008.00	2009.0	2009.0	▇▁▇▁▇

Data Dictionary

Qualitative Data

sex: male and female penguins
island: they have islands to themselves!!
species: Three adorable types!

Quantitative Data

bill_length_mm: The length of the penguins’ bills
bill_depth_mm: See the picture!!
flipper_length_mm: Flippers! Penguins have “hands”!!
body_mass_gm: Grams? Grams??? Why, these penguins are like human babies!!❤️

Business Insights on Examining the penguins dataset

This is a smallish dataset (344 rows, 8 columns).
There are a few missing values in sex(11 missing entries) and all the Quant variables (2 missing entries each).

Plotting Densities

## Set graph theme
theme_set(new = theme_custom())
##
penguins <- penguins %>% drop_na()

gf_density(~body_mass_g, data = penguins) %>%
  gf_labs(title = "Plot A: Penguin Masses", caption = "ggformula")
###
penguins %>%
  gf_density(~body_mass_g, fill = ~species, color = "black") %>%
  gf_refine(scale_color_viridis_d(option = "magma", aesthetics = c("colour", "fill"))) %>%
  gf_labs(title = "Plot B: Penguin Body Mass by Species", caption = "ggformula")
###
penguins %>%
  gf_density(
    ~body_mass_g,
    fill = ~species,
    color = "black",
    alpha = 0.3
  ) %>%
  gf_facet_wrap(vars(sex)) %>%
  gf_labs(title = "Plot C: Penguin Body Mass by Species and facetted by Sex", caption = "ggformula")
###
penguins %>%
  gf_density(~body_mass_g, fill = ~species, color = "black") %>%
  gf_facet_wrap(vars(sex), scales = "free_y", nrow = 2) %>%
  gf_labs(
    title = "Plot D: Penguin Body Mass by Species and facetted by Sex",
    subtitle = "Free y-scale",
    caption = "ggformula"
  ) %>%
  gf_theme(theme(axis.text.x = element_text(angle = 45, hjust = 1)))

## Set graph theme
theme_set(new = theme_custom())
## Remove the rows containing NA (11 rows!)
penguins <- penguins %>% drop_na()

ggplot(data = penguins) +
  geom_density(aes(x = body_mass_g)) +
  labs(title = "Plot A: Penguin Masses", caption = "ggplot")
###
penguins %>%
  ggplot() +
  geom_density(aes(x = body_mass_g, fill = species),
    color = "black"
  ) +
  scale_color_viridis_d(
    option = "magma",
    aesthetics = c("colour", "fill")
  ) +
  labs(
    title = "Plot B: Penguin Body Mass by Species",
    caption = "ggplot"
  )
###
penguins %>% ggplot() +
  geom_density(aes(x = body_mass_g, fill = species),
    color = "black",
    alpha = 0.3
  ) +
  facet_wrap(vars(sex)) +
  labs(title = "Plot C: Penguin Body Mass by Species and facetted by Sex", caption = "ggplot")
###
penguins %>%
  ggplot() +
  geom_density(aes(x = body_mass_g, fill = species),
    color = "black"
  ) +
  facet_wrap(vars(sex), scales = "free_y", nrow = 2) +
  labs(
    title = "Plot D: Penguin Body Mass by Species and facetted by Sex",
    subtitle = "Free y-scale", caption = "ggplot"
  ) %>%
  theme(theme(axis.text.x = element_text(angle = 45, hjust = 1)))

Business Insights from penguin Densities

Pretty much similar conclusions as with histograms. Although densities may not be used much in business contexts, they are better than histograms when comparing multiple distributions! So you should use thems!

Ridge Plots

Sometimes we may wish to show the distribution/density of a Quant variable, against several levels of a Qual variable. For instance, the prices of different items of furniture, based on the furniture “style” variable. Or the sales of a particular line of products, across different shops or cities. We did this with both histograms and densities, by colouring based on a Qual variable, and by facetting using a Qual variable. There is a third way, using what is called a ridge plot. ggformula support this plot by importing/depending upon the ggridges package; however, ggplot itself appears to not have this capability.

Using ggformula
web-r

## Set graph theme
theme_set(new = theme_custom())
##


gf_density_ridges(drv ~ hwy,
  fill = ~drv,
  alpha = 0.3,
  rel_min_height = 0.005, data = mpg
) %>%
  gf_refine(
    scale_y_discrete(expand = c(0.01, 0)),
    scale_x_continuous(expand = c(0.01, 0))
  ) %>%
  gf_labs(title = "Ridge Plot")

Business Insights from mpg Ridge Plots

This is another way of visualizing multiple distributions, of a Quant variable at different levels of a Qual variable. We see that the distribution of hwy mileage varies substantially with drv type.

Case Study-2:

To be created !!

Wait, But Why?

Densities are sometimes easier to compare side by side. That is what Claus Wilke says, at least. Perhaps because they look less “busy” than histograms.
Ridge Density Plots are very cool when it comes to comparing the density of a Quant variable as it varies against the levels of a Qual variable, without having to facet or group.
It is possible to plot 2D-densities too, for two Quant variables, which give very evocative contour-like plots. Try to do this with the faithful dataset in R.

Conclusion

Histograms and Frequency Distributions are both used for Quantitative data variables
Whereas Histograms “dwell upon” counts, ranges, means and standard deviations
Frequency Density plots “dwell upon” probabilities and densities
Ridge Plots are density plots used for describing one Quant and one Qual variable (by inherent splitting)
We can split all these plots on the basis of another Qualitative variable.(Ridge Plots are already split)
Long tailed distributions need care in visualization and in inference making!

Your Turn

Star Trek Books

Which would be the Group By variables here? And what would you summarize? With which function?

Math Anxiety! Hah! Peasants.

References

Winston Chang (2024). R Graphics Cookbook. https://r-graphics.org
See the scrolly animation for a histogram at this website: Exploring Histograms, an essay by Aran Lunzer and Amelia McNamara https://tinlizzie.org/histograms/?s=09
Minimal R using mosaic.https://cran.r-project.org/web/packages/mosaic/vignettes/MinimalRgg.pdf
Sebastian Sauer, Plotting multiple plots using purrr::map and ggplot

R Package Citations

Package	Version	Citation
ggridges	0.5.6	Wilke (2024)
NHANES	2.1.0	Pruim (2015)
TeachHist	0.2.1	Lange (2023)
TeachingDemos	2.13	Snow (2024)
visualize	4.5.0	Balamuta (2023)

Balamuta, James. 2023. visualize: Graph Probability Distributions with User Supplied Parameters and Statistics. https://doi.org/10.32614/CRAN.package.visualize.

Lange, Carsten. 2023. TeachHist: A Collection of Amended Histograms Designed for Teaching Statistics. https://doi.org/10.32614/CRAN.package.TeachHist.

Pruim, Randall. 2015. NHANES: Data from the US National Health and Nutrition Examination Study. https://doi.org/10.32614/CRAN.package.NHANES.

Snow, Greg. 2024. TeachingDemos: Demonstrations for Teaching and Learning. https://doi.org/10.32614/CRAN.package.TeachingDemos.

Wilke, Claus O. 2024. ggridges: Ridgeline Plots in “ggplot2”. https://doi.org/10.32614/CRAN.package.ggridges.

Citation

BibTeX citation:

@online{v.2024,
  author = {V., Arvind},
  title = {\textless Iconify-Icon Icon=“clarity:bell-Curve-Line”
    Width=“1.2em”
    Height=“1.2em”\textgreater\textless/Iconify-Icon\textgreater{}
    {Densities}},
  date = {2024-06-22},
  url = {https://av-quarto.netlify.app/content/courses/Analytics/Descriptive/Modules/26-Densities/},
  langid = {en},
  abstract = {Quant and Qual Variable Graphs and their Siblings}
}

For attribution, please cite this work as:

V., Arvind. 2024. “<Iconify-Icon Icon=‘clarity:bell-Curve-Line’ Width=‘1.2em’ Height=‘1.2em’></Iconify-Icon> Densities.” June 22, 2024. https://av-quarto.netlify.app/content/courses/Analytics/Descriptive/Modules/26-Densities/.

Slides and Tutorials

Setting up R Packages

Plot Theme

What graphs will we see today?

What kind of Data Variables will we choose?

Inspiration

What is a “Density Plot”?

Case Study-1: penguins dataset

Examine the Data

Data Dictionary

Plotting Densities

Ridge Plots

Case Study-2:

Wait, But Why?

Conclusion

Your Turn

R Commands Used Here

References

R Package Citations

Citation

Case Study-1: `penguins` dataset