Statistics and ggplot2: Quick Reference

Posted on Feb 3, 2025 in Statistics

Statistics and ggplot2: Quick Reference

Central Tendency

• Mean: The average of values, affected by outliers.
  • Formula: \(\bar{x} = \frac{\Sigma x_i}{n}\)
• Median: The middle value, robust to outliers.
• Mode: The most frequent value in a dataset.

Variability Metrics

• Range: \(\text{Max} – \text{Min}\)
• Population Variance: \(\sigma^2 = \frac{\Sigma (x_i – \mu)^2}{N}\)
• Sample Variance: \(s^2 = \frac{\Sigma (x_i – \bar{x})^2}{n-1}\) (Bessel’s correction).
• Standard Deviation (SD): The square root of variance.
  • Formula: \(s = \sqrt{s^2}\)

Standard Error (SE)

• Formula: \(SE = \frac{s}{\sqrt{n}}\)
• Measures the precision of the sample mean as an estimate of the population mean.
• Larger sample size = Lower SE.

Central Limit Theorem (CLT)

• For \(n \geq 30\), the sampling distribution of the mean is approximately normal, regardless of population shape.
• Allows normal methods for inference even with non-normal populations.

When to Use Mean vs. Median

• Mean: Symmetrical data with no outliers.
• Median: Skewed data or when there are outliers (e.g., income).

Normal Distribution

• 68%-95%-99.7% Rule:
  • 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD.
• Z-Score: \(z = \frac{x – \mu}{\sigma}\)

Boxplots

• Visualizes the median, spread (IQR), and outliers.
• Useful for detecting skewness and variability.

Histograms

• Displays the frequency distribution of continuous data.
• Shape (e.g., symmetric, skewed) provides insights into data.

ggplot2 Reference

Basics

• Framework:

  ggplot(data, aes(x, y)) + geom_...()

• aes(): Maps variables to aesthetics (e.g., x, y, color).

Common Geoms

• Scatterplot: geom_point() for relationships between variables.
• Line Graph: geom_line() for trends over time.
• Histogram:

  geom_histogram(aes(y = ..density..), bins = 30) + geom_density()

• Bar Plot: geom_bar() for categorical counts.
• Boxplot: geom_boxplot() for variability and outliers.

Customization

• Labels:

  labs(title = "Title", x = "X-axis", y = "Y-axis")

• Color by Category:

  aes(color = variable)

• Faceting:

  facet_wrap(~ category)

• Themes:

  theme(axis.text = element_text(size = 12))

Graph Interpretation

• Scatterplot: Shows correlation (positive, negative, or none).
• Boxplot: Highlights spread, median, and outliers.
• Histogram: Reveals data distribution shape (symmetric, skewed, bimodal).
• Line Graph: Shows trends and patterns over time.

Scales and Limits

• Set Axis Limits:

  scale_x_continuous(limits = c(min, max))

• Log Scale:

  scale_y_log10()

Quick Reference Table