Lecture 8

Data Visualization II: Relationships and Time Trends

Byeong-Hak Choe

bchoe@geneseo.edu

SUNY Geneseo

November 1, 2026

🔗 Relationships between variables

🧩 The same grammar supports a new question

• We have used data, mappings in aes(), and a geom layer to show distributions and compare groups.
• Now the question is: How is engine size related to highway fuel economy?
• A scatterplot maps two numeric variables to horizontal and vertical positions.

Part of the plot	What stays familiar	What changes here
data = mpg	The data frame	We keep the same vehicle rows
mapping = aes(...)	Variables control visual features	Both x and y contain numeric variables
geom_*()	A layer draws the data	geom_point() draws a point for each row

🔵 A scatterplot places each row using two values

ggplot(data = mpg,
       mapping = aes(x = displ,
                     y = hwy)) +
  geom_point() +
  labs(
    x = "Engine size (liters)",
    y = "Highway fuel economy (mpg)"
  )

• displ gives engine size in liters. hwy gives highway miles per gallon.
• One point represents one vehicle in the dataset, as in our earlier mpg charts.

Source: https://ggplot2.tidyverse.org/reference/mpg.html. The mpg dataset describes selected vehicles from 1999 and 2008. Each row describes a particular version of a vehicle model, not an individual car sold. Versions can differ in engine, transmission, drive system, or model year. The rows do not represent individual drivers or current vehicle sales.

👀 Direction, shape, and unusual points

• Direction: larger engines generally accompany lower highway mpg. We call this a negative association.
• Shape and spread: the pattern may curve. Vehicles with similar engine sizes can have different highway mpg.
• Unusual points: a vehicle far from the overall pattern deserves a closer look at its row and context.

A positive association means the variables tend to increase together. No clear association means the plot reveals no obvious pattern. A steep line need not represent a stronger association than a shallow line because changing axis units changes steepness. The plotted variables need not move in the same direction to have an association.

⚠️ Association and cause

• A scatterplot describes how variables vary together in these observations.
• Vehicles also differ in weight, design, and transmission. Those features may relate to both engine size and fuel economy.
• Establishing a causal effect requires an appropriate study design and additional evidence. This chart alone cannot isolate the effect of changing engine size.

🚙 Drive system adds a categorical variable

drv	Drive system
f	Front-wheel drive
r	Rear-wheel drive
4	Four-wheel drive

• These are the same categories used in our grouped histograms and boxplots.
• We can distinguish them within the scatterplot to see where groups overlap.

🎨 Mapped color identifies groups

library(ggthemes)

ggplot(data = mpg,
       mapping = aes(x = displ,
                     y = hwy,
                     color = drv)) +
  geom_point(alpha = 0.5) +
  scale_color_tableau() +
  labs(
    x = "Engine size (liters)",
    y = "Highway fuel economy (mpg)",
    color = "Drive system"
  )

• color = drv inside aes() maps groups to colors. scale_color_tableau() chooses those colors.
• alpha = 0.5 makes points partly transparent. Overlapping points appear darker.

Alpha ranges from 0 for invisible to 1 for opaque. Transparency can reveal overlap but does not make exact duplicate observations individually countable. The scale uses Tableau’s default discrete palette. Color alone does not guarantee that every viewer can distinguish every category.

🖌️ Mappings and fixed appearances

ggplot(data = mpg,
       mapping = aes(x = displ,
                     y = hwy)) +
  geom_point(color = "steelblue", alpha = 0.5)

Code	Meaning
aes(color = drv)	Let a variable determine each point’s color
geom_point(color = "steelblue")	Give every point the same chosen color

• A variable belongs inside aes() when its values should control a visual feature.
• A fixed appearance belongs outside aes(). A color name inside aes() can create an unintended legend.

🪟 Familiar facets separate the groups

ggplot(data = mpg,
       mapping = aes(x = displ,
                     y = hwy)) +
  geom_point(alpha = 0.5) +
  facet_wrap(~ drv)

• Familiar facet_wrap(~ drv) panels use shared axes, just as in our histograms. The groups also cover different engine-size ranges.

Shared scales support comparisons across panels. The groups differ in engine-size ranges and other vehicle characteristics, so the chart alone cannot isolate the effect of drive system.

📉 A fitted line adds a summary layer

ggplot(data = mpg,
       mapping = aes(x = displ,
                     y = hwy)) +
  geom_point(alpha = 0.5) +
  geom_smooth(
    method = "lm", se = FALSE
  ) +
  labs(
    x = "Engine size (liters)",
    y = "Highway fuel economy (mpg)"
  )

• + adds another layer to the same plot. method = "lm" requests a fitted straight line.
• se = FALSE hides the shaded uncertainty band. The line summarizes the overall direction, while the points show variation around it.

The learning target is adding and recognizing a visual summary layer. Detailed regression inference, coefficients, and slope interpretation are outside this unit. This plot has no categorical mapping, so it fits one line across all observations. The line does not establish causation and may not capture a curved pattern well.

✍️ Classwork 14: Relationships

Open Classwork 14: Relationships

• Questions 1–5 apply points, mapped color, facets, and a fitted line to city and highway mpg, comparing the two model years.
• The group-comparison questions in Classwork 14 begin at Question 2. The log-transformation classwork follows the next lecture section.

🌍 Log transformations and relationships

🌍 Historical country data

Country	GDP per person	Life expectancy
Afghanistan	974.6	43.8
Albania	5937.0	76.4
Algeria	6223.4	72.3
Angola	4797.2	42.7

• Each row represents one country in 2007, with 142 countries in this dataset.
• gdpPercap is GDP per person in inflation-adjusted dollars, a measure of economic output per person rather than personal salary.
• lifeExp is life expectancy at birth in years. These historical observations do not describe current conditions.

Dataset documentation: https://jennybc.github.io/gapminder/reference/gapminder.html. The examples use the 2007 subset of the course’s gapminder.tsv file. The documentation describes gdpPercap as GDP per capita in inflation-adjusted US dollars. Countries have equal weight in these plots because each country supplies one row.

📥 Importing the country data

gapminder_2007 <- read_tsv(
  "https://bcdanl.github.io/data/gapminder.tsv"
) |>
  filter(year == 2007)

• read_tsv() reads a file whose columns have tab separators, rather than comma separators.
• filter(year == 2007) keeps a single year, so each country contributes one observation.
• The plotting grammar stays the same when we replace mpg with a different data frame.

🧮 Logarithms express multiplicative differences

Original value \(x\)	\(\log_{10}(x)\)	Why
100	2	\(10^2 = 100\)
1,000	3	\(10^3 = 1{,}000\)
10,000	4	\(10^4 = 10{,}000\)

• Equal tenfold increases become equal one-unit steps after log10().
• In R, log10(x) uses base 10. log(x) uses the natural-log base \(e \approx 2.718\).

log10(c(100, 1000, 10000))  # 2, 3, 4

📊 A log transformation changes the spacing

Original GDP

mapping = aes(x = gdpPercap)

Logged GDP

mapping = aes(x = log10(gdpPercap))

• The same 142 countries remain in both plots. Logging preserves their order but changes the distances between values.

Original country GDP-per-capita values range approximately from 278 to 49,357 US dollars. The raw histogram has a long right tail. The logarithmic transformation reduces the crowding among lower values, but it does not guarantee a symmetric or normal distribution.

🌍 GDP and life expectancy on two scales

Original GDP

Logged GDP

• Higher GDP per person generally accompanies higher life expectancy in these observations. The scale change does not establish a causal relationship.

🧩 A transformation can go inside aes()

ggplot(data = gapminder_2007,
       mapping = aes(x = log10(gdpPercap),
                     y = lifeExp)) +
  geom_point()

• log10(gdpPercap) calculates the horizontal positions from the GDP values.
• lifeExp stays in its original units: years.
• The plot uses transformed positions without replacing the original gdpPercap column.

⚠️ Log transformations require positive values

• For the real-valued logarithms we use here, inputs must be positive. log10(0) returns -Inf, and negative inputs produce NaN.
• A value that looks less extreme after logging still deserves attention. The transformation changes the display rather than correcting a data error.
• Adding 1 automatically changes the transformation and its meaning. First investigate what any zero or negative value represents.

The learning targets are multiplicative spacing, using log10() in aes(), and interpreting the changed display. Students do not need log-slope interpretations, regression coefficients, or percentage-change approximations.

R logarithm documentation: https://stat.ethz.ch/R-manual/R-devel/library/base/html/Log.html

✍️ Classwork 15: Log transformations

Open Classwork 15: Log transformations

• Compare population and life expectancy on original and logged population axes.
• Focus on how the spacing changes and which country differences become easier to see.

📈 Time trends and evidence for an audience

🗓️ Stock-price rows identify a company and a date

Date	Company ticker	Closing price (US$)
2026-10-01	AAPL	330.32
2026-10-01	NVDA	230.86
2026-10-02	AAPL	333.69
2026-10-02	NVDA	233.95

• Each row represents one company on one trading day in October 2026.
• AAPL identifies Apple, and NVDA identifies NVIDIA. Close gives the closing price per share in US dollars.
• A line chart lets us follow these observations in chronological order.

🗓️ Preparing dates and selecting companies

tech_october <- read_csv(
  "https://bcdanl.github.io/data/tech_stocks_2026_10.csv"
) 

tech_two <- tech_october |>
  filter(Ticker %in% c("AAPL", "NVDA"))

apple_october <- tech_two |>
  filter(Ticker == "AAPL")

• as.Date() converts a value such as "2026-10-01" into R’s Date type. read_csv() may already recognize this format.
• %in% checks membership in a set. Here it keeps two companies, and the final filter() keeps only Apple.

📈 Points and a line show one company’s history

ggplot(data = apple_october,
       mapping = aes(x = Date,
                     y = Close)) +
  geom_line(color = "steelblue") +
  geom_point(color = "steelblue") +
  labs(
    x = "Date in October 2026",
    y = "Closing price (US$/share)"
  )

• geom_point() marks the observed values. geom_line() connects them in x-axis order.
• The file contains trading days. A connecting segment does not provide an observed closing price for every calendar day.

⚠️ One line mixes the two companies

# Incorrect grouping
ggplot(data = tech_two,
       mapping = aes(x = Date,
                     y = Close)) +
  geom_line(color = "steelblue") +
  labs(
    x = "Date",
    y = "Closing price (US$/share)"
  )

• The combined data have two observations on each date, one per company.
• This code connects the observations as one group, producing jumps between unrelated company prices.

🧵 Mapped color separates the company lines

ggplot(data = tech_two,
       mapping = aes(x = Date,
                     y = Close,
                     color = Ticker,
                     linetype = Ticker)) +
  geom_line(linewidth = 1) +
  scale_color_tableau() +
  labs(
    x = "Date",
    y = "Closing price (US$/share)",
    color = "Company",
    linetype = "Company"
  )

• Mapping the categorical Ticker variable to color creates a separate line for each company.
• linetype = Ticker also distinguishes the companies when colors are hard to tell apart.

🪟 Explicit grouping works with facets

ggplot(data = tech_two,
       mapping = aes(x = Date,
                     y = Close,
                     group = Ticker)) +
  geom_line(color = "steelblue") +
  facet_wrap(~ Ticker)

• group = Ticker explicitly tells ggplot which rows belong to the same series.
• facet_wrap(~ Ticker) places each company’s history in its own panel, with shared scales by default.
• Fixed color controls appearance. The grouping and panel variables determine which observations appear together.

🔎 A price chart has limits

• A higher price per share does not establish that a company is larger or that its stock is a better investment.
• Price levels do not directly compare percentage returns. That comparison requires a common starting point and an appropriate return calculation.
• This historical chart covers October 2026. It cannot establish why a price moved or predict what happens next.

🧭 Questions, variables, and ggplot layers

Question	Variables	Useful layer
How are values distributed?	One numeric variable	geom_histogram()
How do distributions differ across groups?	Numeric + category	geom_boxplot()
How do categories compare?	Category counts or supplied values	geom_bar() or geom_col()
How do variables vary together?	Two numeric variables	geom_point()
How does a quantity change through time?	Date + numeric variable	geom_line()

• The data frame, aes(), and + remain familiar. The question guides the mappings and layers.

🏷️ Labels for a project audience

labs(
  title = "Apple closing prices in October 2026",
  x = "Date",
  y = "Closing price (US$/share)",
  caption = "Source: Stock Analysis"
)

• labs() changes displayed text without renaming the data columns.
• For classwork and homework, recognize its purpose. You may omit it unless a question specifically requests labels.
• Project charts need clear labels and units, plus a source and relevant time period. labs() is a convenient way to supply them.

📝 A finding needs evidence and a limitation

Worked example: city fuel economy by drive type

• Finding: front-wheel-drive vehicles have higher median city fuel economy in this dataset.
• Evidence: the median is 19 mpg, compared with 14 mpg for four-wheel-drive vehicles and 15 mpg for rear-wheel-drive vehicles.
• Limitation: the groups differ in other vehicle characteristics. This comparison does not isolate a drive-system effect.

This worked interpretation returns to the grouped boxplots in Lecture 7. A useful interpretation identifies the observations and units instead of saying only that a chart shows a trend.

📁 Reproducible charts for the storytelling project

• The R script should recreate the data preparation and charts from the original files, without depending on objects left in the Environment.
• Each chart should support a question with an appropriate comparison. Its labels should make the observations and units clear.
• A recommendation should stay within the evidence. State an important limitation and acknowledge any AI assistance according to course requirements.

✍️ Classwork 16: Time and a data story

Open Classwork 16: Time trends and a data story

• Questions 1–3 compare Microsoft and Meta trading-volume histories, then a short finding supported by evidence and a limitation.
• The final interpretation prepares for the storytelling project.