Foundations of
Data Science

Spring 2026 | Data 2 (399)
Jeffrey M. Girard | Lecture 05c

Roadmap

• Visualize the distribution of a single variable

• Visualize the relationship between two variables

Setup

library(tidyverse)
data("penguins", package = "datasets")

glimpse(penguins)

Rows: 344
Columns: 8
$ species     <fct> Adelie, Adelie, Adelie, Adelie, Adelie, Adelie, Adelie, Ad…
$ island      <fct> Torgersen, Torgersen, Torgersen, Torgersen, Torgersen, Tor…
$ bill_len    <dbl> 39.1, 39.5, 40.3, NA, 36.7, 39.3, 38.9, 39.2, 34.1, 42.0, …
$ bill_dep    <dbl> 18.7, 17.4, 18.0, NA, 19.3, 20.6, 17.8, 19.6, 18.1, 20.2, …
$ flipper_len <int> 181, 186, 195, NA, 193, 190, 181, 195, 193, 190, 186, 180,…
$ body_mass   <int> 3750, 3800, 3250, NA, 3450, 3650, 3625, 4675, 3475, 4250, …
$ sex         <fct> male, female, female, NA, female, male, female, male, NA, …
$ year        <int> 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007…

Distributions

Visualizing Distributions

• Variable distributions are critical in data analysis
  • What are the most and least common values?
  • What are the extrema (min and max values)?
  • Are there any outliers or impossible values?
  • How much spread is there in the variable?
  • What shape does the distribution take?

• Visualization helps us to understand variation
  • It can also communicate it to others
  • The geom chosen depends on whether the variable is discrete or continuous

Common Strategies

• One Discrete \(\rightarrow\) Bar
  • Count each category’s occurrence
  • x = Discrete Categories, y = Count
• One Continuous \(\rightarrow\) Histogram
  • Cut the variable’s range into discrete “bins”
  • Count each bin’s occurrence
  • x = Variable Values, y = Count

Bar Chart

Species Bar Chart

penguins |>                   # start with penguins, pipe to pass tibble
  ggplot(aes(x = species)) +  # map x-axis to species, plus to add to plot
  geom_bar()                  # bar to plot category counts

Species Interpretation

• Three species were observed
• Adelie was most common, then Gentoo, then Chinstrap
• Adelies were observed twice as often as Chinstraps
• Are unequal frequencies expected? Problematic?

Sex Bar Chart

penguins |>               # start with penguins, pipe to pass tibble
  ggplot(aes(x = sex)) +  # map x-axis to sex, plus to add to plot
  geom_bar()              # bar to plot category counts

Sex Interpretation

• Males were slightly more common than Females
• Around 10 observations are missing sex info (NA)
• Are these NAs: expected? problematic? fixable?

Bar Missing Values

penguins |>               # start with penguins, pipe to pass tibble
  drop_na(sex) |>         # drop NAs in sex, pipe to pass tibble
  ggplot(aes(x = sex)) +  # map x-axis to sex, plus to add to plot
  geom_bar()              # bar to plot category counts

Histogram

Mass Histogram

penguins |>                     # start with penguins, pipe to pass tibble
  ggplot(aes(x = body_mass)) +  # map x-axis to body_mass, plus to add to plot
  geom_histogram()              # histogram to plot bin counts

Mass Interpretation

• Body mass ranged from around 2500 to around 6500
• Most values were between 3250 and 4000
• Values from 4000 to 5750 were also pretty common
• At a glance, no values seem impossible or erroneous

Histogram Bins

# break x range into 20 bins
penguins |>
  ggplot(aes(x = body_mass)) +
  geom_histogram(bins = 20)

# break x range into 4 bins
penguins |>
  ggplot(aes(x = body_mass)) +
  geom_histogram(bins = 4)

Histogram Binwidth

# each bin is 20 x-units wide
penguins |>
  ggplot(aes(x = body_mass)) +
  geom_histogram(binwidth = 20)

# each bin is 2000 x-units wide
penguins |>
  ggplot(aes(x = body_mass)) +
  geom_histogram(binwidth = 2000)

Relationships

Visualizing Relationships

• We are often interested in variable relationships
  • Does variation in X go with variation in Y?
  • Do higher X scores go with higher Y scores?
  • Do groups differ on Y score distributions?
  • Do certain groups tend to go together?

• We can extend variation geoms into covariation
  • The geoms we use will heavily depend on…
  • …are the variables continuous or discrete?

Common Strategies

• Two Continuous \(\rightarrow\) Point
  • Use two positional axes, possibly a smoother
  • x = Variable 1 Values, y = Variable 2 Values
• One Discrete, One Continuous \(\rightarrow\) Boxplot
  • Use one positional axis and boxes-and-whiskers
  • x = Discrete Categories, y = Continuous Values
• Two Discrete \(\rightarrow\) Bar (with fill)
  • Use stacked-and-colored bars to show proportions
  • x = Variable 1 Categories, y = Counts, fill = Variable 2 Categories

Scatterplot

Flipper–Mass Scatterplot

penguins |> ggplot(aes(x = flipper_len, y = body_mass)) + geom_point()

Flipper–Mass Interpretation

• As flipper length increases, body mass also increases
• This relationship looks linear (as in a straight line)
• However, there is some spread around that line
• There seem to be more penguins with low body mass and low flipper length

Adding a Smooth

penguins |> 
  ggplot(aes(x = flipper_len, y = body_mass)) + 
  geom_point() +
  geom_smooth()

Boxplot

How to read a boxplot

Species–Mass Boxplot

penguins |> ggplot(aes(x = species, y = body_mass)) + geom_boxplot()

Species–Mass Interpretation

• Adelies and Chinstraps have similar mass distributions
• There are outlier Chinstraps
• Gentoos tend to be larger than Adelies & Chinstraps
• There is overlap: large Adelies & Chinstraps are larger than small Gentoos

Rotated Boxplot

penguins |> ggplot(aes(x = body_mass, y = species)) + geom_boxplot()

Stack Bar Chart

Island–Species Bar Chart

penguins |> ggplot(aes(x = island, fill = species)) + geom_bar()

Island–Species Interpretation

• Adelies are seen on all three Islands
• Gentoos are only seen on Biscoe Island
• Chinstraps are only seen on Dream Island
• Only Adelies are seen on Torgersen Island

Complex Relationships

Three or more variables

ggplot(penguins, aes(x = flipper_len, y = body_mass, 
  color = species, shape = sex)) + geom_point(alpha = 0.7)

Mixed Grouping

# Group by Chick (one line per Chick) but color by Diet
ggplot(ChickWeight, aes(x = Time, y = weight, group = Chick, color = Diet)) + 
  geom_line()