ENVX1002 Statistics in Life and Environmental Sciences
Mar 2026
# Generate normal distribution data
set.seed(123)
normal_data <- rnorm(1000, mean = 0, sd = 1)
# Plot normal distribution
ggplot(data.frame(x = normal_data), aes(x = x)) +
geom_histogram(aes(y = after_stat(density)),
bins = 30,
fill = "skyblue",
colour = "black") +
geom_density(colour = "red") +
labs(title = "Standard Normal Distribution (μ = 0, σ = 1)",
x = "Value",
y = "Density")This pattern makes the normal distribution especially useful: if you know the mean and standard deviation, you already know where most of the data sits.
# Create a standard normal distribution
x <- seq(-4, 4, length.out = 1000)
y <- dnorm(x)
df <- data.frame(x = x, y = y)
# Plot with empirical rule highlighted
ggplot(df, aes(x = x, y = y)) +
geom_line() +
# Add vertical reference lines at standard deviations
geom_vline(xintercept = c(-3, -2, -1, 0, 1, 2, 3), linetype = "dashed", colour = "gray50", alpha = 0.7) +
geom_area(data = subset(df, x >= -1 & x <= 1), fill = "darkblue", alpha = 0.3) +
geom_area(data = subset(df, (x >= -2 & x < -1) | (x > 1 & x <= 2)), fill = "darkgreen", alpha = 0.3) +
geom_area(data = subset(df, (x >= -3 & x < -2) | (x > 2 & x <= 3)), fill = "darkred", alpha = 0.3) +
annotate("text", x = 0, y = 0.2, label = "68%", colour = "darkblue") +
annotate("text", x = 1.5, y = 0.1, label = "95%", colour = "darkgreen") +
annotate("text", x = 2.5, y = 0.05, label = "99.7%", colour = "darkred") +
labs(title = "Normal Distribution: Empirical Rule",
x = "Standard Deviations from Mean",
y = "Density")Many biological measurements follow a normal distribution. If you know your data is roughly normal, you can predict where most values will land and spot anything unusual.
# Simulate plant height data
set.seed(456)
plant_heights <- rnorm(200, mean = 25, sd = 3) # Heights in cm
# Plot the distribution
ggplot(data.frame(height = plant_heights), aes(x = height)) +
geom_histogram(aes(y = after_stat(density)),
bins = 20,
fill = "#66c2a5",
colour = "black") +
geom_density(colour = "#1f78b4", linewidth = 1) +
geom_vline(xintercept = 25, linetype = "dashed", colour = "red") +
annotate("text", x = 25.5, y = 0.05, label = "μ = 25 cm", colour = "red") +
labs(title = "Distribution of Plant Heights in a Population",
x = "Height (cm)",
y = "Density")Here, plant heights cluster around a mean of 25 cm. A plant measuring 35 cm would be more than 3 standard deviations from the mean, making it a clear outlier worth investigating.
The tail stretches to the right, pulling the mean above the median. Common in income data, reaction times, and species abundance counts.
# Generate positive skewed distribution
set.seed(123)
right_skewed <- exp(rnorm(1000, 0, 0.5))
# Calculate statistics
mean_val <- mean(right_skewed)
median_val <- median(right_skewed)
# Plot positive skewed distribution
ggplot(data.frame(x = right_skewed), aes(x = x)) +
geom_histogram(aes(y = after_stat(density)),
bins = 30,
fill = "#66c2a5", # colourblind-friendly green
colour = "black") +
geom_density(colour = "#1f78b4", linewidth = 1) +
# Add vertical lines for mean and median
geom_vline(xintercept = mean_val, colour = "red", linetype = "dashed", linewidth = 1) +
geom_vline(xintercept = median_val, colour = "blue", linetype = "dashed", linewidth = 1) +
# Add annotations
annotate("text", x = mean_val + 0.2, y = 0.5, label = paste("Mean =", round(mean_val, 2)), colour = "red") +
annotate("text", x = median_val - 0.2, y = 0.4, label = paste("Median =", round(median_val, 2)), colour = "blue") +
labs(title = "Positive skew (right-skewed)",
subtitle = "Note that Mean > Median",
x = "Value",
y = "Density")The tail stretches to the left, pulling the mean below the median. Less common, but seen in exam scores near a ceiling or age at retirement.
# Generate skewed distributions
set.seed(123)
left_skewed <- max(right_skewed) - right_skewed + min(right_skewed)
# Calculate statistics
mean_val <- mean(left_skewed)
median_val <- median(left_skewed)
# Plot negative skewed distribution
ggplot(data.frame(x = left_skewed), aes(x = x)) +
geom_histogram(aes(y = after_stat(density)),
bins = 30,
fill = "#8da0cb", # colourblind-friendly blue
colour = "black") +
geom_density(colour = "#1f78b4", linewidth = 1) +
# Add vertical lines for mean and median
geom_vline(xintercept = mean_val, colour = "red", linetype = "dashed", linewidth = 1) +
geom_vline(xintercept = median_val, colour = "blue", linetype = "dashed", linewidth = 1) +
# Add annotations
annotate("text", x = mean_val - 0.2, y = 0.5, label = paste("Mean =", round(mean_val, 2)), colour = "red") +
annotate("text", x = median_val + 0.2, y = 0.4, label = paste("Median =", round(median_val, 2)), colour = "blue") +
labs(title = "Negative skew (left-skewed)",
subtitle = "Note that Mean < Median",
x = "Value",
y = "Density")Compare these three distributions. They all have the same mean and similar spread, but their tails behave very differently.
# Generate distributions with different kurtosis
set.seed(123)
normal <- rnorm(1000, 0, 1) # Mesokurtic
leptokurtic <- rt(1000, df = 5) # t-distribution with 5 df is leptokurtic
platykurtic <- runif(1000, -3, 3) # Uniform distribution is platykurtic
# Calculate kurtosis values (using e1071 package)
library(e1071)
k_normal <- kurtosis(normal)
k_lepto <- kurtosis(leptokurtic)
k_platy <- kurtosis(platykurtic)
# Plot distributions
p1 <- ggplot(data.frame(x = normal), aes(x = x)) +
geom_histogram(aes(y = after_stat(density)), bins = 30,
fill = "#a6cee3", # colourblind-friendly blue
colour = "black") +
geom_density(colour = "#1f78b4", linewidth = 1) + # Darker blue
annotate("text", x = 2, y = 0.3,
label = paste("Kurtosis =", round(k_normal, 2)),
colour = "#1f78b4") +
labs(title = "Mesokurtic (normal)",
subtitle = "Normal distribution with balanced tails",
x = "Value",
y = "Density")
p2 <- ggplot(data.frame(x = leptokurtic), aes(x = x)) +
geom_histogram(aes(y = after_stat(density)), bins = 30,
fill = "#fb9a99", # colourblind-friendly pink
colour = "black") +
geom_density(colour = "#e31a1c", linewidth = 1) + # Darker red
annotate("text", x = 2, y = 0.3,
label = paste("Kurtosis =", round(k_lepto, 2)),
colour = "#e31a1c") +
labs(title = "Leptokurtic (heavy-tailed)",
subtitle = "Sharper peak, heavier tails",
x = "Value",
y = "Density")
p3 <- ggplot(data.frame(x = platykurtic), aes(x = x)) +
geom_histogram(aes(y = after_stat(density)), bins = 30,
fill = "#b2df8a", # colourblind-friendly green
colour = "black") +
geom_density(colour = "#33a02c", linewidth = 1) + # Darker green
annotate("text", x = 0, y = 0.15,
label = paste("Kurtosis =", round(k_platy, 2)),
colour = "#33a02c") +
labs(title = "Platykurtic (light-tailed)",
subtitle = "Flatter peak, thinner tails",
x = "Value",
y = "Density")
# Display plots vertically
library(patchwork)
p1 + p2 + p3The naniar package can show you exactly where values are missing, which is much easier to interpret than scanning raw output.
Use histograms when you want to see how a continuous variable is distributed. They reveal the centre, spread, and any skewness or outliers at a glance.
Bar plots show counts or proportions across groups. Use them for categorical data like species, treatments, or survey responses.
Scatterplots show how two continuous variables relate to each other. They are the go-to plot for spotting correlations, clusters, and outliers.
Boxplots summarise the median, quartiles, and outliers for each group. They are ideal for comparing a continuous variable across categories.
The ggplot() function initialises a plot with data:
This creates an empty plot. We need to add layers to visualise our data.
Aesthetics map variables in the data to visual properties:
We’ve defined which variables go on which axes, but we still need to specify how to represent the data.
Geometries define how the data is represented visually:
Now we can see the relationship between flipper length and body mass!
We can map the species variable to the colour aesthetic:
Notice how ggplot2 automatically creates a legend for the species colours.
We can adjust the size and transparency of points:
Let’s add informative labels and a title:
# Add labels and title
p <- ggplot(penguins_clean, aes(x = flipper_length_mm, y = body_mass_g, colour = species)) +
geom_point(size = 3, alpha = 0.7) +
labs(
title = "Relationship Between Flipper Length and Body Mass",
subtitle = "Palmer Penguins Dataset",
x = "Flipper Length (mm)",
y = "Body Mass (g)",
colour = "Penguin Species"
)
pWe can use a custom colour palette:
# Customize colours
p <- ggplot(penguins_clean, aes(x = flipper_length_mm, y = body_mass_g, colour = species)) +
geom_point(size = 3, alpha = 0.7) +
scale_colour_manual(values = c("darkorange", "purple", "cyan4")) +
labs(
title = "Relationship Between Flipper Length and Body Mass",
subtitle = "Palmer Penguins Dataset",
x = "Flipper Length (mm)",
y = "Body Mass (g)",
colour = "Penguin Species"
)
pFinally, let’s apply a theme to change the overall appearance:
# Apply a theme
p <- ggplot(penguins_clean, aes(x = flipper_length_mm, y = body_mass_g, colour = species)) +
geom_point(size = 3, alpha = 0.7) +
scale_colour_manual(values = c("darkorange", "purple", "cyan4")) +
labs(
title = "Relationship Between Flipper Length and Body Mass",
subtitle = "Palmer Penguins Dataset",
x = "Flipper Length (mm)",
y = "Body Mass (g)",
colour = "Penguin Species"
) +
theme_minimal() +
theme(
legend.position = "bottom",
plot.title = element_text(face = "bold"),
axis.title = element_text(face = "italic")
)
pOne of the strengths of ggplot2 is the ability to add multiple layers:
We can also split the plot into facets by species:
