Neuroscience Animations

John H. Krantz, Hanover College, krantzj@hanover.edu

Central Limit Theorem

Brief description and instructions (DRAFT):

Background:

The Central Limit Theorem is about what we can know about the range of possible means that can be found from samples of a certain size. In a study, you take a sample from a population. To do an inferential statistical test, it is necessary to be able to describe in some fashion how that mean can vary around the actual population mean. Does this sample mean have to be close to the population mean or can it be far off? The central limit theorem allows us to answer this question.

To understand the Central Limit Theorem, it is important to understand the concept of a sampling distribution. A sampling distribution describes all the possible values for a given statistic, e.g., a mean, that can happen for a specific sample size. So the Central Limit Theorem tells us about this distribution for means.

The Central Limit Theorem makes these claims:

The mean of the sampling distribution equals the mean of the population.
If the population is normally distributed, so will the sampling distribution of the mean.
Even if the population is not normally distributed, if the sample is "large enough", the sampling distribution of the mean will still be normally distributed.
The standard deviation of the sampling distribution of the mean (called the standard error)=the standard deviation of the population/the square root of the sample size.

With these claims, we can know how and how much the mean of samples will vary from the true population mean allowing us to do inferential statistics comparing means.

Using the illustration:

When you open the applet, the main portion of the screen will be taken up with 6 graphs arranged in three vertically arranged pairs. Each of these graphs will plot histograms which are plots of how often a given value or range of values happen. Each of the three sets of graphs will sample from a different distribution for the population, this distribution is listed at the top of the upper histogram of the pair. At the beginning of the program, the left pair of histograms will sample from a Normal distribution, the middle from a positively skewed distribution, and right pair from a bimodal distribution.

There are two basic ways to use this illustration: Compare Distributions and Compare Sample Size which can be selected on the left hand side of the screen.

Compare Distributions

Below is a discription for using the acitivity in the Compare Distributions mode.

The top row of histograms will sample from the populations one value at a time (n=1) and thus create a histogram of a sample distribution. In this way you will be able to see the spread of value from the population. The bottom row, while sampling from the same distribution as the top row, will actually take a same of n= some value, indicated on each of the lower histograms and plot the mean of this sample. So these are sampling distributions. When the applet opens, the n=2. You can change the size of the sample on a range from 2 to 256 (each step is twice the smaller sample size) with buttons to the left of the lower row of the histograms.

To start the sampling, press the Start button in the upper left corner of the screen. The number of samples taken (actual number of values for the sample distributions in the top row and the number of samples of size n in the sampling distributions in the lower row) is shown below the Reset button on the upper left corner. As each sample is taken, the value or the mean will be plotted on its proper histogram. You can see the histogram develop as the number of samples increase. To stop the sampling press the Stop button and to start over, press the Reset button. If you change the sample size the next time you start the sampling, the sample will be reset.

There are also advanced option available in the menus. You can change the bin size of the histograms, the speed the samples are taken and the distributions for each of the three pairs of histograms.

Compare Sample Sizes

Below is a description for the Compare Sample Sizes mode of operation emphasizes the differences from the Compare Distributions mode.

There is only one distrubtion visible in the top row, the left distribution. You can pick the distribution for the population with the Left Distribution menu at the top of the screen. The bottom of the screen will show sampling distributions of means sampled from the distribution of the left distribution. The smallest sample size will be on the left graph with each graph to the right having double the sample size of the next graph to the left. The sample size of these graphs can be controlled by the Sample Size buttons on the left hand side of the screen.

Click here to open the applet. It will open a new window that will fill your screen.

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