Variance - GSP

In order to do this section, download this Geometer's Sketchpad file called Variance.sea.hqx for the Macintosh or Variance.zip for the PC.

Note: If you are familiar with Sketchpad, continue with the worksheet below. If not, check out this short Sketchpad Tutorial that will teach you enough about Sketchpad to do this lesson.
The Skecthpad file has five points, P1, P2, P3, P4 and P5, located on the x-axis. We will use these points to represent different data points. You can move these points along the x-axis to change the value of the data point. The points' coordinates are shown in the upper right hand corner so you can place the point where you need to. The dashed line represents the mean of the five data points. Move some of the points along the axis and watch what happens to the mean. Its value is also shown in the upper right hand corner. Ignore the squares for now.

1. Place the points, P1 through P5 at 1,2,3,4,5. What is the mean of this data?

2. Now, place the points at 0,1,3,5,6. Again, what is the mean?

3. Do you think that only the mean can accurately describe these data sets?

4. Can you think of another way to describe this data set?

1. Another way to describe this data set is to measure the deviation of each data point. The deviation of a data point is its distance to the mean.

1. Drag the points to positions 1,2,3,4,5. Find the difference between each data point and the mean.

2. Which points have a negative deviation? a positive deviation? Find the mean of these deviations.

3. Drag the data points to 0,1,3,5,6. Again, find the difference between the data points and the mean and which points have negative and positive deviations.

4. Compare your answers for both data sets. Do you think that this mean of deviations is a good method to describe data sets?

5. Can you think of a way to slightly change it to make it better? Hint: How important is the sign (+ or -) of the deviation?

2. One way to solve the above problem is to take the absolute value of the deviation.

1. Drag the points to 0,2,4,5,6. Find the absolute value of the distance between each data point and the mean.

2. Find the mean deviation for this data.

3. Now, drag the points to 0,1,3,6,7. Again, find the absolute value of the distance between each data point and the mean and the mean deviation.

4. In your own words, describe both of the above data sets in terms of the mean and the mean deviation.

3. Instead of taking the absolute value of the deviation, another way of eliminating the negative sign is to square the deviation.The figures in the Sketchpad file are squares with sides equal to the deviation.

1. Place the points at 1,2,3,4,5. Find the area of each square. Which data points measure the biggest area? the smallest area?

2. In your own words, describe the relation of these points with respect to the mean.

1. Drag the points to 0,1,2,4,5. Find the area of each square.

2. Find the mean of the sum of squares.

3. The mean of the square of the deviations is called the variance. In Sketchpad, this square is shown where it says, Variance. In your own words, describe what the variance measures.

1. Make sure the points are at 0,1,2,4,5.

2. Find the variance of this data set.

3. Next, shift each data point one unit to the right (so they are located at 1,2,3,5,6.) Again, find the variance of this data set.

4. Next, add two to the original data (so they are located at 2,3,4,6,7). One more time, find the variance.

5. Conjecture on what happens to the variance of a data set when a constant is added to each data point.

6. Does this surprise you? Explain why.

1. Drag the points to 0,0.5,1,1.5,2. Find the variance.

2. Next multiply each data point by 2 and move them to their new location. What is the variance?

3. Now multiply each point by 3 and find the variance once more.

4. Conjecture on what happens to the variance of a data set when a constant is multiplied to each data point. Hint: The square of 2 is 4 and the square of 3 is 9.

5. Explain why this makes sense.

1. Michael Jordan and Scottie Pippen are having an argument over who is the better free throw shooter. They had a contest where they took 5 free throws each and did it five times. The results (free throws made) were the following:

Jordan - 4,3,4,3,4
Pippen - 2,5,2,5,3

After the contest they both still think that they are better free throw shooter.

1. Find the mean for each shooter's scores. Could you decide based on your answers who the better shooter is?

2. Find the variance of each shooter's results and use your answer to explain who you would rather have shoot the game winning free throw.

2. Two students who took a statistics class received the following scores (out of 5.0).

Student A - 3.0,4.5,4.0,3.0,4.0
Student B - 2.0,5.0,5.0,2.0,4.5

If you had an upcoming statistics test, who would you rather have as a study partner, A or B? Support your answer by finding the means and variances for both students.