STA2023 Review:
Point Estimation
Data Visualization

June 17, 2025
Tuesday

Introduction: Topics

  • Basic descriptives:
    • Continuous variables:
      • Mean
      • Median
      • Variance and standard deviation
      • Range and interquartile range
    • Categorical variables:
      • Count
      • Overall percentage
      • Row percentage
      • Column percentage

Introduction: Data

  • We will be using data from the kingdom of Equestria (yes, from My Little Pony).

  • Mane Six:

    • Twilight Sparkle (Unicorn \to Alicorn)
    • Applejack (Earth Pony)
    • Fluttershy (Pegasus)
    • Pinkie Pie (Earth Pony)
    • Rainbow Dash (Pegasus)
    • Rarity (Unicorn)

Introduction: Data

Introduction: Data

  • Name: the pony’s name
  • Type: type of pony (Earth, Pegasus, Unicorn, Alicorn)
  • Sex: sex/age of pony (Coal, Filly, Stallion, Mare)
  • Flying speed: average flying speed (km/hr) for winged ponies
  • Friendship: a harmony index from friendship activities (0-10)
  • Magical energy: measured magical energy output (sparkles) for magical ponies
  • Tail shimmer: how much light reflected by the pony’s tail (lux)

Types of Variables: Qualitative

  • A qualitative or categorical variable classifies an observation into one of two or more groups or categories.
    • Nominal: purely qualitative and unordered
    • Ordinal: data can be ranked, but intervals between ranks may not be equivalent
  • Examples:
    • satisfaction rating
    • favorite color
    • type of pet
    • education level
    • blood type

Types of Variables: Quantitative

  • A quantitative or continuous variable takes numerical values for which arithmetic operations such as adding and averaging make sense; typically has a unit of measure.
    • Interval: meaningful differences between values, but no true zero point
    • Ratio: meaningful differences and a true zero point
  • Examples:
    • age (years)
    • temperature (Celsius)
    • daily hours of sleep
    • ACT or SAT score
    • height (inches)

Types of Variables: Example

  • Name: the pony’s name
  • Type: type of pony (Earth, Pegasus, Unicorn, Alicorn)
  • Sex: sex/age of pony (Coal, Filly, Stallion, Mare)
  • Flying speed: average flying speed (km/hr) for winged ponies
  • Friendship: a harmony index from friendship activities (0-10)
  • Magical energy: measured magical energy output (sparkles) for magical ponies
  • Tail shimmer: how much light reflected by the pony’s tail (lux)

Describing Data: Why?

  • Why do we describe data? We want to tell a story!
    • Summarize n observations into a single description
    • Understand what is in the data
    • Spot patterns, missing data, or outliers
    • Compare groups or spot differences or oddities

Describing Data: How?

  • How do we describe data?
    • Numbers
      • Frequency table
      • Mean & standard deviation
      • Median & IQR
    • Graphs
      • Bar charts
      • Box plots
      • Histograms

Point Estimation: Mean

  • Mean: the average of a set of values

\bar{y} = \frac{\sum_{i=1}^n y_i}{n}

  • Find the mean for the flying speeds (km/hr) of 5 ponies: {10, 20, 30, 40, 100}

\bar{y} = \frac{\sum_{i=1}^n y_i}{n} = \frac{10 + 20 + 30 + 40 + 100}{5} = 40

  • The average flying speed for winged ponies is 40 km/hr.

Point Estimation: Median

  • Median: The middle value in an ordered dataset.
    • When we have an even number of observations, we average the two middle.
  • Find the median for the flying speeds (km/hr) of 5 ponies: {10, 20, 30, 40, 100}
    • First, we sort the data: {10, 20, 30, 40, 100}
    • Then, find the middle number: 30
  • The median flying speed for winged ponies is 30 km/hr.

Point Estimation: Variance

  • Variance: A measure of spread; the average of squared differences from the mean.
    • Higher variance = data has more spread.
    • In squared units of the data.

s_y^2 = \frac{\sum_iy_i^2 - (\sum_iy_i)^2/n}{n-1}

  • Find the variance for the flying speeds (km/hr) of 5 ponies: {10, 20, 30, 40, 100}

s_y^2 = \frac{\sum_iy_i^2 - (\sum_iy_i)^2/n}{n-1} = \frac{(10^2+...+100^2)-(10+...+100)^2/5}{4} = 1250

  • The variance is 1250 (km/hr)2

Point Estimation: Standard Deviation

  • Standard Deviation: A measure of spread; the average distance from the mean.
    • Higher standard deviation = data has more spread.
    • Same units as the data.

s_y = \sqrt{s_y^2}

  • Find the standard deviation for the flying speeds (km/hr) of 5 ponies: {10, 20, 30, 40, 100}

s_y = \sqrt{s^2_y} = \sqrt{1250} \approx 35.36

  • The standard deviation is 35.36 km/hr.

Point Estimation: Range

  • Range: difference between the maximum and minimum values

\text{range} = \text{max}(y) - \text{min}(y)

  • Find the range for the flying speeds (km/hr) of 5 ponies: {10, 20, 30, 40, 100}

\begin{align*} \text{range} = \text{max}(y) - \text{min}(y) = 100 - 10 = 90 \end{align*}

  • The range of the flying speeds is 90 km/hr.

Point Estimation: Interquartile Range

  • Interquartile Range (IQR): range of the middle 50% of the data.

\text{IQR} = \text{P}_{75} − \text{P}_{25}

  • Find the IQR for the flying speeds (km/hr) of 5 ponies: {10, 20, 30, 40, 100}
    • Recall that the median is 30.
    • We then find P_{25} using {10, 20} and P_{75} using {40, 100}
    • Thus, P_{25} = 15 and P_{75} = 70.

\begin{align*} \text{IQR} = \text{P}_{75} − \text{P}_{25} = 70 - 15 = 55 \end{align*}

  • The IQR of the flying speeds is 55 km/hr.

Point Estimation: Proportion

  • Proportion: a type of mean for categorical data
    • Often expressed as a percentage
    • Useful for categorical responses

\hat{p} = \frac{\sum_{i=1}^n y_i}{n},

  • Note that in this case,

y_i = \begin{cases} 1 & \text{if in category }i \\ 0 & \text{otherwise} \end{cases}

Point Estimation: Proportion

  • Find the proportion of ponies that have wings in the following sample: {Y, N, Y, Y, N, Y}

  • Count the number of “Y” responses and divide by total:

\hat{p} = \frac{\sum_{i=1}^n y_i}{n} = \frac{4}{6} \approx 0.67

  • The proportion of ponies with wings is 0.667 (or 66.7%).

Point Estimation: Frequency Table

  • Frequency table: A table showing how often each value appears in a dataset.
    • Useful for categorical responses.
    • For each category, i, we report n_i (\%_i)
  • Find the freqency table for the following sample of 8 ponies: {Earth, Pegasus, Unicorn, Earth, Pegasus, Pegasus, Unicorn, Alicorn}
  • Frequencies:
    • Alicorn: n_{\text{A}} = 1
    • Earth: n_{\text{E}} = 2
    • Pegasus: n_{\text{P}} = 3
    • Unicorn: n_{\text{U}} = 2
  • Proportions:
    • Alicorn: \hat{p}_{\text{A}} = 1/8 = 0.125
    • Earth: \hat{p}_{\text{E}} = 2/8 = 0.250
    • Pegasus: \hat{p}_{\text{P}} = 3/8 = 0.375
    • Unicorn: \hat{p}_{\text{U}} = 2/8 = 0.250

Point Estimation: Frequency Table

  • Putting this into a table,

Point Estimation: Contingency Table

  • Contingency table: A table that summarizes two qualitative variables and their overlap.

  • We will not concern ourselves with the derivation, but will rely on R.

  • Consider this data,

Point Estimation: Contingency Table

  • The resulting contingency table would look someting like this:
    • We are using column totals as our denominators.
# A tibble: 4 × 3
  pony_type No        Yes      
  <chr>     <chr>     <chr>    
1 Alicorn   0 (0.0%)  1 (25.0%)
2 Earth     2 (50.0%) 0 (0.0%) 
3 Pegasus   0 (0.0%)  3 (75.0%)
4 Unicorn   2 (50.0%) 0 (0.0%)

Graphs: Box Plots

  • Box plots display the distribution of a continuous variable using the five number summary:

    • Whisker: Minimum
    • Beginning of box: 25th percentile (first quartile; Q1, P25)
    • “Middle” of box: Median (50th percentile, second quartile; Q2, P50)
    • End of box: 75th percentile (third quartile; Q3, P75)
    • Whisker: Maximum

  • We use box and whisker plots to get an idea of the spread and skewness of the data.

  • Note: there are different ways to define the whiskers.

    • I use the min/max as whiskers when sketching by hand.
    • ggplot() uses 1.75 \times IQR.

Graphs: Box Plots

  • Describe this box plot:

Graphs: Box Plots

  • Describe this box plot:

Graphs: Box Plots

  • Describe this box plot:

Graphs: Box Plots

  • Describe this box plot:

Graphs: Histograms

  • Histograms show the distribution of a continuous variable.

    • What is the shape of the distribution?
    • Is the distribution symmetric? Skewed? How skewed?

  • Values are grouped into intervals (“bins”), then the bin height demonstrates how many values fall into that interval.

  • This allows us to quickly see if there are any oddities.

    • Increased proportion of a specific value/bin.
      • Zero inflation? Value used to indicate missing?
    • Any values that are “out in the tail”.
      • Outlier? Data entry error?

Graphs: Histograms

  • Describe the histogram:

Graphs: Histograms

  • Describe the histogram:

Graphs: Histograms

  • Describe the histogram:

Graphs: Histograms

  • Describe the histogram:

Graphs: Histograms

  • Describe the histogram:

Graphs: Bar Graphs

  • Bar graphs display the distribution of categorical data.
    • The frequency or proportion of observations is displayed on the bar graph.
  • Bar graphs usually have categories on the x-axis and counts or proportions on the y-axis.
    • Note that we could flip the axes to create a vertical bar graph.
  • Note that the bars are separated on the x-axis to indicate the lack of continuity.

Graphs: Bar Graphs

  • Consider the bar graph, below.

Graphs: Side-by-Side Bar Graphs

  • Consider the bar graph, below.

Graphs: Stacked Bar Graphs

  • Consider the bar graph, below.

Graphs: Histograms vs Bar Graphs

  • We have now reviewed two “bar style” graphs that we see regularly: histograms and bar graphs.

  • We use histograms to see the distribution of continuous variables.

    • The x-axis represents numeric intervals.
    • The bars touch each other to represent continuity.

  • We use bar graphs to see the distribution of categorical variables.

    • The x-axis represents categories.
    • The bars do not touch each other, implying distinct categories.

Graphs: Scatterplots

  • Scatterplots allow us to look at the relationship between two continuous variables.
    • Each point on the graph represents one observation.
  • What statisticians use scatterplots for:
    • Explore patterns (aka trends or relationships).
      • Linear relationships.
      • Non-linear relationships.
    • Detect clusters of observations.
    • Find oddities in the data (outliers).
  • When we describe the relationship, we are really answering the question, “As x increases, what happens to y?”

Graphs: Scatterplots

  • Consider the scatterplot, below.

Graphs: Scatterplots

  • Consider the scatterplot, below.

Graphs: Scatterplots

  • Consider the scatterplot, below.

Graphs: Scatterplots

  • Consider the scatterplot, below.

Graphs: Scatterplots

  • Consider the scatterplot, below.

Break time!

Introduction to R

  • In this course, we will review formulas, but we will use R for computational purposes.

    • Remember to refer to the lecture notes for specific code needed.
    • Code is also available on this course’s GitHub repository.

  • You can install R and RStudio if you wish; both are free.

  • We also have access to the Posit Workbench (“the server”) through HMCSE.

  • I know that this is probably the first time you are seeing R (or any sort of programming).

    • That is why we have “R lab” time built in to our course.
    • Remember that I am not looking for perfection, but for competency.

Introduction to R

  • Please download today’s activity from Canvas and log into the server.
    • Click on “New session”
    • Click on “Create session”
  • Upload today’s activity to the server:
    • In the bottom right pane, click on the white square with a golden up arrow on in
    • Click on “Choose file”
    • Select the downloaded file
    • Click “Open”
    • Click “OK”
  • Open today’s activity on the server:
    • In the bottom right pane, scroll to the bottom
    • Click on the name of the .qmd file for today’s activity

Introduction to R: .R scripts

  • .R scripts:
    • Only allows code
      • Can comment out code using pound sign
    • Can run code line-by-line
    • Can run multiple lines of code at a time
    • Results output to Console window pane (bottom left)
  • I use .R scripts for my day-to-day analyses

Introduction to R: .qmd file

  • .qmd files:
    • Allow both text and code
      • Can comment out text using html code
      • Can comment out code in chunk using pound sign
    • Uses “code chunks” to evaluate code
      • Button: run all chunks before
      • Button: run this chunk
      • Ctrl+enter / cmd+return: line-by-line
    • Rendering results in .html file
  • I use .qmd files to create sharable documents
    • Reproducible research forever and always

Introduction to R: Disclaimer!

  • My major disclaimer as a biostatistician: I am a statistician first, programmer second.
    • My expertise is in statistics, not programming.
    • I do not know everything about R.
    • I do not claim to write the most efficient code.
    • Our goal is to correctly apply statistics to answer research questions using data.
      • R is a tool for us to apply statistics.
  • My major disclaimer as a professor: yes, I know this is likely the first time you are seeing R or programming in general.
    • We have “R lab” time built into the course.
    • Code you need to answer questions will be provided in lecture.
      • This means you must revisit lecture slides to find the code you need.

Introduction to R: Dr. Seals’s Expectations

  • I expect students to try their best. This includes:

    • referring back to lectures as needed.
    • asking when you have a question.
    • using the resources provided to learn.

  • You must know how to answer questions using R.

  • You will not be expected to write code beyond what is shown in class.

    • Note: Sometimes I include bonus questions…

  • When grading, I am looking for competency.

    • What is the appropriate analysis for the question at hand?
    • What are the assumptions of the analysis? Do we meet them?
    • Are the correct conclusions drawn given the information provided?

Functions in R: base R vs. packages

  • R functions are like baking recipes. They:
    • Take input (ingredients, or data),
    • Does something with it (follow recipe, or perform calculations),
    • Gives back a result (baked good, or statistics).
  • Some functions in R are available as soon as you open RStudio (this is “base R”).
    • e.g., mean(), sd()
  • Other functions are not available and must be called in after you start RStudio (these are “packages”).
    • e.g., after I call in library(tidyverse), I can use summarize(mean(), sd())
    • We will always need library(tidyverse) because of %>% (pipe operator).

Functions in R: tidyverse

  • library(tidyverse) is a collection of R packages designed for data science.
    • All packages share a common philosophy and are meant to work together.
    • This is ideal for using the “same syntax” - I promise it’s better than base R!
  • Core library(tidyverse) packages we will use:
    • library(readr): read in data files
    • library(dplyr): manipulate and summarize data
    • library(ggplot2): create data visualizations
  • If you are interested, there are resources:

Functions in R: Summarizing Continuous Data

  • We will use mean_median() from library(ssstats) to summarize continuous variables.
    • It will return both the mean (standard deviation) and median (IQR).
dataset_name %>% 
  mean_median(var1, var2, ...)
  • We can add group_by() from library(tidyverse) to split the summaries by categories.
dataset_name %>% 
  group_by(grouping_var1, grouping_var2, ...) %>% 
  mean_median(var1, var2, ...)

Functions in R: Summarizing Continuous Data

  • Let’s use mean_median() to summarize the MLP dataset.
mlp_data %>% 
  mean_median(friendship, tail_shimmer, magical_energy)

Functions in R: Summarizing Continuous Data

  • Let’s use mean_median() to summarize the MLP dataset.
mlp_data %>% 
  mean_median(friendship, tail_shimmer, magical_energy)
# A tibble: 3 × 3
  variable       mean_sd      median_iqr   
  <chr>          <chr>        <chr>        
1 friendship     7.6 (1.6)    8.0 (2.0)    
2 magical_energy 9.9 (9.6)    7.0 (11.1)   
3 tail_shimmer   256.6 (65.7) 253.0 (103.0)

Functions in R: Summarizing Continuous Data

  • Let’s use mean_median() to summarize the MLP dataset by pony type.
mlp_data %>% 
  group_by(type) %>%
  mean_median(friendship, tail_shimmer, magical_energy)

Functions in R: Summarizing Continuous Data

  • Let’s use mean_median() to summarize the MLP dataset by pony type.
mlp_data %>% 
  group_by(type) %>%
  mean_median(friendship, tail_shimmer, magical_energy)
# A tibble: 12 × 4
   type    variable       mean_sd      median_iqr   
   <chr>   <chr>          <chr>        <chr>        
 1 Alicorn friendship     7.8 (1.3)    8.0 (2.0)    
 2 Earth   friendship     7.6 (1.6)    8.0 (2.0)    
 3 Pegasus friendship     7.6 (1.6)    8.0 (2.0)    
 4 Unicorn friendship     7.5 (1.6)    8.0 (2.0)    
 5 Alicorn magical_energy 9.0 (8.0)    6.2 (10.9)   
 6 Earth   magical_energy NaN (NA)     NA (NA)      
 7 Pegasus magical_energy NaN (NA)     NA (NA)      
 8 Unicorn magical_energy 9.9 (9.6)    7.0 (11.1)   
 9 Alicorn tail_shimmer   280.1 (64.5) 297.0 (104.0)
10 Earth   tail_shimmer   252.2 (65.2) 246.0 (100.0)
11 Pegasus tail_shimmer   263.5 (67.0) 265.0 (110.0)
12 Unicorn tail_shimmer   261.2 (65.0) 260.0 (100.0)

Functions in R: Summarizing Categorical Data

  • We will use n_pct() from library(ssstats) to summarize categorical variables.

  • For one variable – this returns n_i \ (\%_i):

dataset_name %>% 
  n_pct(var1)
  • For two variables – this returns n_{ij} \ (\%_{\text{col}}):
dataset_name %>% 
  n_pct(var1, var2)

Functions in R: Summarizing Categorical Data

  • Let’s use n_pct() to summarize the MLP dataset.
mlp_data %>% 
  n_pct(type, rows = 4)

Functions in R: Summarizing Categorical Data

  • Let’s use n_pct() to summarize the MLP dataset.
mlp_data %>% 
  n_pct(type, rows = 4)
    type      n (pct)
 Alicorn    41 (1.4%)
   Earth 1678 (58.4%)
 Pegasus  487 (17.0%)
 Unicorn  665 (23.2%)

Functions in R: Summarizing Categorical Data

  • Let’s use n_pct() to summarize the MLP dataset.
mlp_data %>% 
  n_pct(friendship, type, rows = 4)

Functions in R: Summarizing Categorical Data

  • Let’s use n_pct() to summarize the MLP dataset.
mlp_data %>% 
  n_pct(friendship, type, rows = 4)
# A tibble: 4 × 5
  friendship Alicorn  Earth     Pegasus   Unicorn  
       <dbl> <chr>    <chr>     <chr>     <chr>    
1          1 0 (0.0%) 0 (0.0%)  0 (0.0%)  1 (0.2%) 
2          2 0 (0.0%) 5 (0.3%)  1 (0.2%)  1 (0.2%) 
3          3 0 (0.0%) 14 (0.8%) 6 (1.2%)  13 (2.0%)
4          4 2 (4.9%) 54 (3.2%) 14 (2.9%) 16 (2.4%)

Graphs in R: Using ggplot()

  • We will construct data visualizations using library(ggplot2), which loads in when we load library(tidyverse).

  • This package allows us to create a layered visualization.

    • ggplot() creates the base layer.
    • geom_X() creates the individual pieces.
      • geom_point() creates a scatterplot.
      • geom_line() creates connected lines.
      • geom_bar() creates a bar chart.
      • geom_histogram() creates a histogram.

Graphs in R: Using ggplot()

  • We use ggplot() because it is very flexible - it allows us to customize every part of the graph.

    • Note that customization is less important in this course, but incredibly important in real life.

  • The R Graphics Cookbook is a great place to get basic code for graphs.

  • Remember that I do not expect you to memorize code. I do not have the code memorized.

    • Things I regularly ask Google for help with:
      • How to suppress the legend.
      • How to specify the tickmarks on the axis.
      • How to change the font size.

Graphs in R: The ggplot() Layer

  • Calling ggplot() creates the initial layer the graph lasagna.
mlp_data %>% ggplot()

Graphs in R: The ggplot() Layer

  • We specify the aesthetics through aes() in ggplot().
mlp_data %>% ggplot(aes(x = tail_shimmer, y = flying_speed))

Graphs in R: Overriding Defaults

  • We can override plot defaults using additional layers.
mlp_data %>% 
  ggplot(aes(x = tail_shimmer, y = flying_speed)) +
  labs(x = "Tail Shimmer",
       y = "Flying Speed") +
  theme_bw()

Graphs in R: Box Plots

  • Construct a box plot for the tail shimmer of the ponies (tail_shimmer).
mlp_data %>% ggplot(aes(x = tail_shimmer)) +
  geom_boxplot() +
  labs(x = "Tail Shimmer") +
  theme_bw() +
  theme(axis.ticks.y = element_blank(),
        axis.text.y = element_blank())

Graphs in R: Box Plots

  • Construct a box plot for the tail shimmer of the ponies (tail_shimmer).

Graphs in R: Box Plots

  • Construct a box plot for the tail shimmer of the ponies (tail_shimmer).
mlp_data %>% ggplot(aes(y = tail_shimmer)) +
  geom_boxplot() +
  labs(y = "Tail Shimmer") +
  theme_bw() +
  theme(axis.ticks.x = element_blank(),
        axis.text.x = element_blank())

Graphs in R: Box Plots

  • Construct a box plot for the tail shimmer of the ponies (tail_shimmer).

Graphs in R: Histograms

  • Construct a histogram for the flying speed of ponies (flying_speed).
mlp_data %>% ggplot(aes(x = flying_speed)) +
  geom_histogram(bins = 15, 
                 color = "#2E7D32", 
                 fill = "#4CAF50") +
  labs(x = "Flying Speed", 
       y = "Number of Ponies") +
  theme_bw()

Graphs in R: Histograms

  • Describe the histogram of the flying speed of ponies (flying_speed):

Graphs in R: Histograms

  • Construct a histogram for the magical energy of ponies (magical_energy).
mlp_data %>% ggplot(aes(x = magical_energy)) +
  geom_histogram(bins = 15, 
                 color = "#8B6C42", 
                 fill = "#F0E9DD") +
  labs(x = "Magical Energy",
       y = "Number of Ponies") +
  theme_bw()

Graphs in R: Histograms

  • Describe the histogram of the magical energy of ponies (magical_energy):

Graphs in R: Bar Graphs

  • Construct a bar graph for the combined age and sex of ponies.
mlp_data %>%
  count(sex) %>%
  ggplot(aes(x = sex, y = n)) +
  geom_col() +
  labs(x = "Age and Sex of Pony",
       y = "Number of Ponies")+
  theme_bw()

Graphs in R: Bar Graphs

  • Construct a bar graph for the combined age and sex of ponies.

Graphs in R: Bar Graphs

  • Construct a bar graph for the type of pony.
mlp_data %>%
  count(type) %>%
  ggplot(aes(x = type, y = n)) +
  geom_col() +
  labs(x = "Type of Pony",
       y = "Number of Ponies")+
  theme_bw()

Graphs in R: Bar Graphs

  • Construct a bar graph for the type of pony.

Graphs in R: Scatterplots

  • Construct a scatterplot with magical energy (magical_energy) on the x-axis and tail shimmer (tail_shimmer) on the y-axis.
mlp_data %>% ggplot(aes(y = tail_shimmer, x = magical_energy)) +
  geom_point(size = 2) +
  labs(x = "Magical Energy",
       y = "Tail Shimmer") +
  theme_bw()

Graphs in R: Scatterplots

  • Construct a scatterplot with magical energy (magical_energy) on the x-axis and tail shimmer (tail_shimmer) on the y-axis.

Graphs in R: Scatterplots

  • Construct a scatterplot with magical energy (magical_energy) on the x-axis and flying speed (flying_speed) on the y-axis.
mlp_data %>% ggplot(aes(x = magical_energy, y = flying_speed)) +
  geom_point(size = 2) +
  labs(y = "Tail Shimmer",
       x = "Flying Speed (km/h)") +
  theme_bw()

Graphs in R: Scatterplots

  • Construct a scatterplot with magical energy (magical_energy) on the x-axis and flying speed (flying_speed) on the y-axis.

Wrap Up

  • We have covered (“reminded” ourselves of) a lot today!
    • Always remember that I do not expect you to:
      • Memorize code.
      • Produce code in a timed environment.
      • Automatically know how to do these things.
    • I do expect you to:
      • Use your resources (lecture slides, GitHub website, Discord).
      • Try your best.

Wrap Up

  • Today’s lecture:
    • Basic summaization of data.
    • Basic data visualization.
    • Introductory R.
  • Next class:
    • Review of statistical inference.
    • Confidence intervals and hypothesis tests.
      • One sample means.
      • Two sample means.
        • Independent data.
        • Dependent data.

Wrap Up

  • Daily activity: the .qmd we worked on during class.
    • Due date: Monday, June 23, 2025.
  • You will upload the resulting .html file on Canvas.
    • Please refer to the help guide on the Biostat website if you need help with submission.