Jan 02

Central Limit Theorem (CLT) - 10 minute tutorial

by David Harper, CFA, FRM, CIPM

FRM |

The central limit theorem says that the sampling distribution of the sample mean will be (tends to be) approximately normal, regardless of the distribution of the population. Does that sound like boring technical stuff? I hope not! The CLT is a showstopper of uncanny power...it helps explain why we seem to always find ourselves using the normal distribution.

For example, a single six-sided die has a uniform distribution (i.e., the odds of each outcome are equally likely). Now roll, say, nine dice together and add them up (or take their average). That roll is a single sample with expected mean of 3.5 (or, you'd expect their total to be 31.5 because 9 x 3.5 = 31.5). Then roll the nine dice again. That's a second sample. Then again, and so on. The mean or sum of each sample (roll of nine dice), unlike the single die, is not uniformly distributed; it tends to be normally distributed! Intuitively, you expect more often the total of the dice to be nearer to 31 or 32 than you expect to get a total of nine (all ones) or 54 (all sixes). The larger our sample size, the "more normal" the sampling distribution of our sample mean!

clt_1

In the movie below, I use a spreadsheet to prove this. We use the Excel RAND() function which is uniform to produce approximately normal distributions. Further, I also use this function: the sum of twelve RAND()s minus (-) 6. That's a random variable that we can use to approximate a standard normal distribution (i.e., mean of zero and standard deviation of one).

Comments

  1. Manish Sinha 16 Jan 2008

    I am unable to view this 7 minute movie tutorial. Please help.

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