To see for yourself how the central limit theorem works, let's say we have a normal distribution (with mean =100 and standard devation = 20).

To see for yourself how the central limit theorem works, let's say we have a normal distribution (with mean =100 and standard devation = 20). Let's generate some random samples of various sizes from this distribution. We can do this in excel using =norm.inv(rand(),100,20) and it will randomly generate numbers from this distirbution.

I generated four samples of size 5, 10, 20 and 30, and got the means of 124 (n=5); 91 (n=10); 105 (n=20); 103 (n=30). If I continue to increase the sample size, my average values should converge to the mean of 100.

Now you try. Pick a distribution and generate some sample sizes with the number 15, 25, 35, 40 to prove this to yourself. Please post the results in excel using the above information and the information below regarding the central limit theorem and example:

The central limit theorem basically says that as sample size increases, the sampling distribution of the sample will be bell shaped (usually around n=30). This happens because the distribution gets tighter around the mean and less dispersed. For example, let's say we have a uniform distribution,which is essentially a horizontal line as our population distribution. If we take a sample of size 30 from this population, the sampling distribution will NOT be a straight line but be bell shaped. Because of this property, we use z values to tell us how many standard deviations our sampling mean is either above or below a population mean. So the key statistical question we are asking is how close our sample mean (xbar) is to the population mean (mu). We can answer this question if we convert our sampling distribution to a standard normal curve. We do this by using the formula z = (any x - mean)/std dev. z values that are positive tell us that our data value is above the pop mean; z values that are negative tell us that our data value is below the pop mean; and z value of zero tells us we are at the mean. When z = 3 or -3, this means that our is 3 standard deviations either above or below the mean. We know that 99.7% of data within a bell curve is within +/- 3 standard deviations, so data outside of +/- 3 standard deviations has a very low probability (1-.997 = .003 for both tails or 0.003/2 = 0.0015 for either tail). Because of this, we can use z values to identify outliers, or data values that we would not expect to generate if they were randomly generating values.