PSTAT 10 Data Science Principles

Lecture 8: Plotting

John Robin Inston

johninston@ucsb.edu

University of California, Santa Barbara

August 22, 2026

Introduction

🔁 Review: Lecture 7

👈 Last lecture we studied the following topics:

• Tibble Objects;
• Tibble Manipulation;
  • Selecting;
  • Filtering;
  • Mutating;
• Reading Data
  • .txt;
  • .csv; and
  • .xlsx.

👀 Outline: Lecture 8

👇 Today we will continue looking at describing and visualizing external data, specifically:

• Descriptive statistics
• Base R Plotting
• Probability
• Discrete and Continuous Random Variables

Exploring Real Data

📄 .csv Files

File Types Recap

Last lecture we looked at several different file types that data can be stored as, including:

• .txt files
• .csv files
• .xlsx files

Why .csv?

In this course we shall be using .csv files almost exclusively, and so we typically will only need the read.csv() function to load real world data.

Loading Michelson.csv

The Michelson data is a famous dataset measuring the speed of light. I keep my data in a data subfolder, hence the path name below.

Michelson <- read.csv("data/Michelson.csv")
head(Michelson, 3)

  ds12 ds13 ds14 ds15 ds16
1  850  740  900 1070  930
2  850  950  980  980  880
3 1000  980  930  650  760

🌪️ Messy Data Problems

The Reality of Real Data

Real data is rarely clean and well behaved. It is often messy and unintuitive, and it is our job as data scientists and statisticians to understand, interpret and draw meaning from the mess.

Key Tools

To accomplish this goal we begin with several key tools:

• Summary Statistics
• Graphical Tools
• Statistical Measures (Correlation, Standard Deviation)
• Prior Knowledge
• Qualitative Analysis (understanding where the data comes from)

📖 Statistics Terminology

Population & Sample

Data consists of information from observations, counts, measurements or responses.

• The population is the complete collection of all observations.
• The sample is a representative group selected from a population.

Parameters & Statistics

• A parameter describes a population characteristic; a statistic describes a sample characteristic.
• Descriptive statistics summarize data.
• Inferential statistics test hypotheses (PSTAT 126).

📊 Summary Statistics

Summary statistics are the key statistics of a sample that we always generate to get an idea about the shape and location of the data.

Key Statistics

1. Minimum
2. Lower Quartile
3. Median
4. Mean
5. Upper Quartile
6. Maximum
7. Spread Measures (Standard Deviation, Variance)
8. Outliers
9. Variable Correlation

💪 Exercise — Summary Statistics

02:00

The summary() function provides the first 6 statistics. The correlation and standard deviation are given by the cor() and sd() functions respectively.

1. Change the Michelson object to a tibble using the function tibble().
2. Select the column ds16 using the select() function and apply the function summary().

✅ Solution — Summary Statistics

• Convert Michelson to a tibble using tibble().
• Select the ds16 column using select().
• Apply summary() to obtain the key summary statistics.

Michelson <- Michelson |> tibble()

Michelson |>
  select(ds16) |>
  summary()

      ds16      
 Min.   :720.0  
 1st Qu.:795.0  
 Median :840.0  
 Mean   :839.5  
 3rd Qu.:880.0  
 Max.   :960.0  

📦 Boxplots

Boxplots are a graphical visualization of most of the information provided by the summary() function.

boxplot(iris[,-5], col = 2:5, xlab="Length/Width",
        main="Boxplot of Iris Variables")

📈 Descriptive Statistics

Descriptive statistics help to summarize key characteristics of the data such as:

• its location
• its spread
• if its distribution has one peak (unimodal) or many peaks
• how sharp these peaks are (kurtosis)
• whether the data is skewed one way or the other

📌 In this course we only consider location and spread.

⚖️ Measures of Centrality & Spread

Measures of Centrality

• Arithmetic average or mean — use the function mean()
• Data midpoint, a.k.a. the median — use the function median()
• Most frequent value, a.k.a. the mode — use the function mode()

Measures of Spread

• Range (difference between maximum and minimum) — use the function range()
• Interquartile Range (difference between upper and lower quartile)
• Variance and Standard Deviation — use the functions var() and sd()

🧮 Sample Mean & Standard Deviation

Sample Mean

The sample mean is just the arithmetic average. For a sample \((x_1, ..., x_n)\) the sample mean \(\bar{x}\) is given by

\[ \bar{x} = \frac{1}{n}\sum_{i=1}^nx_i=\frac{x_1 +x_2 + \cdots + x_n}{n} \]

📌 This is only an estimate of the (true) population mean, which we denote \(\mu\).

Sample Standard Deviation

The sample standard deviation \(s\) is:

\[ s = \sqrt{\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2} \]

📌 This estimates the (true) population standard deviation \(\sigma\) (recall \(\sigma^2\) is the variance).

🔗 Relation Measures

📌 Relation measures describe how two variables change together — i.e. how does one variable change if we change the value of another?

Intuition

• Both increase together → positive relationship (e.g. height & weight).
• One increases as the other decreases → negative relationship (e.g. price & demand).
• No consistent pattern → unrelated (e.g. shoe size & IQ).

Formal Measures

Covariance measures the joint variability of two variables, and leads to correlation — a measure of the strength of their linear relationship.

cov(iris$Petal.Length, iris$Petal.Width)

[1] 1.295609

cor(iris$Petal.Length, iris$Petal.Width)

[1] 0.9628654

📈 Example — Correlation Plot

Scatter plot with line of best fit.

🧮 Covariance & Correlation

Covariance

The covariance between two variables \(x:= (x_1, ..., x_n)\) and \(y:= (y_1, ..., y_n)\) is given by

\[ \text{Cov}(x,y)=\frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})(y_i-\bar{y}) \]

📌 The covariance can be negative, which indicates a negative linear relationship (one goes up means the other goes down).

Correlation

From the definition of covariance we naturally define correlation, denoted by the Greek letter rho \(\rho\), of two variables \(x\) and \(y\) as

\[ \rho_{XY} = \frac{\text{Cov}(x,y)}{s_x \cdot s_y} \]

We can compute the correlation between two vectors in R using the function cor().

💪 Exercise — Computing Summary Statistics

05:00

We shall be considering the faithful.csv dataset, which contains a list of waiting times between eruptions and the duration of eruptions for the Old Faithful geyser in Yellowstone National Park.

1. Download the data from Canvas and save it to your working directory.
2. Load the data into R and save it to the object faithful. Convert this to a tibble() object.
3. Using the functions mean() and sd(), compute the mean of eruptions and the standard deviation of waiting.
4. Use the functions cov() and cor() to compute both the covariance and the correlation between eruptions and waiting.

✅ Solution — Computing Summary Statistics

• Load faithful.csv and convert it to a tibble.
• Compute the mean of eruptions and the standard deviation of waiting using mean() and sd().
• Compute the covariance and correlation between eruptions and waiting using cov() and cor().

# load data
faithful <- read.csv("data/faithful.csv") |> tibble()
# mean
faithful$eruptions |> mean()
# standard deviation
faithful$waiting |> sd()
# covariance
cov(faithful$eruptions, faithful$waiting)
# correlation
cor(faithful$eruptions, faithful$waiting)

[1] 3.487783

[1] 13.59497

[1] 13.97781

[1] 0.9008112

⚠️ Problems with Descriptive Statistics

The Problem

It is very important to practice caution when using summary and descriptive statistics. They are very simplistic and are easily fooled by more complex data patterns.

Example Data

To demonstrate some of the problems you may run into we consider problem.csv, which can be found on Canvas.

# load problem data
problem <- read.csv("data/problem.csv") |>
  tibble()
# inspect data
problem |> head(8)

# A tibble: 8 × 4
      x     y     a     b
  <int> <int> <int> <dbl>
1     5    45     1 0    
2     2    50     2 0.788
3     5    47     3 1.10 
4     5    46     4 1.39 
5     0    46     5 1.61 
6     5    55     6 1.79 
7     1    54     7 1.95 
8     0    46     8 2.08 

⚠️ Interpreting the Summary

Applying summary() to problem we obtain the following summary statistics:

summary(problem)

       x                y                 a               b        
 Min.   :  0.00   Min.   :   41.0   Min.   : 1.00   Min.   :0.000  
 1st Qu.:  2.25   1st Qu.:   49.0   1st Qu.:12.25   1st Qu.:2.505  
 Median : 46.50   Median :   50.5   Median :21.50   Median :3.068  
 Mean   : 48.87   Mean   :  405.3   Mean   :22.37   Mean   :2.855  
 3rd Qu.: 97.75   3rd Qu.:   53.0   3rd Qu.:32.75   3rd Qu.:3.489  
 Max.   :100.00   Max.   :16368.0   Max.   :44.00   Max.   :3.784  

We might make the following quick observations:

• The variable x is clustered around 48.
• The variable y is clustered around 405.

cor(problem$a, problem$b)

[1] 0.9102097

Furthermore we might conclude that there is a significant linear relationship between variables a and b, as their correlation is 0.91.

⚠️ A Closer Look — Histogram

A closer look at x and y:

par(mfrow = c(1,2))
hist(problem$x, breaks = 30)
hist(problem$y, breaks = 30)

⚠️ A Closer Look — Scatter Plot

The relationship between a and b does not appear to be linear!

Plotting in Base R

🖼️ Types of Plots

There are a wide variety of plots which all have their uses for data visualization and interpretation, including:

• Scatterplots
• Boxplots
• Histograms
• Barcharts
• Pie Charts
• Q-Q plots

Example Data

For an example we consider the airquality data in the datasets library.

# load data
airquality <- datasets::airquality |>
  tibble()
airquality |> head(3)

# A tibble: 3 × 6
  Ozone Solar.R  Wind  Temp Month   Day
  <int>   <int> <dbl> <int> <int> <int>
1    41     190   7.4    67     5     1
2    36     118   8      72     5     2
3    12     149  12.6    74     5     3

😬 An “Unprofessional” Plot

A Basic Scatter Plot

We produce a scatter plot of Wind against Temp using the plot() function.

plot(airquality$Wind, airquality$Temp)

A very basic scatter plot!

• The axes are not labelled and there is no title.
• There is no formatting.

🎨 A More Professional Plot

This is fine but not very professional. Fortunately, plot() has a lot of arguments we can use to tune our plot.

plot(airquality$Wind, airquality$Temp,
  main = "Wind and Temperature in NYC, 1973",
  xlab = "Wind (mph)",
  ylab = "Temperature (degrees F)",
  pch = 20,
  col = "darkgreen",
  panel.first = grid()
)

A more professional plot!

🎨 Further Improvements

What We Changed

• Defined a plot title using main="...";
• Defined x and y axis labels using xlab="..." and ylab="...";
• Formatted our point type to be solid pch=20 and dark green col="darkgreen".

Line of Best Fit

We also might want to overlay a line of best fit using the linear model lm() function.

We will not cover the lm() function in this course, but in summary it fits a linear regression model between the x and y variables and we use this model to plot the line of best fit.

We add this line to our plot using the function abline().

📉 Line of Best Fit

# A more professional plot with best fit line
plot(airquality$Wind, airquality$Temp,
  main = "Wind and Temperature in NYC, 1973",
  xlab = "Wind (mph)",
  ylab = "Temperature (degrees F)",
  pch = 20,
  col = "darkgreen",
  panel.first = grid()
)
abline(lm(airquality$Temp ~ airquality$Wind), col = "red")

📉 Line of Best Fit

A more professional plot with a line of best fit!

📐 Formula Notation

Notice in lm() we used R’s formula notation y ~ x. It is recommended to use formula notation for your plots as it makes your code clearer.

Key Syntax

• y ~ x reads as “y modeled by x” — the response goes on the left of ~, predictors go on the right.
• The data = ... argument lets you refer to columns by name directly, without repeating dataframe$ in front of every variable.
• Multiple predictors can be combined with +, e.g. y ~ x1 + x2.

# A more professional plot with best fit line
plot(Temp ~ Wind, data = airquality,
  main = "Wind and Temperature in NYC, 1973",
  xlab = "Wind (mph)",
  ylab = "Temperature (degrees F)",
  pch = 20,
  col = "darkgreen"
)
grid()
abline(lm(airquality$Temp ~ airquality$Wind), col = "red")

📐 Formula Notation

A more professional plot with a line of best fit!

〰️ Plotting Continuous Functions

Discretizing

• Computers think in discrete terms and cannot represent continuous time.
• To plot continuous functions, e.g. sin() or exp(), we discretize them — redefining the interval as a sequence with very fine discrete jumps.
• For example, on the interval \([0,10]\) we could define:

x <- seq(0, 10, by = 0.01)
head(x)

[1] 0.00 0.01 0.02 0.03 0.04 0.05

Cosine Wave Example

• We compute cos(x) and specify type="l", which plots a smooth line between the points rather than the points themselves.

plot(x, cos(x),
  main = "Cosine Wave",
  xlab = "x",
  ylab = "cos(x)",
  col = "darkgreen",
  type = "l"
)

〰️ Plotting Continuous Functions

Cosine Wave

➕ Overlaying Lines

Using lines()

• We can overlay several lines one on top of the other using the lines() function.
• lines() takes broadly the same arguments as plot() (e.g. col, type, lty).

Adding a Legend

• We add a legend using the function legend().
• The first argument (e.g. "bottomleft") sets the position of the legend.
• legend = c(...) gives the labels for each line.
• col = ... and lty = ... should match the plotted lines so the legend correctly identifies each one.

# A more professional plot with best fit line
plot(x, cos(x),
  main = "Cosine Wave",
  xlab = "x", ylab = "f(x)",
  col = "darkgreen", type = "l"
)
lines(x, sin(x), col = "red")
legend("bottomleft",
  legend = c("sin(x)", "cos(x)"),
  col = c("darkgreen", "red"), lty = 1)

➕ Overlaying Lines

Cosine & Sin Waves

🔁 Iterating Function Plotting

For a bit of fun we conclude by demonstrating how a for loop can be used to iteratively produce complicated plots:

• We first plot the base cosine wave, then loop i from 1 to 100.
• Each iteration draws a phase-shifted sine wave sin(x + i/10) using lines().
• Setting col = i cycles through R’s default color palette, giving each new line a different color.
• The result is a dense fan of overlapping waves, all built up from a single simple plotting call repeated inside the loop.

# A more professional plot with best fit line
plot(x, cos(x),
  main = "Cosine Waves",
  xlab = "x",
  ylab = "f(x)",
  col = "darkgreen",
  type = "l"
)
for(i in 1:100){
  lines(x, sin(x+(i/10)),
  col = i)
}

🔁 Iterating Function Plotting

Lots and lots of waves!

💪 Exercise — Plotting in R

05:00

Produce plots for \(x^3\) and \(e^x\) for x between 1 and 10.

1. Define your x as a discretized sequence of values.
2. Define x_cubed as \(x^3\) and exp_x as \(e^x\).
3. Construct a plot of x against x_cubed, adjusting formatting as desired.
4. Add an additional line to the plot representing \(e^x\).
5. Add an appropriate legend.

✅ Solution — Plotting in R

• Discretize x between 1 and 10, then define x_cubed as \(x^3\) and x_exp as \(e^x\).
• Plot x against x_cubed as a line, then overlay x_exp using lines().
• Add a legend identifying each line.

# define variables
x <- seq(1, 10, by = 0.01)
x_cubed <- x^3
x_exp <- exp(x)
# produce plot
plot(
  x, x_cubed,
  main = "Exercise Plot",
  xlab = "x", ylab = "f(x)",
  type = "l", col = "darkgreen"
)
lines(x, x_exp, col = "red", lty = 2)
legend(
  "topleft",
  legend = c("x^3", "exp(x)"),
  col = c("darkgreen", "red"), lty = 1:2
)

✅ Solution — Plotting in R

Exercise Plot

Summary

✅ Topics Covered

🤔 Today we looked into:

• Descriptive statistics
• Base R Plotting

📅 Next Class

🤩 Next class we will temporarily leave behind data science topics as we study probability theory! The next four classes will be (roughly) split as follows:

1. Introduction to Probability
2. Discrete Random Variables
3. Continuous Random Variables
4. Monte Carlo Methods