0. set up

Code

# cleaning
library(tidyverse)

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.0     ✔ readr     2.1.4
✔ forcats   1.0.0     ✔ stringr   1.5.0
✔ ggplot2   3.4.1     ✔ tibble    3.2.0
✔ lubridate 1.9.2     ✔ tidyr     1.3.0
✔ purrr     1.0.1     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the ]8;;http://conflicted.r-lib.org/conflicted package]8;; to force all conflicts to become errors

Code

# visualization
library(showtext)

Loading required package: sysfonts
Loading required package: showtextdb

Code

font_add_google("Lato", "Lato")
showtext_auto()

1. 68-95-99.7 rule

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 = 22),
        axis.line.y = element_blank(),
        axis.title.y = element_blank(),
        axis.text.y = element_blank(),
        axis.ticks = element_blank())

2. central limit theorem

Code

# randomly select 10000 numbers from a uniform distribution for the population
uniform <- runif(10000, min = 2, max = 8)

# make a histogram for the population
uniformdf <- as.data.frame(uniform)

ggplot(uniformdf, aes(x = uniform)) +
  geom_histogram(breaks = seq(2, 8, length.out = 41), fill = "firebrick", alpha = 0.7, color = "firebrick") +
  geom_vline(xintercept = mean(uniform), linewidth = 2) +
  annotate("text", x = 4, y = 290, label = "mean = 4.967", size = 10) +
  scale_x_continuous(breaks = seq(from = 2, to = 8, by = 1)) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, 305)) +
  labs(x = "Continuous value", y = "Count") +
  theme_bw() +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18))

Code

# for() loop to 
store2 <- c()
store5 <- c()
store15 <- c()
store30 <- c()
store50 <- c()

for(i in 1:100) {
  
  store2[i] <- mean(sample(uniform, 2, replace = FALSE))

}


for(i in 1:100) {
  
  store5[i] <- mean(sample(uniform, 5, replace = FALSE))

}
for(i in 1:100) {
  
  store15[i] <- mean(sample(uniform, 15, replace = FALSE))

}

for(i in 1:100) {
  
  store30[i] <- mean(sample(uniform, 30, replace = FALSE))

}

for(i in 1:100) {
  
  store50[i] <- mean(sample(uniform, 50, replace = FALSE))

}

df <- cbind(store2, store5, store15, store30, store50) %>% 
  as.data.frame()
  
ggplot(df) +
  geom_histogram(aes(x = store2), bins = 10, alpha = 0.7, fill = "chocolate1", color = "chocolate1") +
  coord_cartesian(xlim = c(2, 8), ylim = c(0, 30)) +
  scale_y_continuous(expand = c(0, 0)) +
  geom_vline(xintercept = mean(store2)) +
  geom_vline(xintercept = mean(uniform), color = "red") +
  labs(x = "Sample means", y = "Count") +
  theme_bw() +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        plot.margin = unit(c(0.5, 0.5, 0.1, 0.1), "cm"))

Code

ggplot(df) +
  geom_histogram(aes(x = store5), bins = 10, alpha = 0.7, fill = "blue3", color = "blue3") +
  coord_cartesian(xlim = c(2, 8), ylim = c(0, 30)) +
  scale_y_continuous(expand = c(0, 0)) +
  geom_vline(xintercept = mean(store5)) +
  geom_vline(xintercept = mean(uniform), color = "red") +
  labs(x = "Sample means", y = "Count") +
  theme_bw() +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        plot.margin = unit(c(0.5, 0.5, 0.1, 0.1), "cm"))

Code

ggplot(df) +
  geom_histogram(aes(x = store15), bins = 12, alpha = 0.7, fill = "darkorchid4", color = "darkorchid4") +
  coord_cartesian(xlim = c(2, 8), ylim = c(0, 30)) +
  scale_y_continuous(expand = c(0, 0)) +
  geom_vline(xintercept = mean(store15)) +
  geom_vline(xintercept = mean(uniform), color = "red") +
  labs(x = "Sample means", y = "Count") +
  theme_bw() +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        plot.margin = unit(c(0.5, 0.5, 0.1, 0.1), "cm"))

Code

ggplot(df) +
  geom_histogram(aes(x = store30), bins = 12, alpha = 0.7, fill = "lightseagreen", color = "lightseagreen") +
  coord_cartesian(xlim = c(2, 8), ylim = c(0, 30)) +
  scale_y_continuous(expand = c(0, 0)) +
  geom_vline(xintercept = mean(store30)) +
  geom_vline(xintercept = mean(uniform), color = "red") +
  labs(x = "Sample means", y = "Count") +
  theme_bw() +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        plot.margin = unit(c(0.5, 0.5, 0.1, 0.1), "cm"))

Code

ggplot(df) +
  geom_histogram(aes(x = store50), bins = 12, alpha = 0.7, fill = "violetred3", color = "violetred3") +
  coord_cartesian(xlim = c(2, 8), ylim = c(0, 30)) +
  scale_y_continuous(expand = c(0, 0)) +
  geom_vline(xintercept = mean(store50)) +
  geom_vline(xintercept = mean(uniform), color = "red") +
  labs(x = "Sample means", y = "Count") +
  theme_bw() +
  theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        plot.margin = unit(c(0.5, 0.5, 0.1, 0.1), "cm"))

3. z- vs t-distribution

Code

ggplot(data.frame(x = -5:5), aes(x)) +
  stat_function(geom = "line", n = 1000, fun = dnorm, args = list(mean = 0, sd = 1), linewidth = 1, color = "darkorange") +
  annotate("text", x = 2.5, y = 0.4, label = "normal", color = "darkorange", size = 6) +
  stat_function(geom = "line", n = 1000, fun = dt, args = list(df = 1), linewidth = 1, color = "#856F33") +
  annotate("text", x = 3, y = 0.32, label = "t-distribution (small n)", color = "#856F33", size = 6) +
  stat_function(geom = "line", n = 1000, fun = dt, args = list(df = 10), linewidth = 1, color = "#56E9E7") +
  annotate("text", x = 3, y = 0.37, label = "t-distribution (large n)", color = "#56E9E7", 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"))

3. math notation

\[ SE_{\bar{x}} = \frac{s}{\sqrt{n}} \]

4. qqplot examples

Code

as_tibble(nhtemp) %>% 
  ggplot(aes(x = x)) +
  geom_histogram(breaks = seq(47, 55, length.out = 9), fill = "turquoise3", color = "#000000") +
  scale_x_continuous(breaks = seq(47, 55, length.out = 9), expand = c(0, 0)) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, 23)) +
  theme_classic() +
  labs(x = "Bins", y = "Count") +
    theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        text = element_text(family = "Lato"),
        plot.margin = unit(c(0.1, 1, 0.1, 0.1), "cm"))

Code

ggplot(as_tibble(nhtemp)) +
  stat_qq(aes(sample = x), color = "turquoise3", size = 3) +
  theme_classic() +
  labs(x = "Theoretical", y = "Sample") +
      theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        text = element_text(family = "Lato"),
        plot.margin = unit(c(0.1, 1, 0.1, 0.1), "cm"))

Don't know how to automatically pick scale for object of type <ts>. Defaulting
to continuous.

Code

as_tibble(sunspots) %>% 
  ggplot(aes(x = x)) +
  geom_histogram(breaks = round(seq(0, 260, length.out = 30)), fill = "tomato2", color = "#000000") +
  scale_x_continuous(breaks = round(seq(0, 260, length.out = 30)), expand = c(0, 0)) +
  scale_y_continuous(expand = c(0, 0), limits = c(0, 480)) +
  theme_classic() +
  labs(x = "Bins", y = "Count") +
    theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        text = element_text(family = "Lato"),
        plot.margin = unit(c(0.1, 1, 0.1, 0.1), "cm"))

Code

ggplot(as_tibble(sunspots)) +
  stat_qq(aes(sample = x), color = "tomato2", size = 3) +
  theme_classic() +
  labs(x = "Theoretical", y = "Sample") +
      theme(panel.grid = element_blank(),
        axis.text = element_text(size = 18),
        axis.title = element_text(size = 18),
        text = element_text(family = "Lato"),
        plot.margin = unit(c(0.1, 1, 0.1, 0.1), "cm"))

Don't know how to automatically pick scale for object of type <ts>. Defaulting
to continuous.

Citation

BibTeX citation:

@online{bui2023,
  author = {Bui, An},
  title = {Lecture 03 Figures},
  date = {2023-04-17},
  url = {https://an-bui.github.io/ES-193DS-W23/lecture/lecture-03_2023-04-17.html},
  langid = {en}
}

For attribution, please cite this work as:

Bui, An. 2023. “Lecture 03 Figures.” April 17, 2023. https://an-bui.github.io/ES-193DS-W23/lecture/lecture-03_2023-04-17.html.