Sampling Distribution of the Mean Tool

Instructions for using the Sampling Distribution of the Mean Tool

Sampling Distribution of the Mean: An online, printable lecture.

Sampling Distribution of the Mean Tool Instructions

Note: These instructions are abstracted from and can be supplemented by the full web lecture on the Sampling Distributions available through another link on this page.

These instructions are for the SDM Tool only. They assume you know about or have read the accompanying web lecture on sampling distributions.

Top two panels

Upper Right Hand Panel. You'll find two distributions in the upper right hand panel of the SDM Tool. The black distribution is the population. The red distribution is the sampling distribution of the mean. They have the same mu so the two distributions are centered at the same value (because, of course, they have the same center). The population and SDM differ only in that they have different standard deviations. And so the population (which has a larger sigma) is lower and wider and the SDM (which has a smaller sigma) is thinner and taller.

Upper Left Hand Panel. Looking at the upper left hand panel, you see an interface where you can enter information. There is a place where you can set population mu depending on what's given in the homework or test problem. You can also set the population standard deviation. Finally, you can set the sample size, n.

Sample size, n, along with population mu and sigma, are the three really important pieces of information you need to get from a word problem. You need to set all three of them to use the SDM Tool. As an example to work with, set population mu = 100, sigma = 5, and n = 10. Set mu, sigma, and n. PRESS UPDATE.

The SDM tool will immediately and automatically give you the mean and standard deviation of the SDM. (Remember that the standard deviation of the SDM is also called the standard error of the mean or SEM.)

Lower two panels

Press the "Sample" button in the lower right hand corner panel.

Lower Left Hand Panel. Sample scores will appear in the lower left hand panel when you press the "Sample" button. (The "Sample" button is in the lower right panel.)

The sample scores are called empirical data since they correspond to the data collected in a research project. The SDM Tool calculates the mean of the scores automatically for you. This mean is the empirical mean that a scientist would calculate in a research project. It is always important to distinguish between the theoretical population mean, mu, and the empirical research mean.

New sets of empirical data will appear every time you press the "Sample" button. Notice that (empirical) individual scores are written in black. That is because they are sampled from the (theoretical) population which is black. In contrast, the (empirical) sample mean is written in red. That is because it is sampled from the (theoretical) SDM which is red.

Click the "Sample" button many times. Notice how the sample data and the sample mean change with each new sample you take. The theoretical populations are constant and unchanging. The empirical data change with every sample. Just stare at the data as you click or just stare at the sample mean. Notice how their values change.

Lower Right Panel. As you click "Sample" many times, also notice the lower right panel. Every time you click "sample" a small red hatch mark appears representing that mean. Each empirical mean is a different value so the hatch marks are placed in different places along the number line. If two means have values very close to each other the hatch marks are piled on top of one another. Click "Sample" quickly many times to get a sense of this.

If you click quickly, over and over, these hatch marks will eventually stack up and begin to take the shape of the Sampling Distribution of the Mean. This shows you that across huge numbers of samples there is an empirical frequency distribution of sample means which looks roughly like the theoretical SDM (shown above it in the upper right hand panel).

1 to 10,000 Samples per click

There's a pop-up menu in the lower right panel just below where the red hatch marks appear. There's a small white window. Next to the window is an arrow. Click on the arrow. A menu will pop down. This menu will give you the choice of how many samples you can draw with a single click. You can draw 1, 5, 10, 50, 100, 500, 1,000, or 10,000 samples with a single click. This allows you to easily see the evolution of the shape of the frequency distribution of sample means.

Play with taking large numbers of samples with a single click. You'll notice that the shape of the empirical distribution of sample means quickly conforms to the normal shape of the theoretical SDM.

How many samples are necessary before you think the frequency distribution of empirical means closely takes on the shape of the the theoretical SDM? Do you need hundreds of samples? thousands? tens of thousands?

A central idea in using this tool is to compare the theoretical SDM (upper right panel) with the distribution of sample means which pile up empirically (lower right panel).

[Note: The empirical data and mean in the lower left panel only change when the number of samples per click is set to 1. This is because when more than one sample is collected with a single click, the computer samples means directly from the SDM rather than sampling n different scores from the population and then computing the sample mean. This allows it to get 1,000 or even 10,000 mean very quickly.]