Lecture 7

Data Visualization I: Distributions and Group Comparisons

Byeong-Hak Choe

bchoe@geneseo.edu

SUNY Geneseo

October 21, 2026

📊 One numeric distribution

🧭 A useful chart answers a question

• Data visualization represents data through positions, lengths, colors, and other visual features.
• Why use it? Charts make patterns easier to see than a long table of values and help us explain findings to others.
• A distribution describes how values occur across a variable’s range: which values are common, uncommon, or absent.
• Variation means how much the observed values differ from one another. Values may cluster closely or spread widely.
• Our first question: How does city fuel economy vary across vehicles in the dataset?

🚗 Each row in mpg describes a vehicle

manufacturer	model	year	cty	drv	class
audi	a4	1999	18	f	compact
audi	a4	1999	21	f	compact
audi	a4	2008	20	f	compact
audi	a4	2008	21	f	compact

• mpg comes with ggplot2. It contains data on 234 vehicles from 1999 and 2008.
• Each row describes a particular version of a vehicle model, not an individual car sold.

Source: https://ggplot2.tidyverse.org/reference/mpg.html, also available through ?mpg. The vehicle versions can differ in engine, transmission, drive system, or model year. The dataset supports learning visualization, rather than describing today’s vehicle market.

📏 Variable names need meanings and units

Column	Meaning	Type for this comparison
cty	City fuel economy in miles per gallon	Numeric
drv	Drive system, coded 4, f, or r	Categorical
class	Vehicle class, such as compact or SUV	Categorical

• A higher miles-per-gallon (mpg) value means the vehicle travels farther per gallon of fuel.
• A numeric value has a meaningful magnitude. A category identifies a group, even if its label looks like a number.

🧰 ggplot2 is part of the tidyverse

library(tidyverse)
mpg <- mpg    # ggplot2::mpg

?mpg  # description of the `mpg` data.frame

• library(tidyverse) loads ggplot2 along with packages such as dplyr.
• mpg is available after loading ggplot2. This example needs no CSV import.
• The same plotting structure also works with a data frame we import ourselves.

📊 A histogram shows a distribution of a numeric variable

ggplot(data = mpg,
       mapping = aes(x = cty)) +
  geom_histogram()

• A histogram groups numeric values into equal-width intervals called bins, arranged along the horizontal axis.
• The height of each bar represents the number of observations in each bin. Here, the vertical axis counts vehicles.

🧩 Data, aesthetic mappings, and geometric objects

Essential component	Its job in our histogram
Data frame: data = mpg	Supplies the observations in rows and variables in columns
Aesthetic mappings: mapping = aes(x = cty)	Connect variables to visual features, such as position or color
Geometric object: geom_histogram()	Chooses how to draw the data. This geom groups values into bins and draws bars

• Geom functions (geom_*()) draw objects such as bars, points, or lines. We choose the geom to match our question.
• aes() is short for aesthetics. Here, x = cty maps city MPG to horizontal position. The geom_histogram() calculates the vertical counts automatically.
• data = and mapping = are argument names. R lets us omit these names when we supply the arguments in that order.

Both ggplot(data = mpg, mapping = aes(x = cty)) and ggplot(mpg, aes(x = cty)) are valid. Keep the names visible while students learn the roles of the two arguments. Omitting names does not mean omitting the data or mapping themselves.

➕ + connects the plot’s parts

ggplot(data = mpg,
       mapping = aes(x = cty)) +
  geom_histogram()

• Put + at the end of a line when the plotting expression continues on the next line.
• Each + adds a plot component. We can add more layers or settings with another +, such as labs() for labels.

At the R console

A + at the R console prompt means R is waiting for the rest of an unfinished command. Do not type that prompt into your code.

The console continuation prompt is not part of the code students should type. Common causes include an unfinished parenthesis, quote, or plotting expression. Histogram documentation: https://ggplot2.tidyverse.org/reference/geom_histogram.html

🖌️ A fixed appearance belongs outside aes()

ggplot(data = mpg,
       mapping = aes(x = cty)) +
  geom_histogram(
    fill = "steelblue",
    color = "black"
  )

• fill controls the inside of the bars. color controls their outlines.
• Both settings stay outside aes() because they give all bars one appearance. aes(x = cty) maps a data column.

🏷️ labs() provides readable labels

ggplot(data = mpg,
       mapping = aes(x = cty)) +
  geom_histogram(fill = "steelblue",
                 color = "white") +
  labs(x = "City fuel economy (MPG)",
       y = "Number of vehicles")

• labs() changes displayed text. The column name remains cty.
• Recognize what it does. It is optional in classwork and homework answers, while project charts need clear labels and units.

📏 Bin width changes the visible detail

Bin width: 1 MPG

geom_histogram(binwidth = 1)

Bin width: 5 MPG

geom_histogram(binwidth = 5)

• binwidth sets each interval’s width. These bins are 1 MPG or 5 MPG wide.
• Distribution shape is sensitive to bin width. Narrow bins reveal detail. Wide bins can hide peaks or gaps. The 234 rows stay the same.

📊 Uniform, unimodal, and bimodal

Uniform

Roughly similar bar heights across bins.

Unimodal

One clear peak (mode).

Bimodal

Two clear peaks separated by a dip.

Each panel uses 100 constructed values. All three panels use the same bin width and axes. These labels describe the illustrated patterns, rather than formally testing a population distribution. Two peaks may suggest a mixture of groups, but the peaks alone do not establish that explanation.

🧭 Skewness, mean, median, and mode

• Like a histogram, a smooth curve shows distribution shape. Skewness is asymmetry: one tail extends farther than the other.
• Left (negative) skew has a longer tail toward smaller values. Right (positive) skew has a longer tail toward larger values. Follow the longer tail, not the side with most values.

The smooth curves are schematic distribution examples, not histograms. All three examples have one peak. Peak count and skew describe different features, so a unimodal distribution can be symmetric or skewed.

For an assessment, ask: “For each distribution shown, put the mean, median, and mode in order from smallest to largest.” The displayed ordering holds for familiar examples such as a lognormal distribution and its reflection; a normal distribution has coincident mean, median, and mode. Skewness, even with a single peak, does not guarantee the ordering for every possible distribution. Keep that qualification in the teaching notes rather than making exceptions part of the freshman exercise. Symmetry with a unique central mode and an existing mean gives equality.

Sources: https://jse.amstat.org/v13n2/vonhippel.html; https://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htm; https://www.itl.nist.gov/div898/handbook/eda/section3/eda351.htm

Course illustration: ../lec_figs/skewness-danl-examples-labeled.png, edited from ../lec_figs/skewness-danl-examples.png to add the labels Negatively skewed and Positively skewed while preserving the diagram. The earlier figure is retained.

🔎 Reading the city-MPG distribution

• Typical values: most vehicles cluster roughly between 12 and 22 MPG in city driving.
• Spread and shape: values range from 9 to 35 MPG, with a longer tail toward higher MPG. This is right skew.
• Unusual values: high-MPG vehicles deserve a closer look at their rows and context. A histogram alone cannot tell us why they differ.

The typical-value band is an approximate reading of the main cluster of histogram bars, not an exact median. The endpoints 9 and 35 MPG come from checking the data rather than treating the edges of bins as exact observed values. The mpg data contain selected vehicles, rather than a random sample of all cars on the road.

✍️ Classwork 10: Distributions

Classwork 10: Histograms and Distribution Shapes

• Use histograms to examine city fuel economy and distribution shapes.

🚙 Distributions across groups

🚙 drv identifies the drive system

Value of drv	Meaning
4	Four-wheel drive
f	Front-wheel drive
r	Rear-wheel drive

• Our next question: How does city fuel economy differ across drive systems?
• The numeric variable remains cty. The categorical variable drv tells us which group each row belongs to.

🪟 Facets separate the distributions

ggplot(data = mpg,
       mapping = aes(x = cty)) +
  geom_histogram(binwidth = 2) +
  facet_wrap(~ drv)

• facet_wrap(~ drv) means “make panels by drive system.” Each row appears in its group’s panel.

📐 Shared bins and axes support comparisons

• Every panel uses 2-MPG bins and the same axes. The same position represents the same value across panels.
• Front-wheel-drive vehicles tend toward higher city MPG in these observations.
• The groups contain different numbers of vehicles. Taller count bars can reflect a larger group, as well as how values concentrate within it.

Try a different x-axis range

In the previous plot, replace facet_wrap(~ drv) with:

facet_wrap(~ drv, scales = "free_x")

Each panel now has its own x-axis range, while the y-axis stays shared. Does this make the groups easier to inspect or harder to compare?

Default facet scales are fixed. Keep the bin width consistent across panels. Counts are 103 four-wheel-drive, 106 front-wheel-drive, and 25 rear-wheel-drive observations. Comparing shares or densities could address unequal group sizes, but density scaling is outside the required programming scope here.

👁️ Design with colorblindness in mind

• Color-vision deficiency can make some colors difficult to distinguish.
• Among people with Northern European ancestry, inherited red–green deficiency affects about 8% of males and 0.5% of females.

The image illustrates three simulated forms of color vision deficiency. It is not an exhaustive classification or a diagnostic test, and students do not need to memorize the medical labels. Most people with color vision deficiency do not see only black and white.

The prevalence figures apply to inherited red–green color vision deficiency among populations with Northern European ancestry. They are not worldwide estimates for every form of color vision deficiency.

The following comparisons isolate three design choices using the same illustrative weekly sales data. Color, direct labels, and visual cues can identify the same groups. These are design previews, not additional programming requirements here.

Sources: https://medlineplus.gov/genetics/condition/color-vision-deficiency/; https://www.nei.nih.gov/eye-health-information/eye-conditions-and-diseases/color-blindness/types-color-vision-deficiency; https://www.w3.org/WAI/WCAG22/Understanding/use-of-color.html

Illustration: course asset ../lec_figs/types-of-color-blindness.png, supplied in the earlier lecture material.

🎨 Tip 1: Colorblind-friendly colors

Weekly sales. Red–green simulation below.

Without this tip

Original: Store A Store B

With this tip

Original: Store A Store B

Red and green look similar in this simulation.

These are constructed classroom data, not observations from actual stores: Store A has weekly sales of 12, 16, 20, and 18 thousand dollars; Store B has 15, 13, 18, and 23 thousand dollars. The same eight observations and axes appear in both charts. Only the selected colors change.

The keys labeled Original show red/green on the left and blue/orange on the right before any simulation. The bars and chart legends below show their simulated appearance. Both charts approximate severe deuteranopia using the Machado, Oliveira, and Fernandes (2009) severity-100 matrix, applied in linear sRGB. This illustrates one color-vision condition, not everyone’s perception, and is not a diagnostic test. Read the caption before comparing the charts. Labels are still useful even with more distinguishable colors.

Sources: https://doi.org/10.1109/TVCG.2009.113; https://github.com/njsmith/colorspacious/blob/master/colorspacious/cvd.py; https://www.w3.org/WAI/WCAG22/Understanding/use-of-color.html

🏷️ Tip 2: Text labels or separate panels

Weekly sales, shown without color. Can you identify each store?

Without this tip

With this tip

Text labels identify each line directly. Separate panels (e.g., facet_wrap()) identify groups with panel labels.

Use the same constructed sales data as in the previous comparison. Both panels deliberately remove hue differences. The improvement here is direct text labels, not new colors, new values, or different axes. Ask which store has the highest sales in week 4. Separate panels, such as the earlier facet_wrap() examples, could also show each store with its own labeled panel.

Source: https://www.w3.org/WAI/WCAG22/Understanding/use-of-color.html

🔷 Tip 3: Point shapes and line types

Weekly sales, shown without color. Which line belongs to each store?

Without this tip

With this tip

Point shapes and line types distinguish groups even when colors look alike.

Use the same constructed data and the same gray color in both panels. Store A uses circles and a solid line on the right; Store B uses triangles and a dashed line. The legend names the stores in both charts, so only the distinguishing marks change. For scatterplots, use different point shapes; for line charts, use different line types. A line chart with points can use both. Students are seeing the design principle, not learning additional R syntax on this slide.

Source: https://www.w3.org/WAI/WCAG22/Understanding/use-of-color.html

🧰 ggthemes: color scales

• The scale_*_colorblind() and scale_*_tableau() functions are provided by the ggthemes package.

install.packages("ggthemes")  # Install once
library(ggthemes)            # Load in each R session

• Tableau is business intelligence (BI) software for exploring data and building interactive dashboards.
  • scale_fill_tableau() and scale_color_tableau() are inspired by Tableau’s color palette.

Mapped feature	Tableau functions	Colorblind-friendly functions
fill: interiors	scale_fill_tableau()	scale_fill_colorblind()
color: points or lines	scale_color_tableau()	scale_color_colorblind()

• These scales apply a colorblind-friendly palette (a set of colors) to the mapped categories.

The scale_colorblind() and scale_tableau() functions come from the ggthemes package, not from ggplot2 or the tidyverse; the * stands for fill or color, as in the table. The tidyverse does not automatically load ggthemes. Install the package once on the computer, then load it in each new session where these scales are used. ggthemes supplies additional color scales in R; students do not need to install or open Tableau software.

Match the scale name to the mapped aesthetic: fill functions change interiors, while color functions change points or lines. Replace scale_fill_tableau() with scale_fill_colorblind() to use colorblind-friendly fill colors; replace scale_color_tableau() with scale_color_colorblind() for points or lines. Use these functions without arguments in this introduction. Changing the scale changes colors, not the data. Keep labels, facets, or other visual cues so color is not the only way to identify groups.

Sources: https://www.tableau.com/dashboard/business-intelligence-dashboard; https://help.tableau.com/current/tableau/en-us/tableau_product_overview.htm; https://jrnold.github.io/ggthemes/reference/scale_color_tableau.html; https://jrnold.github.io/ggthemes/reference/colorblind.html; https://www.w3.org/WAI/WCAG22/Understanding/use-of-color.html

🎨 Mapping fill and choosing a palette

ggplot(data = mpg,
       mapping = aes(x = cty, fill = drv)) +
  geom_histogram(binwidth = 2, color = "white",
                 show.legend = FALSE) +  # Hide the legend (color key)
  facet_wrap(~ drv) +
  scale_fill_colorblind()   # Use a colorblind-friendly color palette for fills

• fill = drv maps groups to colors. scale_fill_colorblind() chooses the colorblind-friendly color palette.
• Discuss: Do we need different fill colors when the facet labels already identify each drive system?

This repeats the familiar faceted histograms before introducing a new geom. Each drive-system category controls the interiors of its bars. The scale changes colors, not the observations, bins, or counts. show.legend = FALSE hides a redundant legend because the facet labels already identify the groups. The same palette is used in the following boxplots and bar charts. Its nonblack fills keep the boxplots’ black median lines visible.

Sources: https://jrnold.github.io/ggthemes/reference/tableau_color_pal.html; https://jrnold.github.io/ggthemes/reference/scale_color_tableau.html; https://ggplot2.tidyverse.org/reference/geom_histogram.html

📦 Boxplots compare a numeric variable across categories

ggplot(data = mpg,
       mapping = aes(x = drv,
                     y = cty,
                     fill = drv)) +
  geom_boxplot(show.legend = FALSE) +
  scale_fill_tableau() +
  labs(x = "Drive system",
       y = "City fuel economy (MPG)")

• We now map both axes: a category on x and its numeric values on y.
• geom_boxplot() calculates a separate summary for each drive-system group. fill = drv inside aes() gives each group its own color.

⚖️ Histograms and boxplots provide different detail

Display	Especially useful for	Main limitation
Histograms	Peaks, gaps, tails, and distribution shape	Bin width changes visible detail
Boxplots	Comparing group medians and spread compactly	Can hide peaks and gaps within groups

• Both displays summarize the same city-MPG observations.
• The chart choice depends on which features the reader needs to compare.

✍️ Classwork 11: Distribution of a Numeric Variable Across Groups

Classwork 11: Distribution of a Categorical Variable Across Groups

• Compare drive systems using faceted histograms and boxplots.

📊 Distribution of a categorical variable

🚗 geom_bar() counts categories

ggplot(data = mpg,
       mapping = aes(y = class,
                     fill = class)) +
  geom_bar(show.legend = FALSE) +
  scale_fill_tableau() +
  labs(x = "Number of vehicles",
       y = "Vehicle class")

• A categorical distribution shows how many observations belong to each category. Here, our question is: Which vehicle classes are most common in mpg?
• geom_bar() counts rows automatically. Map class to y for horizontal bars and to fill inside aes() for category colors.

show.legend = FALSE hides the redundant color legend. The fill mapping still assigns a color to each vehicle class. Students do not need to memorize the legend setting.

🧱 Bar charts and histograms group values differently

	Histogram	Bar chart
Variable	Numeric, such as city MPG	Categorical, such as vehicle class
Grouping	Numeric intervals called bins	Named categories
Order	Follows the numeric scale	Chosen to support the comparison

• Histograms show a numeric distribution using adjacent intervals.
• A bar chart compares categories through bar length. Its numeric value axis should begin at zero.

🔢 count() creates a table of counts

class_counts <- mpg |>
  count(class)

class	n
2seater	5
compact	47
midsize	41
minivan	11
pickup	33
subcompact	35
suv	62

• count(class) creates one row per vehicle class.
• The new column n gives the number of original rows in that class.
• The seven counts add to 234, the total number of vehicles in mpg.

📊 geom_col() uses numbers already in a column

ggplot(data = class_counts,
       mapping = aes(x = n,
                     y = class,
                     fill = class)) +
  geom_col(show.legend = FALSE) +
  scale_fill_tableau() +
  labs(x = "Number of vehicles",
       y = "Vehicle class")

• Use class_counts data.frame from the previous slide: one row per vehicle class, with its count in n.
• x = n sets each bar’s length. geom_col() uses these supplied values, while geom_bar() counts rows automatically.

The previous slide created class_counts with mpg |> count(class). Its seven rows summarize the 234 vehicles. Each bar uses one n value, so the plot shows the same class counts as the earlier geom_bar() chart. Mapping class to y keeps the bars horizontal. The next slide changes only the category order with fct_reorder(class, n).

show.legend = FALSE hides the redundant color legend because the axis already names each class. On an already summarized table, default geom_bar() would count summary rows rather than use n as the bar length.

Source: https://ggplot2.tidyverse.org/reference/geom_bar.html

🪜 Sorted bars make rankings easier to read

ggplot(data = class_counts,
  mapping = aes(
    x = n,
    y = fct_reorder(class, n),
    fill = class)) +
  geom_col(show.legend = FALSE) +
  scale_fill_tableau() +
  labs(x = "Number of vehicles",
       y = "Vehicle class")

• Using the same table and counts, fct_reorder(class, n) orders the categories using their corresponding counts.
• Here, the largest count appears at the top. Horizontal bars leave room for the category names.

✍️ Classwork 12: Distribution of a Categorical Variable

Classwork 12: Distribution of a Categorical Variable

• Compare category counts and plot a prepared summary table.

✈️ Categorical distributions across groups

✈️ Airlines across three NYC-area airports

flights <- nycflights13::flights

airline_flights <- flights |>
  filter(carrier == "AA" |
         carrier == "DL" |
         carrier == "UA")

Column	Categories used
origin	EWR: Newark JFK: John F. Kennedy LGA: LaGuardia
carrier	AA: American DL: Delta UA: United

• Each row is one 2013 flight record. We keep only these three airlines at the three NYC-area airports.
• Our question: How do airline counts and shares differ across airports? These records do not count passengers.

Source: nycflights13::flights, documented in ?nycflights13::flights. Students have already used this data frame for filtering. The vertical bars in filter() mean “or.” The new data frame contains 139,504 records, with all nine airport-and-airline combinations present.

These are historical records from 2013. Some rows have no recorded actual departure time, so describe row counts as flight records rather than completed departures. The selected airlines are not all airlines in the source data. EWR is Newark, in the NYC metropolitan area.

🔢 count(origin, carrier) counts combinations

airport_airline_counts <- airline_flights |>
  count(origin, carrier)

origin	carrier	n
EWR	AA	3487
EWR	DL	4342
EWR	UA	46087
JFK	AA	13783
JFK	DL	20701
JFK	UA	4534
LGA	AA	15459
LGA	DL	23067
LGA	UA	8044

• Each row is one airport-and-airline combination. n counts its flight records.

count(origin, carrier) creates nine rows, one for each observed combination of the three airports and three selected airlines. The counts total 139,504. Adding the three counts for an airport gives its selected-airline denominator.

Source: https://dplyr.tidyverse.org/reference/count.html

📊 Stacked bars: totals and their breakdown

ggplot(data = airline_flights,
       mapping = aes(x = origin,
                     fill = carrier)) +
  geom_bar()

• fill = carrier gives each airline a color. geom_bar() stacks the airline counts within each airport.
• The total bar height counts records for AA, DL, and UA combined at that airport.
• Use this chart for airport totals and their airline breakdown. Some segments start at different heights, making their counts harder to compare.

Stacking is the default when fill divides a category into groups. Each colored segment counts flight records for one airline. The selected-airline totals are 53,916 at EWR, 39,018 at JFK, and 46,570 at LGA. These totals exclude all other carriers.

The bottom segment shares a zero baseline across bars. Higher segments can start at different cumulative counts. The next chart makes airline-count comparisons easier by placing every bar at zero. Counts of flight records do not measure passengers or ticket sales.

Source: https://ggplot2.tidyverse.org/reference/geom_bar.html

📊 position = "dodge": grouped counts

ggplot(data = airline_flights,
       mapping = aes(x = origin,
                     fill = carrier)) +
  geom_bar(position = "dodge")

• Our question: Which airline has the most flight records at each airport?
• position = "dodge" puts the airline bars side by side. All bars start at zero, and their heights are flight-record counts.

Airline is the categorical variable being compared, and each origin airport defines a comparison group. The counts are EWR: AA 3,487, DL 4,342, UA 46,087; JFK: AA 13,783, DL 20,701, UA 4,534; LGA: AA 15,459, DL 23,067, UA 8,044. United has the largest selected-airline count at EWR, while Delta has the largest at JFK and LGA.

All nine combinations occur in the data, so the ordinary dodge setting produces equal widths. Changing the position does not change any count. These are historical flight records, not passenger counts or current market shares.

Source: https://ggplot2.tidyverse.org/reference/geom_bar.html

💯 position = "fill": within-group proportions

ggplot(data = airline_flights,
       mapping = aes(x = origin,
                     fill = carrier)) +
  geom_bar(position = "fill") +
  labs(y = "Proportion")

• fill = carrier assigns colors. position = "fill" rescales each airport’s selected-airline total to 1 (100%).
• Each segment shows an airline’s share within that airport. Its denominator is AA + DL + UA records at that airport.
• Compare the airline mix here. Equal total heights hide differences in the number of flight records.

The two uses of fill do different jobs. Inside aes(), fill maps the carrier codes to colors. Inside geom_bar(), position = “fill” rescales stacked counts. A height of 0.75 means 75% of the selected-airline records at that airport.

At JFK, American’s share is 13,783 divided by 39,018, about 35.3%. At LGA, it is 15,459 divided by 46,570, about 33.2%. The denominator is specific to each airport and includes only the three selected airlines. These proportions are not shares of all airlines or passengers.

Source: https://ggplot2.tidyverse.org/reference/geom_bar.html

🧭 Bar-chart choice depends on the question

Stacked counts

Use: totals and breakdown.

Watch out: unaligned segments.

Side by side

Use: subgroup counts.

Watch out: crowded bars with many categories.

100% stacked

Use: within-group proportions.

Watch out: hidden totals.

All panels use the same 139,504 selected-airline records: 53,916 at EWR, 39,018 at JFK, and 46,570 at LGA. The first two panels share the same count scale from zero to 55,000. The third uses proportions from zero to one. Each airline retains the same color in all panels.

Stacked counts emphasize airport totals and their airline breakdown. Side-by-side bars align airline counts at zero. A 100% stacked chart compares the airline mix within each airport and hides differences in group size. All three displays exclude carriers other than AA, DL, and UA.

With three airlines, middle segments can lack aligned edges even in a 100% stacked chart. Students should choose a display based on whether the question concerns total records, airline counts, or within-airport proportions.

Source: https://ggplot2.tidyverse.org/reference/geom_bar.html

🧩 The same grammar supports different questions

Question	Mapping	Drawing layer
How does city MPG vary?	aes(x = cty)	geom_histogram()
How does city MPG differ by drive system?	aes(x = drv, y = cty, fill = drv)	geom_boxplot()
How many vehicles are in each class?	aes(y = class, fill = class)	geom_bar()
How do airline counts differ across airports?	aes(x = origin, fill = carrier)	geom_bar(position = "dodge")
What counts are in our class-count table?	aes(x = n, y = class, fill = class)	geom_col()

• We choose the data and mappings, then add the layer suited to our question.

✍️ Classwork 13: Distribution of a Categorical Variable Across Groups

Classwork 13: Distribution of a Categorical Variable Across Groups

• Compare side-by-side counts with within-group proportions, then check a two-variable count table.