# Complete Means Demonstration

### How do I do the Complete Means Demonstration?

1. Enter the size of the population (2-5).  For example, if you select Population size = 5, the population will consist of the numbers 1, 2, 3, 4, and 5.  If you select Population Size = 3, the population will consist of the numbers 1, 2, and 3.

2. Enter the sample size (1-5).  For example, if you select sample size = 2, the computer will randomly select 2 values from the population each time it draws a sample.

3. Choose OK

4. Pressing the number keys 1 - 9 will cause a delay of from 1 /10 to 8 seconds , respectively, for each sample.  If you press the 0 (zero) key, the demonstration will finish as fast as possible.  Pressing the Escape key will abort the demonstration.

### What will I see in the Complete Means Demonstration?

The left hand side of your screen will consecutively display all of the possible samples of the size that you selected, given the parameters of the population that you selected, and their mean.  Each time the computer draws a sample and calculates the mean, it will plot the mean in a distribution that is displayed on the right hand side of your screen.  When the distribution of sample means is complete, the computer superimposes a theoretical normal curve onto it.

Also displayed with this distribution is the number of samples that the computer can draw given the parameters that you selected, and the sample size.  (The number of possible samples that the computer can select is equal to the size of the population raised to a power equal to the sample size.  For example, if you choose a Population Size = 3, and Sample Size = 2, the number of possible samples is 3 to the power 2, = 9 samples.)

### Purpose of the Complete Means Demonstration

The purpose of the Complete Means Demonstration is demonstrate important relationships between a population distribution and a distribution of sample means drawn from that population.  These relationships are:

I) that the mean of the population and the distribution of sample means will be the same,

ii) that the standard deviation of the sample mean distribution will be equal to the population standard deviation divided by the square root of the sample size you select, and

iii) that the distribution of sample means (calculated from sample sizes greater than 1) will be approximately normally distributed.

In this demonstration, you are limited to a small population and a small sample size in order that you can reasonably view the whole process of building the distribution of sample means from a population.  An understanding of the relationships outlined above will be critical to your understanding of many inferential statistical techniques where you will be dealing with populations and sample sizes much greater than those in this demonstration.  The Central Limit Demonstration is an extension of this demonstration, with the difference that the population and sample size are much greater.

### Plain English Translation

When you run the Complete Means Demonstration, you will see that the distribution of the sample means is i) normal in shape, ii) has the same mean as the population and iii) has a standard deviation equal to the population standard deviation divided by the square root of the sample size that you select. Let`s go through a concrete example.

We will choose a Population Size of 3, and a sample size of 2.

Our population consists of the numbers 1, 2, and 3.  The mean of this population is 2:

The standard deviation of the population is 0.82

Here are all the possible samples of size 2, with their means in parentheses, that can be drawn from this population (the computer will show you these samples when you run the demonstration).

1,1 (1)

1,2 (1.5)

1,3 (2)

2,1 (1.5)

2,2 (2)

2,3 (2.5)

3,1 (2)

3,2 (2.5)

3,3 (3)

The mean of the sample means is  ( 1+ 1.5 + 2 + 2.5 + 3 ) / 9 = 2  (the same mean as the population).

The standard deviation of the sample means is 0.58

One more step is required to show point iii) that the distribution of the sample means has a standard deviation equal to the population standard deviation divided by the square root of the sample size that you select.