The central limit theorem

Introduction to Statistics in R

Maggie Matsui

Content Developer, DataCamp

Rolling the dice 5 times

die <- c(1, 2, 3, 4, 5, 6)

# Roll 5 times
sample_of_5 <- sample(die, 5,
                      replace = TRUE)
sample_of_5

1 3 4 1 1

mean(sample_of_5)

2.0

six sided die

# Roll 5 times and take mean
sample(die, 5, replace = TRUE) %>% mean()

4.4

sample(die, 5, replace = TRUE) %>% mean()

3.8

Repeat 10 times:

sample_means <- replicate(10, sample(die, 5, replace = TRUE) %>% mean())

sample_means

3.8 4.0 3.8 3.6 3.2 4.8 2.6 3.0 2.6 2.0

Sampling distribution of the sample mean

histogram of 10 sample means

replicate(100, sample(die, 5, replace = TRUE) %>% mean())

2.8 3.2 1.8 4.6 4.0 2.8 4.4 2.4 3.4 2.8 4.2 3.4 ... 2.2 3.8 3.6 3.8 4.4 4.8 2.4

histogram of 100 sample means

sample_means <- replicate(1000, sample(die, 5, replace = TRUE) %>% mean())

histogram of 1000 sample means

The sampling distribution of a statistic becomes closer to the normal distribution as the number of trials increases.

histograms of 10, 100, and 1000 sample means, where higher number of sample means has a more bell-curve shaped distribution

* Samples should be random and independent

replicate(1000, sample(die, 5, replace = TRUE) %>% sd())

Distribution of 1000 sample standard deviations of 5 die rolls

sales_team <- c("Amir", "Brian", "Claire", "Damian")

sample(sales_team, 10, replace = TRUE)

"Claire" "Brian"  "Brian"  "Brian"  "Damian" "Damian" "Brian"  "Brian" 
"Amir"   "Amir"

sample(sales_team, 10, replace = TRUE)

"Amir"   "Amir"   "Claire" "Amir"   "Amir"   "Brian"  "Amir"   "Claire" 
"Claire" "Claire"

Distribution of sample proportions also looks normal

# Estimate expected value of die
mean(sample_means)

3.48

# Estimate proportion of "Claire"s
mean(sample_props)

0.26

Sampling distribution of sample means with dashed line down the middle

Introduction to Statistics in R