Week 1 figures - Lectures 1 and 2

central tendency
spread
regression
histogram
jitterplot
box-and-whisker plot
probability mass
probability density
probability distribution
normal distribution
z-score calculation
t-distribution
uniform distribution
binomial distribution
poisson distribution
Author
Affiliation

An Bui

UC Santa Barbara, Ecology, Evolution, and Marine Biology

Lecture date

April 1, 2024

Modified

June 13, 2024

0. set up

Code 
# cleaning
library(tidyverse)
theme_set(theme_classic() +
            theme(panel.grid = element_blank(),
                  axis.text = element_text(size = 18),
                  axis.title = element_text(size = 18),
                  text = element_text(family = "Lato")))

# visualization
library(patchwork)

1. Math

sample mean

\[ \bar{y} = \frac{1}{n}\sum_{i = 1}^ny_i \]

sample variance

\[ s^2 = \frac{\sum(y_i - \bar{y})^2}{n - 1} \]

sample standard deviation

\[ s = \sqrt{\frac{\sum(y_i - \bar{y})^2}{n - 1}} \]

coefficient of variation

\[ CV = \frac{\sigma}{\mu} \]

z-score for selecting a single individual

\[ z = \frac{y_i - \mu}{\sigma} \]

2. Mean and median

For data following a symmetrical distribution, the mean and median tend to be similar.

Code 
set.seed(1)
rnorm(n = 40, mean = 6, sd = 1) %>% 
  as_tibble() %>% 
  ggplot(aes(x = value)) +
  geom_density() +
  geom_vline(aes(xintercept = mean(value)), color = "blue") +
  annotate("text", x = 5.75, y = 0.5, label = "mean", color = "blue") +
  geom_vline(aes(xintercept = median(value))) +
  annotate("text", x = 6.5, y  = 0.5, label = "median") +
  scale_x_continuous(limits = c(2.5, 10)) +
  scale_y_continuous(limits = c(0, 0.5)) +
  labs(x = "Sculpin lengths (cm)") +
  theme(axis.text.y = element_blank(),
        axis.title.y = element_blank(),
        axis.ticks.y = element_blank(),
        axis.line.y = element_blank())

3. Range

Code 
set.seed(1)
narrow <- rnorm(n = 30, mean = 6, sd = 1) %>% 
  as_tibble() %>% 
  mutate(y = 0) %>% 
  ggplot(aes(x = value, y = y)) +
  geom_jitter(shape = 21) +
  geom_point(aes(x = mean(value), y = 0), color = "blue", size = 3) +
  scale_x_continuous(limits = c(0, 15)) +
  scale_y_continuous(limits = c(-0.5, 0.5)) +
  theme(axis.line.y = element_blank(),
        axis.ticks.y = element_blank(),
        axis.text.y = element_blank(),
        axis.title.y = element_blank()) +
  labs(x = "Sculpin lengths (cm)")
# min: 3.78
# max: 7.60

set.seed(1)
wide <- rnorm(n = 30, mean = 6, sd = 2) %>% 
  as_tibble() %>% 
  mutate(y = 0) %>% 
  ggplot(aes(x = value, y = y)) +
  geom_jitter(shape = 21) +
    geom_point(aes(x = mean(value), y = 0), color = "blue", size = 3) +
  scale_x_continuous(limits = c(0, 15)) +
  scale_y_continuous(limits = c(-0.5, 0.5)) +
  theme(axis.line.y = element_blank(),
        axis.ticks.y = element_blank(),
        axis.text.y = element_blank(),
        axis.title.y = element_blank()) +
  labs(x = "Sculpin lengths (cm)")
# min: 1.57
# max: 9.19

narrow + wide

How would you describe this data?

Code 
set.seed(1)
ex1 <- rf(n = 100, df1 = 30, df2 = 10)
mean(ex1)
[1] 1.321259
Code 
median(ex1)
[1] 1.163362
Code 
ex1 %>% 
  enframe() %>% 
  ggplot(aes(x = value)) +
  geom_histogram(bins = 9,
                 color = "#000000",
                 fill = "orange") +
  scale_y_continuous(expand = c(0, 0)) +
  labs(x = "Hermit crab shell length (cm)")

Code 
set.seed(1)
ex2 <- rnorm(n = 100, mean = 25, sd = 5)
mean(ex2)
[1] 25.54444
Code 
median(ex2)
[1] 25.56955
Code 
ex2 %>% 
  enframe() %>% 
  ggplot(aes(x = value)) +
  geom_histogram(bins = 9,
                 color = "#000000",
                 fill = "darkgreen") +
  scale_y_continuous(expand = c(0, 0)) +
  labs(x = "Octopus arm length (cm)")

4. anemone regression example

Code 
# number of arms 
arms <- seq(from = 40, to = 100, by = 1)

# diameter: anemones can be up to 8 cm long
set.seed(10)
diam <- rnorm(length(arms), mean = seq(from = 1, to = 5, length = length(arms)), sd = 1) 

# create a data frame
df <- cbind(diam, arms) %>% 
  as.data.frame()

ggplot(df, aes(x = arms, y = diam)) +
  geom_point(size = 2) +
  labs(x = "Number of arms", y = "Diameter (cm)")

Code 
ggplot(df, aes(x = arms, y = diam)) +
  geom_point(size = 2) +
  # just using geom smooth for the purposes of visualization
  geom_smooth(method = "lm", se = FALSE, linewidth = 2) +
  labs(x = "Number of arms", y = "Diameter (cm)") +
  theme_bw() +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        text = element_text(family = "Lato"))

5. histogram example

The Rice rule guidelines for the calculating the number of bins in a histogram:

\[ bins = 2n^{1/3} \]

where \(n\) is the number of observations. This is an example of a histogram that does follow the rice rule, where the bin number is 8.

Code 
ggplot(df, aes(x = diam)) +
  scale_x_continuous(breaks = seq(from = 0, to = 8, by = 1)) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, 19), breaks = seq(from = 0, to = 18, by = 3)) +
  geom_histogram(breaks = seq(from = 0, to = 8, by = 1), color = "#000000", fill = "lightblue") +
  labs(x = "Anemone diameter (cm)", y = "Count") +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        text = element_text(family = "Lato"))

These histograms do not, and it proves difficult to see the distribution:

Code 
ggplot(df, aes(x = diam)) +
  scale_x_continuous(breaks = seq(from = 0, to = 8, by = 1)) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, 19), breaks = seq(from = 0, to = 18, by = 3)) +
  geom_histogram(color = "#000000", fill = "lightblue") +
  labs(x = "Anemone diameter (cm)", y = "Count") +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        text = element_text(family = "Lato"))

Code 
ggplot(df, aes(x = diam)) +
  scale_x_continuous(breaks = seq(from = 0, to = 8, by = 1)) +
  scale_y_continuous(expand = c(0, 0)) +
  geom_histogram(color = "#000000", fill = "lightblue", bins = 3) +
  labs(x = "Anemone diameter (cm)", y = "Count") +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        text = element_text(family = "Lato"))

6. jitter plot and box and whisker plot example

Code 
set.seed(1)

pretend_lengths <- cbind(
  juveniles = rnorm(20, mean = 2, sd = 0.5), 
  females = rnorm(20, mean = 8, sd = 1), 
  males = rnorm(20, mean = 4, sd = 1)
) %>% 
  as_tibble() %>% 
  pivot_longer(cols = 1:3)

ggplot(pretend_lengths, aes(x = name, y = value, color = name)) +
  geom_jitter(width = 0.1, alpha = 0.8, size = 2) +
  scale_color_manual(values = c("darkgreen", "cornflowerblue", "orange")) +
  labs(y = "Weight (g)") +
  theme(axis.title.x = element_blank(),
        legend.position = "none")

Code 
ggplot(pretend_lengths, aes(x = name, y = value, color = name, fill = name)) +
  geom_boxplot(alpha = 0.8) +
  scale_color_manual(values = c("darkgreen", "cornflowerblue", "orange")) +
  scale_fill_manual(values = c("darkgreen", "cornflowerblue", "orange")) +
  labs(y = "Weight (g)") +
  theme(axis.title.x = element_blank(),
        legend.position = "none")

7. Probability mass example

Code 
ggplot(data.frame(x = 1:55), aes(x)) +
  stat_function(geom = "bar", n = 55, fun = dpois, args = list(lambda = 10), fill = "coral") +
  scale_y_continuous(expand = c(0, 0), limits = c(0, 0.13)) +
  coord_cartesian(xlim = c(0, 22)) +
  labs(x = "Mussel clump size (count)", y = "Probability mass") +
  theme_bw() +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        text = element_text(family = "Lato"))

8. Probability density example

Code 
ggplot(data.frame(x = 1:20), aes(x)) +
  stat_function(geom = "line", n = 100, fun = dnorm, args = list(mean = 10, sd = 2), linewidth = 1) +
  stat_function(geom = "area", fun = dnorm, args = list(mean = 10, sd = 2), xlim = c(12, 14), fill = "turquoise3") +
  geom_vline(xintercept = 12, lty = 2, color = "grey", linewidth = 1) +
  geom_vline(xintercept = 14, lty = 2, color = "grey", linewidth = 1) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, 0.22)) +
  # coord_cartesian(xlim = c(0, 22)) +
  labs(x = "Individual mussel weight (g)", y = "Probability density")

9. probability distribution

Code 
set.seed(1)
normdist <- rnorm(n = 100000, mean = 0, sd = 1) %>% 
  as_tibble(rownames = "x")

ggplot(normdist) +
  geom_histogram(aes(x = value, after_stat(density)), fill = "white", color = "black", bins = 100) +
  stat_function(fun = dnorm, args = list(mean = 0, sd = 1), color = "blue", linewidth = 2) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, 0.42)) +
  labs(x = "Continuous value", y = "Density")

10. normal distribution

Code 
ggplot(data.frame(x = -10:25), aes(x)) +
  stat_function(geom = "line", n = 1000, fun = dnorm, args = list(mean = 0, sd = 1), linewidth = 1, color = "darkorange") +
  annotate("text", x = 4.5, y = 0.4, label = "\U03BC = 0, \U03C3 = 1", color = "darkorange", size = 6) +
  stat_function(geom = "line", n = 1000, fun = dnorm, args = list(mean = 15, sd = 3), linewidth = 1, color = "blue") +
  annotate("text", x = 16, y = 0.15, label = "\U03BC = 15, \U03C3 = 3", color = "blue", size = 6) +
  stat_function(geom = "line", n = 1000, fun = dnorm, args = list(mean = 5, sd = 5), linewidth = 1, color = "darkgreen") +
  annotate("text", x = 7, y = 0.1, label = "\U03BC = 5, \U03C3 = 5", color = "darkgreen", size = 6) +
  scale_x_continuous(breaks = seq(-10, 25, 5)) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, 0.42)) +
  labs(x = "Continuous value", y = "Density")

11. z-score calculation

figure

We’ll use \(z = -1.23\) for this example.

Code 
# z-score
q <- -1.23

ggplot(data.frame(x = -4:4), aes(x)) +
  # zscore
  geom_linerange(x = q, ymin = 0, ymax = 0.19) +
  # area under the curve
  stat_function(geom = "area", fun = dnorm, args = list(mean = 0, sd = 1), xlim = c(-4, -1.23), fill = "turquoise3") +
  # Z distribution curve
  stat_function(geom = "line", n = 1000, fun = dnorm, args = list(mean = 0, sd = 1), linewidth = 1.5, color = "darkorange") +
  scale_y_continuous(expand = c(0, 0), limits = c(0, 0.45)) +
  theme(axis.text.y = element_blank(),
        axis.ticks.y = element_blank(),
        axis.title = element_blank(),
        axis.line.y = element_blank())

calculation

Code 
pnorm(q, mean = 0, sd = 1)
[1] 0.1093486

You can compare this with the Z-score table.

chiton example

What is the probability of selecting a chiton that is less than 6 ft long given a normally distributed population with \(\mu = 12\) g with \(\sigma = 3\) g?

Code 
# calculate the z-score
chiton_z <- (6 - 12)/3
  
# calculate the probability under the curve
pnorm(chiton_z, mean = 0, sd = 1)
[1] 0.02275013

12. 68-95-99.7 rule

In a normal distribution, 68% of values lie within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.

Code 
labels <- c(
  "", "\U03BC - 3\U03C3", "\U03BC - 2\U03C3", "\U03BC - \U03C3", "\U03BC", "\U03BC + \U03C3", "\U03BC + 2\U03C3", "\U03BC + 3\U03C3", ""
)

ggplot(data.frame(x = -4:4), aes(x)) +
  geom_linerange(x = 1, ymin = 0, ymax = 0.24) +
  geom_linerange(x = -1, ymin = 0, ymax = 0.24) +
  geom_linerange(x = 2, ymin = 0, ymax = 0.055) +
  geom_linerange(x = -2, ymin = 0, ymax = 0.055) +
  geom_linerange(x = 3, ymin = 0, ymax = 0.005) +
  geom_linerange(x = -3, ymin = 0, ymax = 0.005) +
  geom_linerange(x = 0, ymin = 0, ymax = 0.399) +
  stat_function(geom = "line", n = 1000, fun = dnorm, args = list(mean = 0, sd = 1), linewidth = 1.5, color = "darkorange") +
  scale_x_continuous(labels = labels, breaks = seq(-4, 4, by = 1)) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, 0.41)) +
  labs(x = "") +
  theme_classic() +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 24),
        axis.line.y = element_blank(),
        axis.title.y = element_blank(),
        axis.text.y = element_blank(),
        axis.ticks = element_blank())

13. Student’s t distribution

Code 
ggplot(data.frame(x = -10:10), aes(x)) +
  stat_function(geom = "line", n = 1000, fun = dt, args = list(df = 1), linewidth = 1, color = "#856F33") +
  annotate("text", x = 3.5, y = 0.3, label = "\U03BD = 1", color = "#856F33", size = 6) +
  stat_function(geom = "line", n = 1000, fun = dt, args = list(df = 3), linewidth = 1, color = "#E6821C") + 
  annotate("text", x = 3.5, y = 0.35, label = "\U03BD = 3", color = "#E6821C", size = 6) +
  stat_function(geom = "line", n = 1000, fun = dt, args = list(df = 5), linewidth = 1, color = "#56E9E7") +
  annotate("text", x = 3.5, y = 0.37, label = "\U03BD = 5", color = "#56E9E7", size = 6) +
  stat_function(geom = "line", n = 1000, fun = dt, args = list(df = 100), linewidth = 1, color = "#04B37F") +
    annotate("text", x = 3.5, y = 0.4, label = "\U03BD = 100", color = "#04B37F", size = 6) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, 0.42)) +
  labs(x = "Continuous value", y = "Density") +
  theme_bw() +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        text = element_text(family = "Lato"))

14. Uniform distribution

Code 
ggplot(data.frame(x = 0:10), aes(x)) +
  stat_function(geom = "line", n = 1000, fun = dunif, args = list(min = 2, max = 8), linewidth = 1, color = "firebrick4") +
  annotate("text", x = 2, y = 0.172, label = "a = 2", color = "firebrick4", size = 6) + 
  annotate("text", x = 8, y = 0.172, label = "b = 8", color = "firebrick4", size = 6) + 
  scale_x_continuous(breaks = seq(0, 10, 2)) +
  scale_y_continuous(expand = c(0, 0), limits = c(-0.001, 0.18)) +
  labs(x = "Continuous value", y = "Density") +
  theme_bw() +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        text = element_text(family = "Lato"))

14. Binomial distribution

Code 
ggplot(data.frame(x = 1:20), aes(x)) +
  stat_function(geom = "line", n = 20, fun = dbinom, args = list(size = 20, p = 0.1), color = "black") +
  stat_function(geom = "point", n = 20, fun = dbinom, args = list(size = 20, p = 0.1), color = "#6D9929", size = 3) +
  annotate("text", x = 5.5, y = 0.29, label = "n = 20, p = 0.1", color = "#6D9929", size = 6) +
  stat_function(geom = "line", n = 20, fun = dbinom, args = list(size = 20, p = 0.4), color = "black") +
  stat_function(geom = "point", n = 20, fun = dbinom, args = list(size = 20, p = 0.4), color = "#4A76E5", size = 3) +
  annotate("text", x = 8, y = 0.2, label = "n = 20, p = 0.4", color = "#4A76E5", size = 6) +
  stat_function(geom = "line", n = 20, fun = dbinom, args = list(size = 20, p = 0.7), color = "black") +
  stat_function(geom = "point", n = 20, fun = dbinom, args = list(size = 20, p = 0.7), color = "#E67960", size = 3) +
  annotate("text", x = 15, y = 0.21, label = "n = 20, p = 0.7", color = "#E67960", size = 6) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, 0.32)) +
  labs(x = "Number of successes", y = "Mass") +
  theme_bw() +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        text = element_text(family = "Lato"))

15. Poisson distribution

Code 
ggplot(data.frame(x = 1:20), aes(x)) +
  stat_function(geom = "line", n = 20, fun = dpois, args = list(lambda = 1), color = "black") +
  stat_function(geom = "point", n = 20, fun = dpois, args = list(lambda = 1), color = "coral", size = 4) +
  annotate("text", x = 3, y = 0.37, label = "\U03BB = 1", color = "coral", size = 6) +
  stat_function(geom = "line", n = 20, fun = dpois, args = list(lambda = 4), color = "black") +
  stat_function(geom = "point", n = 20, fun = dpois, args = list(lambda = 4), color = "darkgreen", size = 4) +
  annotate("text", x = 6, y = 0.2, label = "\U03BB = 4", color = "darkgreen", size = 6) +
  stat_function(geom = "line", n = 20, fun = dpois, args = list(lambda = 10), color = "black") +
  stat_function(geom = "point", n = 20, fun = dpois, args = list(lambda = 10), color = "turquoise", size = 4) +
  annotate("text", x = 14, y = 0.12, label = "\U03BB = 10", color = "turquoise", size = 6) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, 0.42)) +
  labs(x = "Discrete value", y = "Mass")

Citation

BibTeX citation:
@online{bui2024,
  author = {Bui, An},
  title = {Week 1 Figures - {Lectures} 1 and 2},
  date = {2024-04-01},
  url = {https://spring-2024.envs-193ds.com/lecture/lecture_week-01.html},
  langid = {en}
}
For attribution, please cite this work as:
Bui, An. 2024. “Week 1 Figures - Lectures 1 and 2.” April 1, 2024. https://spring-2024.envs-193ds.com/lecture/lecture_week-01.html.