library(tidyverse)
set.seed(301)
flips <- sample(c(0, 1), 5000, replace = TRUE)
tibble(
n = seq_along(flips),
running = cummean(flips)
) %>%
ggplot(aes(n, running)) +
geom_line(color = "#2E86C1") +
geom_hline(yintercept = 0.5,
color = "#E74C3C",
linetype = "dashed") +
coord_cartesian(ylim = c(0.3, 0.7)) +
labs(title = "Proportion of heads settles at 0.5",
x = "Number of flips", y = NULL) +
theme_minimal(base_size = 11)Probability, Distributions and Concepts
Statistical Inference: probability and distributions
Diamonds Histogram Plot 1
Binwidth of 0.1
Diamonds Histogram Plot 2
Binwidth of 0.2
Diamonds Histogram Plot 3
Binwidth of 0.3
Diamonds Histogram Plot 4
Binwidth of 0.6
Diamonds Histogram Plot 5
Binwidth of 0.8
Diamonds Histogram Plot 6
Binwidth of 1
Diamonds Histogram Plot 7
Binwidth of 10
Is our data a single huge square!?
Frequency of Heads in Coin Flips
Frequency of Heads in Coin Flips Plot
Continuous Distribution Example
library(tidyverse)
# Scatter plot of continuous data
# (Engine displacement vs. Highway MPG)
ggplot(data = mpg, aes(x = displ, y = hwy)) +
geom_point(color = "steelblue", size = 2.5, alpha = 0.7) +
labs(
title = "Continuous Data: Engine Size vs. Fuel Efficiency",
x = "Engine Displacement (Liters)",
y = "Highway Miles per Gallon"
) +
theme_minimal()
Discrete Distribution Example
library(tidyverse)
# Bar chart of discrete data (Count of diamonds by cut quality)
ggplot(data = diamonds, aes(x = cut)) +
geom_bar(fill = "steelblue", color = "black", alpha = 0.7) +
labs(
title = "Discrete Data: Count of Diamonds by Cut",
x = "Cut Quality (Category)",
y = "Count"
) + theme_minimal()
The Normal Distribution Plot 🔔
The 68-95-99.7% rules for a normal distribution:
- 68%: Approximately 68% of observations fall within 1 standard deviation of the mean (μ ± 1σ).
- Green shading: Approximately 68% of the distribution, from −1 to +1.
- 95%: Approximately 95% of observations fall within 2 standard deviations of the mean (μ±2σ).
- Purple shading: The wider interval from −2 to +2, containing approximately 95% of the distribution; it overlaps the green region.
- 99.7%: Approximately 99.7% of observations fall within 3 standard deviations of the mean (μ±3σ).
- The ±3 standard deviation interval is not shaded in the original code.
The Binomial Distribution 🪙
Binomial: the count of successes in n independent trials, each with probability p.
[1] 0.2023312# The probability of getting 10 or more correct answers
# out of 20 questions when randomly guessing on a
# multiple-choice quiz with 4 options per question?
# P(10 or more correct). Note, `lower.tail = FALSE`
# calculates the probability of getting more than
# 9 correct, rather than 9 or fewer.
pbinom(9, 20, 0.25, lower.tail = FALSE)[1] 0.01386442
Poisson Distribution Plot
tibble(k = rep(0:15, 3),
lambda = rep(c(1, 4, 9), each = 16)) %>%
mutate(p = dpois(k, lambda)) %>%
ggplot(aes(k, p, fill = factor(lambda))) +
geom_col(position = "identity", alpha = 0.6) +
scale_fill_manual(values = c("#27AE60", "#3498DB", "#E67E22")) +
labs(title = "Poisson gets more symmetric as lambda (an average) grows",
fill = "lambda", x = "Events", y = "Probability") +
theme_minimal(base_size = 11)
Uuuhhh, What is the Central Limit Theorem?
Important
The central limit theorem establishes that when independent events are added the sum tends toward a normal distribution (a bell curve) even if the original variables themselves are not normally distributed.
For Example
If you repeatedly roll a die and take the average of the rolls, the distribution of those averages will approach a normal distribution as the number of rolls increases, regardless of the uniform distribution of a single die roll.
An Example Central Limit Theorem?
Random 0s and 1s were generated, and then their means calculated for sample sizes ranging from 1 to 512. Note that as the sample size increases the tails become thinner and the distribution becomes more concentrated around the mean. Ref: https://www.ml-science.com/central-limit-theorem
Population vs Sample
Population — everyone you care about
Parameters (usually unknown):
- μ (mu) = population mean
- σ (sigma) = population sd
Sample — who you actually measured
Statistics (computed from data):
- x̄ = sample mean
- s = sample sd
The Central Question
Your sample mean is not the population mean. How far off might it be? Everything in Weeks 8-10 answers that question. 🎯
Warning
Who is in your population? A “national” survey that samples only landline owners has a population you did not intend. Sampling frames encode bias before a single number is computed. ⚖️
The Sampling Distribution Part 2
set.seed(301)
# Repeat "take a sample of 25, record the mean" 5,000 times
sample_means <- replicate(5000, mean(sample(population, 25)))
ggplot(tibble(m = sample_means), aes(m)) +
geom_histogram(bins = 40, fill = "#3498DB", color = "white") +
geom_vline(xintercept = mean(population), color = "#E74C3C", linewidth = 1) +
labs(title = "5,000 sample means, n = 25 each",
x = "Sample mean", y = "Count") +
theme_minimal(base_size = 12)
Visualizing the Difference
set.seed(301)
# Our "population": 100,000 adults with mean height 170cm, sd 10
population <- rnorm(100000, mean = 170, sd = 10)
# Take a single sample of 25 people
sample_mean <- mean(sample(population, 25))
ggplot(tibble(x = population), aes(x)) +
geom_histogram(bins = 50, fill = "#3498DB", color = "white") +
geom_vline(xintercept = mean(population), color = "#E74C3C", linewidth = 1) +
geom_vline(xintercept = sample_mean, color = "#2E8B57", linewidth = 1) +
labs(title = "Population vs. Sample",
x = "Height (cm)", y = "Count") +
theme_minimal(base_size = 12)
Visualizing the Difference Plot
Another View Sample Means != Population Mean Plot!!
Is There a Way to Predict The Error in Our Sample Mean?
Standard Error: Plot
se_table <- tibble(n = c(4, 16, 25, 100, 400, 1600)) %>%
mutate(standardError = 10 / sqrt(n))
ggplot(se_table, aes(n, standardError)) +
geom_line(color = "#8E44AD", linewidth = 1) +
geom_point(size = 3, color = "#8E44AD") +
labs(title = "Diminishing returns: quadruple n to halve the error",
x = "Sample size", y = "Standard error") +
theme_minimal(base_size = 12)
The Central Limit Theorem Plot
