{"id":49,"date":"2011-09-14T11:53:14","date_gmt":"2011-09-14T11:53:14","guid":{"rendered":"http:\/\/www.clayford.net\/statistics\/?p=49"},"modified":"2023-08-18T07:05:25","modified_gmt":"2023-08-18T11:05:25","slug":"evidence-of-the-truth-of-the-central-limit-theorem","status":"publish","type":"post","link":"https:\/\/www.clayford.net\/statistics\/evidence-of-the-truth-of-the-central-limit-theorem\/","title":{"rendered":"Evidence of the truth of the Central Limit Theorem"},"content":{"rendered":"

The Central Limit Theorem is really amazing if you think about it. It says that the sum of a large number of independent random variables will be approximately normally distributed almost regardless of their individual distributions. Now that’s a mouthful and perhaps doesn’t sound terribly amazing. So let’s break it down a bit. “A large number of independent random variables” means any random variables from practically any distribution. I could take 6 observations from a binomial distribution, 2 from a uniform and 3 from a chi-squared. Now sum them all up. That sum has an approximate normal distribution. In other words, if I were to repeatedly take the observations I stated before and calculate the sum (say a 1000 times) and make a histogram of my 1000 sums, I would see something that looks like a Normal distribution. We can do this in R:<\/p>\n

# Example of CLT at work\r\ntot <- vector(length = 1000)\r\nfor(i in 1:1000){\r\n  s1 <- rnorm(10,32,5)\r\n  s2 <- runif(12)\r\n  s3 <- rbinom(30,10,0.2)\r\n  tot[i] <- sum(s1,s2,s3)\r\n}\r\nhist(tot)<\/pre>\n
\"\"<\/a><\/p>\n
See what I mean? I took 10 random variables from a Normal distribution with mean 32 and standard deviation 5, 12 random variables from a uniform (0,1) distribution, and 30 random variables from a Binomial (10,0.2) distribution, and calculated the sum. I repeated this a 1000 times and then made a histogram of my sums. The shape looks like a Normal distribution, does it not?<\/p>\n
Now this is NOT a proof of the Central Limit Theorem. It's just evidence of its truth. But I think it's pretty convincing evidence and a good example of why the Central Limit Theorem is truly central to all of statistics.<\/p>\n","protected":false},"excerpt":{"rendered":"
The Central Limit Theorem is really amazing if you think about it. It says that the sum of a large… Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8,13],"tags":[],"class_list":["post-49","post","type-post","status-publish","format-standard","hentry","category-simulation","category-using-r"],"_links":{"self":[{"href":"https:\/\/www.clayford.net\/statistics\/wp-json\/wp\/v2\/posts\/49","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.clayford.net\/statistics\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.clayford.net\/statistics\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.clayford.net\/statistics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.clayford.net\/statistics\/wp-json\/wp\/v2\/comments?post=49"}],"version-history":[{"count":2,"href":"https:\/\/www.clayford.net\/statistics\/wp-json\/wp\/v2\/posts\/49\/revisions"}],"predecessor-version":[{"id":862,"href":"https:\/\/www.clayford.net\/statistics\/wp-json\/wp\/v2\/posts\/49\/revisions\/862"}],"wp:attachment":[{"href":"https:\/\/www.clayford.net\/statistics\/wp-json\/wp\/v2\/media?parent=49"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.clayford.net\/statistics\/wp-json\/wp\/v2\/categories?post=49"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.clayford.net\/statistics\/wp-json\/wp\/v2\/tags?post=49"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}