Review of Statistical Estimation

Introduction

• In this lecture, we will review estimation
  • Continuous variables
    • Mean
    • Median
    • Percentiles / quartiles
    • Variance and standard deviation
    • Interquartile range
  • Categorical variables
    • Count
    • Percentage
• We will also discuss exploring data graphically

R: Introduction

• 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 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 instead for competency.

Today’s Data: Palmer Penguins

• Today we will be demonstrating the basics using the Palmer Penguins dataset, available through R.

penguins <- palmerpenguins::penguins

Types of Variables

• Between any two possible values, there are an infinite number of possible values.

• These typically arise from measurement. (Height, weight, etc.)

• The possible values can usually be listed.

• These typically arise from categorizing (work vs. home) or counting.

Types of Variables

Types of Continuous Variables

• True zero point indicates the absence of the quantity being measured.

• All arithmetic operations (addition, subtraction, multiplication, division) are meaningful.

• The zero point does not indicate a true absence.

• A 1 unit difference always represents the same amount.

Types of Discrete Variables

• We understand which value is “greater” or “less,” but not by how much.

• Arithmetic is not meaningful.

• Categories are used merely as labels or names.

• No arithmetic or ordering operations are meaningful.

Measures of Centrality: Mean

• R syntax:

dataset_name %>% summarize(mean(variable_name, na.rm = TRUE))

Measures of Centrality: Mean

• Let’s find the average weight (body_mass_g) of the penguins.

penguins %>% summarize(mean(body_mass_g, na.rm = TRUE))

• Let’s find the average flipper length (flipper_length_mm) of the penguins.

penguins %>% summarize(mean(flipper_length_mm, na.rm = TRUE))

Measures of Centrality: Median

• R syntax:

dataset_name %>% summarize(median(variable_name, na.rm = TRUE))

Measures of Centrality: Median

• Let’s find the median weight (body_mass_g) of the penguins.

penguins %>% summarize(median(body_mass_g, na.rm = TRUE))

• Let’s find the median flipper length (flipper_length_mm) of the penguins.

penguins %>% summarize(median(flipper_length_mm, na.rm = TRUE))

Measures of Spread: Variance and Standard Deviation

• When we have a mound-shaped and symmetric distribution, most observations will fall within 2 standard deviations of the mean.

• Variance results in units², which typically does not make sense.

• Standard deviation is the square root of the variance, measuring spread in the original units of the data.

• R syntax:

dataset_name %>% summarize(var(variable_name, na.rm = TRUE), 
                           sd(variable_name, na.rm = TRUE))

Measures of Spread: Variance and Standard Deviation

• Let’s find the variance and standard deviation of the weight (body_mass_g) of the penguins.

penguins %>% summarize(var(body_mass_g, na.rm = TRUE),
                       sd(body_mass_g, na.rm = TRUE))

• Let’s find the variance and standard deviation of the flipper length (flipper_length_mm) of the penguins.

penguins %>% summarize(var(flipper_length_mm, na.rm = TRUE),
                       sd(flipper_length_mm, na.rm = TRUE))

Measures of Spread: Interquartile Range

• R syntax:

dataset_name %>% summarize(IQR(variable_name))

Measures of Spread: Interquartile Range

• Let’s find the IQR of the weight (body_mass_g) of the penguins.

penguins %>% summarize(IQR(body_mass_g, na.rm = TRUE))

• Let’s find the IQR of the flipper length (flipper_length_mm) of the penguins.

penguins %>% summarize(IQR(flipper_length_mm, na.rm = TRUE))

Mean & Standard Deviation vs. Median & IQR

• When should we use the mean vs. the median to describe the center of the distribution?

  • Mound-shaped and symmetric \to \bar{x} & s.
  • Not mound-shaped and symmetric \to M & \text{IQR}.

• … How do we know the shape of the distribution?

• We will explore histograms.

Graphs: Histograms

Graphs: Histograms

Graphs: Histograms

Graphs: Histograms

Graphs: Histograms (R code)

• We are using the ggplot2 package for graphing.
  • It will always start with ggplot().
  • We will then layer elements on top.
• R syntax:

dataset_name %>% 
  ggplot(aes(x=variable_name)) + 
  geom_histogram() 

Graphs: Histograms

• Let’s look at the histogram of penguin weight (body_mass_g):

penguins %>% 
  ggplot(aes(x=body_mass_g)) + 
  geom_histogram() 

Graphs: Histograms

• Let’s look at the histogram of penguin weight (body_mass_g):

penguins %>% 
  ggplot(aes(x=body_mass_g)) + 
  geom_histogram() +
  labs(x = "Body Mass (g)",
       y = "Number of Penguins",
       title = "Penguin Weight Distribution") +
  theme_bw()

Wrap Up

• Today we reviewed estimation.

• Next week, we will review statistical inference.

  • Confidence intervals
  • Hypothesis testing

• Get to know you quiz - complete with RStudio.

  • .qmd \to Quarto
  • .R \to R script

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