1. Select Variance Demonstration from the main menu.
2. Choose the total Number of Samples you want the computer to randomly select (a number between 1,000 and 50,000).
3. Choose the Sample Size - the number of observations that you want the computer to select for each sample (a number between 2 and 50).
4. Choose the population from which the computer draws its samples (normal, uniform, triangular or v-shaped).
5. Choose OK and the computer will build the distribution of sample variance scores. Before the computer adds a point to the distribution, it shows you the sample values, the sum of the sample values, and the sample variance score
6. 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. Press the Escape key if you want to terminate the sampling. When the End key is pressed, the computer displays a dialog box, "Pause Again?", and pauses until any key is pressed or 'YES' or 'NO' is clicked -- If 'YES' was clicked or the End key was pressed again, the computer pauses again.
Your selections will be displayed in the top corners of the window, the sample values that contribute to each mean and the variance score will appear at the bottom of the screen, and the Distribution of Variance Scores will begin to form. If you have selected a sample size between 1 and 10, then, for each sample, the computer displays the values that it has randomly selected as well as their mean and variance. For sample sizes greater than 10 only the variance is displayed. When the distribution of variance scores is complete, the computer superimposes an idealized chi-square curve onto it. The population variance is displayed in red on the horizontal axis.
The purpose of the Sample Variance Demonstration is to demonstrate that Sample Variance Distributions are positively skewed, especially for small sample sizes.
The Distribution of Sample Variance is positively skewed, especially for small sample sizes.
Construct 4 Distributions of Sample Variance by a) choosing the normal population as the population from which to draw the samples each time, b) keeping the number of samples constant in each demonstration (10,000), and c) varying the sample size for each distribution. Make your first distribution with a sample size of 5, your second with 10, your third with 30, and your fourth with 50.
You will see that increasing your sample size decreases the skew of the distribution, but that even for large sample sizes, the distribution is still positively skewed (the peak of the distribution is to the left of the population variance, indicated in red on the horizontal axis). What this means in practice is that when you calculate a sample variance, that variance will more likely be an underestimate of the population variance than an overestimate. You can also see that the bigger the sample size, the more likely it is that the sample variance will more closely approximate the population variance.
Remember that the definitional formula for calculating the sample variance is
(The compuational formula is
When you run the Sample Variance Demonstration, you select a population from which to draw a number of samples (between 1000 and 50000), and the sample size. Let's say you select 10000 samples of size 3 from the Normal Distribution. The computer randomly selects 3 values between -5 and +5, and calculates the sample variance. For example, we'll say that the computer randomly draws the following values: -0.88, -0.71 and +0.63. The computer will show you these values, as well as the sum of these values (-0.96), and the sample variance 0.69.
The computer will plot 0.69 in the Distribution of Sample Variance, select another 3 values, plot their sample variance and so on, until it has randomly generated 10000 samples and plotted all of these sample variances. The resulting distribution is the Distribution of Sample Variance for 10000 samples of size = 3, drawn from a Normal Population with values ranging from -5 through to +5. Note that this distribution is positively skewed, that is, a large number of values occur to the left of the population variance (printed in red on the horizontal axis), with fewer values to the right of the population variance.