File:Empirical CLT.svg

Summary[edit]

DescriptionEmpirical CLT.svg	English: This figure demonstrates the central limit theorem. It illustrates that increasing sample sizes result in sample means which are more closely distributed about the population mean. It also compares the observed distributions with the distributions that would be expected for a normalized Gaussian distribution.
Date	12 November 2014
Source	@ Wikipedia Commons
Author	User:Gerbem (2011) & User:Chen-Pan_Liao

Licensing[edit]

This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/3.0CC BY-SA 3.0 Creative Commons Attribution-Share Alike 3.0 truetrue

R code to generate this figure (not including all details; using other software to modify)[edit]

# fixed random seed
set.seed(123)

# sample mean of U(0, 500)
m.unif.30  <- rowMeans(matrix(runif(500*250, 0, 100), nrow=500)[,1:30])
m.unif.100 <- rowMeans(matrix(runif(500*250, 0, 100), nrow=500)[,1:100])
m.unif.250 <- rowMeans(matrix(runif(500*250, 0, 100), nrow=500)[,1:250])

# approximated sample mean based on CLT
sigma <- sqrt((1/12)*(100-0)^2)
m.norm.30   <- rnorm(500, mean(m.unif.30),  sigma/sqrt(30))
m.norm.100  <- rnorm(500, mean(m.unif.100), sigma/sqrt(100))
m.norm.250  <- rnorm(500, mean(m.unif.250), sigma/sqrt(250))

# plot
par(mfrow = c(2,3), mgp = c(2,0.7,0), width = 10, height = 6)
hist(
	m.unif.30,
	main ="N = 30", xlab="Sample mean", ylab="Observed frequency",
	xlim=c(30,70), ylim=c(0,200)
)
hist(
	m.unif.100,
	main ="N = 100", xlab="Sample mean", ylab="Observed frequency", 
	xlim=c(30,70), ylim=c(0,200)
)
hist(
	m.unif.250,
	main ="N = 250", xlab="Sample mean", ylab="Observed frequency", 
	xlim=c(30,70), ylim=c(0,200)
)
hist(
	m.norm.30,
	main ="N = 30", xlab="Approximated Sample mean", ylab="Observed frequency", 
	xlim=c(30,70), ylim=c(0,200)
)
hist(
	m.norm.100,
	main ="N = 100", xlab="Approximated Sample mean", ylab="Observed frequency", 
	xlim=c(30,70), ylim=c(0,200)
)
hist(
	m.norm.250,
	main ="N = 250", xlab="Approximated Sample mean", ylab="Observed frequency", 
	xlim=c(30,70), ylim=c(0,200)
)