Data visualization: quantitative data

CMSACamp 2026

Department of Statistics & Data Science
Carnegie Mellon University

Background

Quantitative data

Two different versions of quantitative data:

Discrete: countable and has clear space between values (i.e. whole number only)

Examples: number of goals scored in a game, number of children in a family

Continuous: can take any value within some interval

Examples: price of houses in Pittsburgh, water temperature, wind speed

Data

We will explore ultra trail running data courtesy of TidyTuesday and Benjamin Nowak of the International Trail Running Association (ITRA).

The data was pre-processed for this lecture (steps shown in appendix).

library(tidyverse)
ggplot2::theme_set(ggplot2::theme_light(base_size = 20)) # setting the ggplot theme

runs <- read_csv("https://raw.githubusercontent.com/36-SURE/2026/main/data/ultra_running.csv") 
race_stats_2019 <- read_csv("https://raw.githubusercontent.com/36-SURE/2026/main/data/ultra_races_2019.csv")

Western States Endurance Run

For the majority of the lecture, we will be using race results from the Western States Endurance Run.

wser <- runs |>
  filter(event == "Western States Endurance Run")

Highlights from Western States 2024

1D quantitative data

Summarizing 1D quantitative data

Center: mean, median, number and location of modes

Spread: range, variance, standard deviation, IQR, etc.

Shape: skew vs symmetry, outliers, heavy vs light tails, etc.

Compute various statistics in R with summary(), mean(), median(), quantile(), range(), sd(), var(), etc.

Example: Summarizing the finishing time (in hours) of Western States runners

summary(wser$time_in_hours)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  14.50   22.89   26.51   25.42   28.68   30.00

sd(wser$time_in_hours, na.rm = TRUE)

[1] 3.742693

Boxplots visualize summary statistics

Pros:

Displays outliers, percentiles, spread, skew
Useful for side-by-side comparison

Cons:

Does not display the full distribution shape
Does not display modes

wser |> 
  ggplot(aes(x = time_in_hours)) +
  geom_boxplot() +
  theme(axis.text.y = element_blank())+
  labs(x= 'Time (hours)')

Histograms display 1D continuous distributions

\(\displaystyle \text{# total obs.} = \sum_{j=1}^k \text{# obs. in bin }j\)

Pros:

Displays full shape of distribution
Easy to interpret

Cons:

Have to choose number of bins and bin locations (will revisit later)

wser |> 
  ggplot(aes(x = time_in_hours)) +
  geom_histogram()+
  labs(x= "Time (hours)")

Density plots are a smoothed, continuous version of a histogram

Pros:

Bin-free
Excellent for comparing multiple categories at once
Easy to view full shape of distribution

Cons:

No individual data points or frequency count
Smooth curve can sometimes extend into ranges where no real data exists

wser |> 
  ggplot(aes(x = time_in_hours)) +
  geom_density()+
  labs(x= "Time (hours)")

We will talk more about density estimation on Monday!

Display the data points directly with beeswarm plots

Pros:

Displays each data point
Easy to view full shape of distribution

Cons:

Can be overbearing with large datasets
Which algorithm for arranging points?
- A few different methods

library(ggbeeswarm)
wser |> 
  ggplot(aes(x = time_in_hours, y = "")) +
  geom_beeswarm(cex = 1, size = 0L)+
  labs(x= "Time (hours)", y="")

Smooth summary with violin plots

Pros:

Displays full shape of distribution
Can easily layer…

wser |> 
  ggplot(aes(x = time_in_hours, y = "")) +
  geom_violin()+
  labs(x= "Time (hours)", y ="")

Smooth summary with violin plots + box plots

Pros:

Displays full shape of distribution
Can easily layer… with box plots on top

Cons:

Summary of data via density estimate (no sample size)
Mirror image is duplicate information
May be harder to interpret for general audience

wser |> 
  ggplot(aes(x = time_in_hours, y = "")) +
  geom_violin() +
  geom_boxplot(width = 0.25)+
  labs(x= "Time (hours)", y = "")

What do visualizations of continuous distributions display?

Probability that continuous variable \(X\) takes a particular value is 0

e.g. \(P(\) time_in_hours \(= 24) = 0\) (why?)

For continuous variables, the cumulative distribution function (CDF) is \[F(x) = P(X \leq x)\]

For \(n\) observations, the empirical CDF (ECDF) can be computed based on the observed data \[\hat{F}_n(x) = \frac{\text{# obs. with variable} \leq x}{n} = \frac{1}{n} \sum_{i=1}^{n} I (x_i \leq x)\]

where \(I()\) is the indicator function, i.e. ifelse(x_i <= x, 1, 0)

Display full distribution with ECDF plot

Pros:

Displays all of your data at once
As \(n \rightarrow \infty\), the ECDF \(\hat F_n(x)\) converges to the true CDF \(F(x)\)

Cons:

What are the cons?

wser |> 
  ggplot(aes(x = time_in_hours)) +
  stat_ecdf()+
  labs(x= "Time (hours)")

Zooming in on our ECDF

On the previous plot, we have enough data points that the ECDF looks like a smooth curve
Keep in mind, however, while the CDF is a curve, the ECDF is a step function
- Each jump size of \(1/n\) is a single observed point

wser |> 
  ggplot(aes(x = time_in_hours)) +
  stat_ecdf()+
  coord_cartesian(xlim = c(16, 17), ylim = c(0.01, 0.03))+
  labs(x= "Time (hours)")

Rug plots display raw data

Pros:

Displays raw data points
Useful supplement for summaries and 2D plots

Cons:

Can be overbearing for large datasets

wser |> 
  ggplot(aes(x = time_in_hours)) +
  geom_rug(alpha = 0.3)+
  labs(x= "Time (hours)")

Rug plots supplement other displays

wser |> 
  ggplot(aes(x = time_in_hours, y = "")) +
  geom_violin() +
  geom_rug(alpha = 0.3)+
  labs(x= "Time (hours)")

wser |> 
  ggplot(aes(x = time_in_hours)) +
  stat_ecdf() +
  geom_rug(alpha = 0.3)+
  labs(x= "Time (hours)")

2D quantitative data

Summarizing 2D quantitative data

Direction/trend (positive, negative)
Strength of the relationship (strong, moderate, weak)
Linearity (linear, non-linear)

Big picture

Scatterplots are by far the most common visual
Regression analysis is by far the most popular analysis (coming in t - 2 weeks!)
Relationships may vary across other variables, e.g., categorical variables

Note: for this section, we will be using our race_stats_2019 dataset to compare 80+ ultramarathon races that took place in 2019!

Making scatterplots

Use geom_point(): each point is a race
Displaying the joint (bivariate) distribution
Adjust transparency with alpha to visualize overlap
- Provides better understanding of joint frequency
- Especially important with larger datasets
- See also: ggblend

race_stats_2019 |> 
  ggplot(aes(x = elevation_gain, y = mean_finish_time)) +
  geom_point(color = "navy", size = 4, alpha = 0.5)+
  scale_x_continuous(labels = scales::label_comma())+
  labs(x = "Elevation Gain (m)", y = "Avg finish time (hrs)")

Summarizing 2D quantitative data

Correlation coefficient

cor(race_stats_2019$elevation_gain, 
    race_stats_2019$mean_finish_time, 
    use = "complete.obs")

[1] 0.8232547

Note: the default correlation you get from cor() is Pearson correlation coefficient

Other correlations:

Displaying trend line: linear regression
(a preview)

Display regression line for
mean finish time ~ elevation gain
95% confidence intervals by default
Estimating the conditional expectation of mean finish time | elevation gain
- \(\mathbb{E}[\) mean finish time \(\mid\) elevation gain \(]\)

race_stats_2019 |> 
  ggplot(aes(x = elevation_gain, y = mean_finish_time)) +
  geom_point(color = "navy", size = 4, alpha = 0.5)+
  geom_smooth(method = "lm", linewidth = 2)+
  labs(x = "Elevation Gain (m)", y = "Avg finish time (hrs)")

Summarizing 2D quantitative data

Add rug plots to supplement scatterplot

race_stats_2019 |> 
  ggplot(aes(x = elevation_gain, y = mean_finish_time)) +
  geom_point(color = "navy", size = 4, alpha = 0.5) +
  geom_rug(alpha = 0.4)+
  labs(x = "Elevation Gain (m)", y = "Avg finish time (hrs)")

Pairs plot

library(GGally)
race_stats_2019 |> 
  ungroup() |>
  select(elevation_gain, elevation_loss, mean_finish_time, n_finishers) |> 
  ggpairs(aes(alpha=0.3)) # note: alpha in aes() here, even though it is not a variable

Continuous by categorical data

Continuous by categorical: side by side plots

wser <- wser |>
  mutate(age_grp = case_when(age <= 39 ~ "22-39", 
                             age > 39 & age <= 49 ~ "40-49", 
                             age > 49 & age <= 59 ~ "50-59", 
                             age > 59 ~ "60+"))
  
wser |> 
  ggplot(aes(x = time_in_hours, y = age_grp))+
  geom_boxplot()+
  labs(x = "Time (hours)", y = "Age group")

Continuous by categorical: ECDF

wser |> 
  ggplot(aes(x = time_in_hours, color = age_grp)) +
  stat_ecdf(linewidth = 1) +
  theme(legend.position = "bottom")+
  scale_color_viridis_d()+
  labs(x = "Time (hours)", color = "Age group")

Continuous by categorical: density plot

 wser |> 
  ggplot(aes(x = time_in_hours, color = age_grp)) +
  geom_density() +
  scale_color_viridis_d()+
  labs(color = "Age group", x = "Time (hours)")

Continuous by categorical: ridgeline plot

For more, see this tutorial

library(ggridges)
wser |> 
  ggplot(aes(x = time_in_hours, y = age_grp, fill = age_grp)) +
  geom_density_ridges(scale = 1.5)+
  scale_fill_viridis_d()+
  labs(x = "Time (hours)", y = "Age group")+
  theme(legend.position = "none")

What about for histograms?

 wser |> 
  ggplot(aes(x = time_in_hours, fill = age_grp)) +
  geom_histogram(bins = 15) +
  scale_fill_viridis_d(alpha = 0.6)+
  labs(fill = "Age group", x = "Time (hours)")

What about facets?

Difference between facet_wrap and facet_grid

wser |>
  ggplot(aes(x = time_in_hours)) +
  geom_histogram(bins = 15) +
  facet_wrap(~ age_grp, nrow = 1)+
  labs(x= "Time (hours)")

Appendix: Pre-processing steps

ultra_rankings <- read_csv('https://raw.githubusercontent.com/rfordatascience/tidytuesday/master/data/2021/2021-10-26/ultra_rankings.csv')
race <- read_csv('https://raw.githubusercontent.com/rfordatascience/tidytuesday/master/data/2021/2021-10-26/race.csv')

# join race results with course information 
runs <- ultra_rankings |> 
  # include race information for each race completed by a runner
  left_join(race, by = "race_year_id") |> 
  # remove incomplete races or races < ~100 miles
  filter(!is.na(time_in_seconds) & distance > 150) |> 
  # convert time in seconds to hours
  mutate(time_in_hours = (time_in_seconds/60)/60) 

# 2019 ultramarathon race course information 
race_stats_2019 <- runs |> 
  # only keep results from 2019 (courses can change year to year)
  filter(year(date) == 2019) |>
  # a race may have multiple events, so we group by both variables 
  group_by(race, event) |> 
  # number of finishers for particular race event 
  mutate(n_finishers = n()) |> 
  # only keep race events with at least 50 finishers
  filter(n_finishers >= 50) |> 
  # average time of finishers for particular race event
  mutate(mean_finish_time = mean(time_in_hours)) |>
  # collapse across individuals: we only want course information
  distinct(race, event, elevation_gain, elevation_loss, n_finishers, mean_finish_time)

Data visualization: quantitative data

Background

Quantitative data

Data

Western States Endurance Run

1D quantitative data

Summarizing 1D quantitative data

Boxplots visualize summary statistics

Histograms display 1D continuous distributions

Density plots are a smoothed, continuous version of a histogram

Display the data points directly with beeswarm plots

Smooth summary with violin plots

Smooth summary with violin plots + box plots

What do visualizations of continuous distributions display?

Display full distribution with ECDF plot

Zooming in on our ECDF

Rug plots display raw data

Rug plots supplement other displays

2D quantitative data

Summarizing 2D quantitative data

Making scatterplots

Summarizing 2D quantitative data

Displaying trend line: linear regression(a preview)

Summarizing 2D quantitative data

Pairs plot

Continuous by categorical data

Continuous by categorical: side by side plots

Continuous by categorical: ECDF

Continuous by categorical: density plot

Continuous by categorical: ridgeline plot

What about for histograms?

What about facets?

What about facets?

Appendix: Pre-processing steps

Displaying trend line: linear regression
(a preview)