> Central Limit Theorem Illustrated

Central Limit Theorem Screen

The complete screen image is contained at the bottom of this page. Simulation controls are described below that image. After you have read this material, you will have an informed understanding of the animations on the other CLT-I page.

The red histogram shows the observations in the most recent random sample. You can count 10 observations.

Another part of the screen indicates the observations were sampled from a normal distribution.

Material in red says something about the one sample of observations.

The green histogram shows the 6,632 means each based on a random sample of 10 observations.

Material in green indicates information about many samples.

Notice the means are distributed normally. That would have been the case even had the parent distribution have been something other than Normal.

Information presented in blue relates to the parent population and is a theoretical value, not an empirical value.

The top blue line is centered on the population mean and extends one standard deviation below the mean to one standard deviation above the mean.

The top blue line shows graphically that single observations in the top (red) histogram should fall within the range of the line about 2 times in 3 (actually 68%). Note that 6 or 7 of the 10 observations in the displayed sample are within this range--about what would be expected. For this simulation, the population standard deviation was set to be 10.0 units.

The top red line shows the stnadard deviation calculated for the current data set. I'm sure you notice red line is almost identical to the the blue line showing the sample standard deviation accurately estimated the population standard deviation (based on the current sample). For this sample of 10 observations, the sample standard deviation was calculated to be 14.768. This calculation used the (n - 1) divisor since we want to compare it to the value of the population standard deviation.
The second blue line (population standard error of the mean) is calculated from the population standard deviation. Whatever the population standard deviation is, divide it by the square root of sample size to get the standard error of the mean: 15 ÷ √10 = 4.7434.

The populaton standard error of the mean (SEM) describes the standard deviation of sample means for a particular size. So about 2/3 of the means of samples of size 10 should be within the shorter blue bar. You can get a better idea of this by examining the entire screen shown at the bottom of the page.

There are two estimates of the population SEM. One estimate calculates the standard deviation of all of the means (4.6303) which is within a tenth of a unit of the population SEM.

The other estimate is shown in pink and is the standard deviation of the current sample divided by the square root of sample size. This value, for this sample, is 4.670, again close to the true value. This item is colored pink to make it stand out from all the other information on the screen. It's that important.

This is the important truth to retain from this simulation: a single sample reveals how much the mean of a sample is likely to jump around from one sample to the next.

The final item displayed is the location of the mean of the current sample. In this instance the sample mean (94.12) is more than one SEM (based on the standard deviation of the sample) from the population mean AND it is less than two SEM from the population mean.

The material in black relates to the way the simulation displays information.

These are the values mentioned earlier on the page.

©2006 by Burrton Woodruff. All rights reserved. Modified 10/24/06