2  Graphs and Data Visualization

2.1 Introduction

• Graphs reveal information about the center, shape, and spread of distributions.
• They can also show typical and extreme outcomes, associations, and differences between groups.
• Different data types require different graph types.

2.2 Categorical Data

Categorical data types:

• Nominal: no natural order (e.g., eye color, blood type).
• Ordinal: ordered, but spacing not meaningful (e.g., therapy response).
• Interval: ordered and evenly spaced (e.g., fruit count).

2.2.1 Bar Chart Example

Bar chart is used for displaying categorical variables.

colors <- c('Red', 'Pink', 'White')
counts <- c(80, 60, 40)
bar_df <- data.frame(Color = colors, Count = counts)
ggplot(bar_df, aes(x = Color, y = Count)) +
  geom_bar(stat = 'identity', fill = 'steelblue') +
  theme_minimal() +
  labs(title = 'Poinsettia Colors')

• Question: What does the bar plot tell you about the color distribution for this poinsettia variety?

2.3 Quantitative Data

• Measured on a numeric scale: e.g., weights, heights, cholesterol levels.
• Use histograms, boxplots, dotplots, density plots.

2.3.1 Histogram Example (Tip Amounts)

Histograms classify values into bins of equal width

• Heights of bars represent relative frequencies

data("tips", package="reshape2")
ggplot(tips, aes(x = tip)) +
  geom_histogram(binwidth = 1, fill = 'orange', color = 'black') +
  theme_minimal() +
  labs(title = 'Histogram of Tip Amounts', x = 'Tip ($)', y = 'Count')

• Question: What does this histogram tell you about tips at this restaurant?

Tip

• Center
• Shape
• Spread

2.3.2 Try Different Bin Widths

bw_list <- c(0.25, 0.5, 1.0)
plots <- lapply(bw_list, function(bw) {
  ggplot(tips, aes(x = tip)) +
    geom_histogram(binwidth = bw, fill = 'lightblue', color = 'black') +
    ggtitle(paste('Bin Width =', bw)) +
    theme_minimal()
})
library(gridExtra)
do.call(grid.arrange, c(plots, ncol = 1))

2.3.3 Boxplot

Boxplot is graphical display of the five number data summary (minimum, Q1, median, Q3, maximum). It is good for comparing samples from different populations.

ggplot(tips, aes(y = tip)) +
  geom_boxplot(fill = 'lightgreen') +
  theme_minimal() +
  labs(title = 'Boxplot of Tip Amounts', y = 'Tip ($)')

2.4 One Quantitative & One Categorical

Side by side box plots and dot plots can be used to compare distributions of a quantitative response variable for different levels of a categorical variable.

ggplot(tips, aes(x = day, y = tip)) +
  geom_boxplot(fill = 'skyblue') +
  theme_minimal() +
  labs(title = 'Tip Amount by Day of Week')

2.5 Two Quantitative Variables

Scatterplots convey information about associations between quantitative variables and also about unusual observations.

tips$bill <- tips$total_bill
ggplot(tips, aes(x = bill, y = tip)) +
  geom_point() +
  geom_smooth(method = 'lm', se = FALSE, col = 'red') +
  theme_minimal() +
  labs(title = 'Tip vs Bill Amount', x = 'Bill ($)', y = 'Tip ($)')

• How is the tip related to the bill?
• Which variable is the response? Which is the explanatory variable?
• What type of relationship would you expect?

2.6 Practice Problems

2.6.1 Exercise 1: Bar Chart from Survey

Create a bar chart for the following survey results:

Pet Type	Count
Dog	15
Cat	10
Fish	3
Bird	2

Use ggplot2 package to visualize the data and describe what the chart tells you about pet preferences.

library(ggplot2)
pet_df <- data.frame(
  Pet = c("Dog", "Cat", "Fish", "Bird"),
  Count = c(15, 10, 3, 2)
)
ggplot(pet_df, aes(x = Pet, y = Count, fill = Pet)) +
  geom_bar(stat = "identity") +
  theme_minimal() +
  labs(title = "Pet Types", x = "Pet Type", y = "Count") +
  scale_fill_brewer(palette = "Set2")

2.6.2 Exercise 2: Simulated Histogram

Simulate 300 observations from a normal distribution with mean 70 and standard deviation 10.

1. Plot a histogram using binwidth = 5 and 10.
2. Overlay a density curve using geom_density().
3. Describe the center, spread, and shape of the distribution.

set.seed(4750)
x <- rnorm(300, mean = 70, sd = 10)
# Histogram with binwidth 5
p1 <- ggplot(data.frame(x), aes(x = x)) +
  geom_histogram(
    binwidth = 5,
    fill = "skyblue",
    color = "white",
    alpha = 0.7
  ) +
  geom_density(aes(y = after_stat(count) * 5),
               color = "red",
               linewidth = 1.2) +
  labs(title = "Histogram (binwidth = 5) with Density", x = "Value", y = "Count") +
  theme_minimal()

# Histogram with binwidth 10
p2 <- ggplot(data.frame(x), aes(x = x)) +
  geom_histogram(
    binwidth = 10,
    fill = "orange",
    color = "white",
    alpha = 0.7
  ) +
  geom_density(aes(y =  after_stat(count) * 10),
               color = "red",
               linewidth = 1.2) +
  labs(title = "Histogram (binwidth = 10) with Density", x = "Value", y = "Count") +
  theme_minimal()

library(patchwork)
p1 / p2

2.6.3 Exercise 3: Boxplots by Group

Simulate exam scores for two groups of students (Group A and Group B), each with 50 observations.

1. Generate scores from rnorm(50, mean=80, sd=5) for Group A and rnorm(50, mean=75, sd=7) for Group B.
2. Create a combined data.frame and make a boxplot comparing the two groups.
3. Interpret differences in central tendency and variability.

set.seed(4750)
group_a <- rnorm(50, mean = 80, sd = 5)
group_b <- rnorm(50, mean = 75, sd = 7)

scores <- data.frame(
  Score = c(group_a, group_b),
  Group = rep(c("A", "B"), each = 50)
)

library(ggplot2)
ggplot(scores, aes(x = Group, y = Score, fill = Group)) +
  geom_boxplot() +
  theme_minimal() +
  labs(title = "Exam Scores by Group", y = "Score")

Interpretation:

• Central tendency: Group A has a higher median and mean score than Group B.
• Variability: Group B shows a wider spread (greater interquartile range and more outliers) compared to Group A, which aligns with the larger standard deviation used in its simulation.

2.6.4 Exercise 4: Scatterplot with Regression

Simulate a dataset of 100 observations where x ~ runif(100, 0, 100) and y = 0.5 * x + rnorm(100, 0, 5).

1. Create a scatterplot of y vs x.
2. Add a regression line.
3. Describe the relationship and interpret the slope.

set.seed(4750)
x <- runif(100, 0, 100)
y <- 0.5 * x + rnorm(100, 0, 5)
data <- data.frame(x = x, y = y)

ggplot(data, aes(x = x, y = y)) +
  geom_point(color = "steelblue", alpha = 0.7) +
  geom_smooth(method = "lm", se = TRUE, color = "red") +
  labs(title = "Scatterplot of y vs x with Regression Line",
       x = "x",
       y = "y") +
  theme_minimal()

Interpretation:

• There is a strong positive linear relationship between x and y, as expected from the model.
• The slope of the regression line is close to 0.5, indicating that, on average, each 1-unit increase in x results in a 0.5-unit increase in y.
• The scatter around the regression line reflects the normal error term with standard deviation 5.

2.7 Two Quantitative & One Categorical

• Grouping: a graph consisting of a single panel with multiple variables differentiated using different visual characteristics such as color, shape, and size.
• Faceting: a graph consisting of several separate panels, with one for each level of the faceted variable, or combination of two faceted variables.

ggplot(tips, aes(x = bill, y = tip, color = sex)) +
  geom_point(alpha = 0.6) +
  geom_smooth(method = 'lm', se = FALSE) +
  labs(title = 'Tip vs Bill by Gender')

ggplot(tips, aes(x=bill, y=tip)) + 
  geom_point(alpha=0.6) + 
  geom_smooth(method = 'lm', se = FALSE) + 
  facet_wrap(~ sex, ncol=2) + 
  labs(title = 'Tip vs Bill by Gender')

• Does the relationship differ for men and women?

2.8 Two Quantitative & Two Categorical Variables

ggplot(tips, aes(x = bill, y = tip, color = sex)) +
  geom_point(alpha = 0.6) +
  geom_smooth(method = "lm", se = FALSE) +
  facet_grid(smoker ~ .) +
  labs(title = "Tip vs Bill by Sex and Smoking Status",
       x = "Bill ($)", y = "Tip ($)")

ggplot(tips, aes(x = bill, y = tip, color = sex)) +
  geom_point(alpha = 0.6) +
  geom_smooth(method = "lm", se = FALSE) +
  facet_grid(smoker ~ sex) +
  labs(title = "Tip vs Bill by Sex and Smoking Status",
       x = "Bill ($)", y = "Tip ($)")

2.9 Three Variables

2.9.1 3D Scatterplot

library(plotly)
data(mtcars)
# Example with mtcars
plot_ly(
  data = mtcars,
  x = ~mpg, y = ~hp, z = ~wt,
  type = 'scatter3d',
  mode = 'markers',
  color = ~as.factor(cyl)
)

2.9.2 Bubble Chart

ggplot(mtcars, 
       aes(x=mpg, y=hp, size=wt)) + 
  geom_point(alpha=.5, 
             fill="red", 
             color="black",
             shape=21) + 
  scale_size_continuous(range = c(1, 5)) 

2.10 Scatterplot Matrix

A scatterplot matrix (sometimes called a “pairs plot”) is used to visualize the pairwise relationships between several quantitative variables in a dataset, all at once.

• Each row and column represents one variable.
• The off-diagonal panels show scatterplots for each pair of variables (e.g., x vs y, x vs z, y vs z).
• The diagonal often shows the distribution of each variable (as a histogram, density, or boxplot).
• You can optionally color points by a group or class variable.

data(mtcars)
library(GGally)
GGally::ggpairs(
  mtcars,
  columns = c("mpg", "hp", "wt")
)

pairs(mtcars[,c("mpg", "hp", "wt")])

cor(mtcars[,c("mpg", "hp", "wt")])

           mpg         hp         wt
mpg  1.0000000 -0.7761684 -0.8676594
hp  -0.7761684  1.0000000  0.6587479
wt  -0.8676594  0.6587479  1.0000000

2.11 Parallel Coordinates

A parallel coordinates plot is a powerful visualization tool used for exploring and comparing multivariate data (data with several quantitative variables). It’s especially valuable when you want to see how patterns, clusters, or outliers appear across many variables at once.

# Convert cyl to factor for coloring
mtcars$cyl <- as.factor(mtcars$cyl)

GGally::ggparcoord(
  mtcars,
  columns = c(1,4,6), # mpg, hp, wt
  groupColumn = "cyl",
  scale = "uniminmax",    # Normalize to [0,1] for fair comparison
  showPoints = TRUE,
  alphaLines = 0.6
) +
  scale_color_brewer(palette = "Dark2", name = "Cylinders") +
  labs(title = "Parallel Coordinates Plot: mtcars (by cylinders)")

Interpretation:

• Each line is a car.
• The color indicates the number of cylinders.
• Cars with more cylinders generally have lower mpg and higher hp and weight.

2.12 Exercises

2.12.1 Exercise 1: Ames Housing Data

Load the Ames Housing data into R and answer the following questions:

Ames = read.csv("Ameshousing.csv")
head(Ames)

  SalePrice Bedrooms LotArea LivingArea GarageArea Neighborhood
1    270000        4   11792       2283        632      Gilbert
2    377500        3   14892       1746        758      Gilbert
3    337500        3   12456       1718        786      NridgHt
4    462000        4   14257       2772        754      NridgHt
5    489900        2   14803       2084       1220      NridgHt
6    555000        2   15431       2402        672      NridgHt

1. Summary Statistics

• How many homes were included in this study?
  
• What are the names of the variables for which information was collected?
• Compute the median living area, the mean living area and the standard error for the mean living area for all of the houses in the data set.
• Compute the median living area, the mean living area and the standard error for the mean living area for the houses in each neighborhood.
• Compute the number of houses, mean sale price and the standard error of the mean sale price for houses with living areas less than 1800 square feet.
• Summarize in a paragraph the information in the correlation matrix about associations between sales price, lot size, living area, garage area, and number of rooms from the sample correlation matrix.

View Solution

dim(Ames)[1]

[1] 168

names(Ames)

[1] "SalePrice"    "Bedrooms"     "LotArea"      "LivingArea"   "GarageArea"  
[6] "Neighborhood"

mean_LivingArea <- mean(Ames$LivingArea)
mean_LivingArea

[1] 1700.107

median_LivingArea <- median(Ames$LivingArea)
sderr_LivingArea <- sd(Ames$LivingArea)/ sqrt(length(Ames$LivingArea))
sderr_LivingArea

[1] 32.32775

mean_LivingArea <- tapply(Ames$LivingArea, Ames$Neighborhood, mean)
mean_LivingArea

 CollgCr  Gilbert  NridgHt 
1565.921 1658.966 1880.921 

median_LivingArea <- tapply(Ames$LivingArea, Ames$Neighborhood, median)
median_LivingArea

CollgCr Gilbert NridgHt 
 1591.5  1560.0  1743.0 

sd_LivingArea <- tapply(Ames$LivingArea, Ames$Neighborhood, sd)
sderr_LivingArea <- sd_LivingArea / sqrt(tapply(Ames$LivingArea, Ames$Neighborhood, length))
sderr_LivingArea

 CollgCr  Gilbert  NridgHt 
42.72812 56.21652 57.40376 

Ames2 <- Ames[ Ames$LivingArea<1800, ]
n <- dim(Ames2)[1]
cat("number of houses with living area < 1800: ", n)

number of houses with living area < 1800:  116

mean_Saleprice <- mean(Ames2$SalePrice)
cat("Mean Sale Price for Houses with Living Area < 1800: ", mean_Saleprice)

Mean Sale Price for Houses with Living Area < 1800:  215756.7

sderr_SalePrice <- sd(Ames2$SalePrice)/ sqrt(length(Ames2$SalePrice))
cat("Stderror for Mean Sale Price: ", sderr_SalePrice)

Stderror for Mean Sale Price:  4693.031

2. Graphical Summaries

• Create a scatterplot with a smooth curve passed through the points.
• Fit a least squares regression line to the data in the plot.
• What does the regression line indicate about sales prices of houses increase with living area?
• What information is provided by the plot with smooth curve?
• Add a new variable to the data frame that contains information on sales price divided by the living area in the house, i.e., the sales price per square foot of living area.
• Create a histogram for the price per square foot of living space.
• What does this histogram reveal about the distribution of costs per square foot of living space for houses sold in the Ames area? The description should be based on the shape, center and spread of the distribution.
• Create separate plots of sales prices versus living space categorized for each neighborhood
• Construct side-by-side box plots to compare prices per square foot of living space across neighborhoods.
• Describe the relationship between sales price and total living area of house changes across the three neighborhoods.

View Solution

library(ggplot2)
Ames %>% 
  ggplot(aes(x=LivingArea, y=SalePrice)) + 
    geom_point(alpha=.7) + 
    geom_smooth() + 
    labs(
      x="Living Area (sq ft)",
      y="Sale Price (dollars)",
      title="Sale Price vs Total Living Area"
    )

lm(SalePrice ~ LivingArea, data = Ames)

Call:
lm(formula = SalePrice ~ LivingArea, data = Ames)

Coefficients:
(Intercept)   LivingArea  
   -21769.1        159.5  

plot( Ames$LivingArea, Ames$SalePrice, 
      xlab="Living Area (sq ft)", 
      ylab="Sale Price (dollars)",
      main = "Sale Price vs Total Living Area")
abline(lm(SalePrice~LivingArea, data=Ames), lty=1)

• The slope of the regression suggests that the mean sale price of homes in the Ames area goes up by about 159.50 dollars for each additional square foot of living space.
• The smooth curve indicates that there is an approximate straight line relationship between the mean home price and the living area of homes for homes with between 800 and 2000 square feet of living area, but the relationship curves up for larger home sizes. Also the variation in sale prices tends to increase with the amount of living area in the house.

Ames$pricesqft  <-  Ames$SalePrice/Ames$LivingArea 
head(Ames)

  SalePrice Bedrooms LotArea LivingArea GarageArea Neighborhood pricesqft
1    270000        4   11792       2283        632      Gilbert  118.2654
2    377500        3   14892       1746        758      Gilbert  216.2085
3    337500        3   12456       1718        786      NridgHt  196.4494
4    462000        4   14257       2772        754      NridgHt  166.6667
5    489900        2   14803       2084       1220      NridgHt  235.0768
6    555000        2   15431       2402        672      NridgHt  231.0575

Ames %>% ggplot() + 
  geom_histogram(aes(pricesqft), binwidth =20) 

• This histogram indicates that the distribution of costs per square foot of living space for houses sold in the Ames area is centered around $145 per square foot. The distribution is skewed to the right and it appears to be bimodal with one mode around $125 per square foot and another near $165 pre sqaure foot. Most houses cost between $100 and $200 per square foot of living space.

Ames %>% 
  ggplot(aes(x=LivingArea, y=SalePrice)) + 
  geom_point() + 
  geom_smooth() + 
  facet_grid( . ~ Neighborhood) + 
  labs(
    x="Living Area (sq ft)",
    y="Sale Price (dollars)",
    title="Sales Price vs Total Living Area"
  )

Ames %>% 
  ggplot(aes(x=LivingArea, y=SalePrice)) + 
  geom_point() + 
  geom_smooth() + 
  facet_grid(Neighborhood ~ .) + 
  labs(
    x="Living Area (sq ft)",
    y="Sale Price (dollars)",
    title="Sales Price vs Total Living Area"
  )

Ames %>% 
  ggplot(aes(x=Neighborhood, y=pricesqft)) + 
  geom_boxplot() + 
  labs(y="Sales Price per Square Foot")

• In all three neighborhoods sales prices tend to be larger for house with more living area. In the College Circle (CollgCr) and Northridge Heights (NridgHt) neighborhoods, there are strong linear relationships between these two variables. Because the slope is larger for houses in the Northridge Heights neighborhood, the price per square foot of living space is higher. There are more expensive houses in the Northridge Heights neighborhood. In Gilbert, the trend in sales prices is not so close to a straight line. There is little trend in sales prices for houses with less than 1,750 square feet of living space. There is also little trend in the sales prices for houses with living space above 1,900 square feet, but those houses are more expensive than house with less then 17,50 square feet. There is one relatively expensive house with about 1,750 square feet of living space that appears to be an outlier. Perhaps this house has an extremely large lot size or some additional buildings on the property. This should be investigated.

2.12.2 Exercise 2: Music Clips Data

The music clips data is posted in music-plusnew-sub.csv. The data file has five quantitative variables containing audio information from 62 songs. The first two columns (Artist, Type) describe the artist and type of music. The raw data come from a time series for the sound produced by each music clip (track). For each time series the variance of amplitude, average amplitude, maximum amplitude, and two additional variables calculated from the spectral decomposition of the time series are calculated. The Type variable classifies the tracks as either Rock, Classical or New Wave, and there are 5 tracks that are not identified.

• Read the data into a data frame, indicating that the row names are in column 1 of the data file and that column is not a variable.
• Obtain information on the dimensions of the data frame. Also list the column names. List the first six columns odf data.
• First select a subset of the data that contains only classical and rock music.
• For classical and rock music make histograms for the avergae amplitude variable (LAve) faceted by Type. Set the binwidth to units of 10. How do the distributions of average amplitude values differ between classical and rock music?
• Make a scatterplot of LVar vs LAve, with points colored by the type of music. Describe differences between the patterns of the points on the plot corresponding to Rock and Classical music.
• Select three music types. The other songs have missing values for the music type.
• Make a parallel coordinate plot
• Reorder how the variables appear on the plot.

View Solution

dat = read.csv("music-plusnew-sub.csv", row.names=1)  

head(dat) 

              Artist Type     LVar      LAve  LMax   LFEner     LFreq
Dancing Queen   Abba Rock 17600756 -90.00687 29921 105.9210  59.57379
Knowing Me      Abba Rock  9543021 -75.76672 27626 102.8362  58.48031
Take a Chance   Abba Rock  9049482 -98.06292 26372 102.3249 124.59397
Mamma Mia       Abba Rock  7557437 -90.47106 28898 101.6165  48.76513
Lay All You     Abba Rock  6282286 -88.95263 27940 100.3008  74.02039
Super Trouper   Abba Rock  4665867 -69.02084 25531 100.2485  81.40140

str(dat)

'data.frame':   62 obs. of  7 variables:
 $ Artist: chr  "Abba" "Abba" "Abba" "Abba" ...
 $ Type  : chr  "Rock" "Rock" "Rock" "Rock" ...
 $ LVar  : num  17600756 9543021 9049482 7557437 6282286 ...
 $ LAve  : num  -90 -75.8 -98.1 -90.5 -89 ...
 $ LMax  : int  29921 27626 26372 28898 27940 25531 14699 8928 22962 15517 ...
 $ LFEner: num  106 103 102 102 100 ...
 $ LFreq : num  59.6 58.5 124.6 48.8 74 ...

summary(dat)

    Artist              Type                LVar                LAve        
 Length:62          Length:62          Min.   :   293608   Min.   :-98.063  
 Class :character   Class :character   1st Qu.:  2844213   1st Qu.: -6.253  
 Mode  :character   Mode  :character   Median :  8210359   Median : -5.662  
                                       Mean   : 19951792   Mean   : -7.807  
                                       3rd Qu.: 24547475   3rd Qu.:  1.962  
                                       Max.   :129472199   Max.   :216.232  
      LMax           LFEner           LFreq       
 Min.   : 2985   Min.   : 83.88   Min.   : 41.41  
 1st Qu.:16200   1st Qu.:101.69   1st Qu.: 99.18  
 Median :24431   Median :104.35   Median :175.29  
 Mean   :22486   Mean   :104.03   Mean   :231.39  
 3rd Qu.:29918   3rd Qu.:108.15   3rd Qu.:315.12  
 Max.   :32766   Max.   :114.00   Max.   :877.77  

table(dat$Type)

Classical  New wave      Rock 
       24         3        30 

df.sub = dat %>% 
  dplyr::filter(Type=="Rock" | Type=="Classical")

ggplot(df.sub, aes(LAve)) + 
  geom_histogram(binwidth=10) + 
  facet_wrap( ~ Type, ncol=1)

ggplot(df.sub, aes(x=LVar, y=LAve, color=Type)) + 
  geom_point()

dat2 = dat %>% 
  filter(Type=="Rock" | Type=="Classical" | Type=="New wave")
head(dat2)

              Artist Type     LVar      LAve  LMax   LFEner     LFreq
Dancing Queen   Abba Rock 17600756 -90.00687 29921 105.9210  59.57379
Knowing Me      Abba Rock  9543021 -75.76672 27626 102.8362  58.48031
Take a Chance   Abba Rock  9049482 -98.06292 26372 102.3249 124.59397
Mamma Mia       Abba Rock  7557437 -90.47106 28898 101.6165  48.76513
Lay All You     Abba Rock  6282286 -88.95263 27940 100.3008  74.02039
Super Trouper   Abba Rock  4665867 -69.02084 25531 100.2485  81.40140

GGally::ggparcoord(dat2, columns=3:7,
                   groupColumn = "Type",
                   title="Parallel Coordinate Plot: Music Types")

GGally::ggparcoord(dat2, columns=c(4,3,5,6,7),
                   groupColumn = "Type",
                   title="Parallel Coordinate Plot: Music Types")