N otes on Chapter 5
pp 39 - 49 of your lab book

This lab is actually kind of fun! What we are going to do is throw darts at a target (fun) and then analyze the data we get with the statistical error analysis we learned last meeting (not so fun).

Analyzing the data
As we learned last time, Gaussian distributions are handy for characterizing data with random error. In this experiment, you want to compare actual data with random error to a Gaussian distribution. The data with random errors is the distribution of darts. You expect that the more darts you throw, the closer your distribution comes to a Gaussian.  Recall from last time that a normalized Gaussian has the form:

F(x) represents the probability that you will get the value x on the next trial. W, the width of the curve, is also the standard deviation. P, the peak value of the curve, is also the average value. What we will do is get the standard deviation and average of the actual data, and plug that into this formula to see how close our real data comes to a Gaussian.

Since f(x) is the probability that you will get the value x, and the probability is:

the number of times you saw the value x
the total number of trials

You can find the number of times you see the value x by multiplying f(x) by N, the total number of trials. If you do this, you will get the formula in the book (4.2, pg. 30). However, if you do this, the Gaussian is no longer normalized (the area is not equal to 1).

Procedure
1. Obtain 5 - 10 darts per group.
2. Stand about 6-8 feet from the board. Throw your darts at the board.  The center of the board is what you are aiming for (the line between bin 24 and 25).
3. Count the number of darts which land in each bin on the board.  Do not count darts which land outside the sheet.
4. Continue throwing darts until you have reached a total of 200 thrown darts.
5. Write the total number of darts in each bin, the average bin number, your standard deviation, and your SDM on the chalkboard in the space alloted to you.
6. Plot the experimental number of darts versus the bin number.
7. Calculate the average and standard deviation for the class data.
8. With the standard deviation, calculate the number of darts that would have landed in the bin if the class data was a perfect Gaussian.

Safety notice:

1. The darts are sharp! Be very careful with them.  
2. Do not throw darts when a group next to you is retrieving their darts from the board.
3. Do not walk in the space between the lab tables and the dart boards.
4. Anyone behaving in an unsafe manner will be thrown out of the lab and will recieve a zero for the lab!

Assignments:

1. Answer questions 1, 4, and 6 on page 49.
2. The report format has been modified from its original format slightly for this lab: the point distribution is now: Objective (10 points), Procedure (10 points), Data (10 points), Calculation (40 points), Conclusion (20 points), Questions (10 points). As you can see, the points normally assigned to Discussion/Analysis is now in Calculation, the points from Other went into questions.

Calculations: Make sure you always follow the calculation style described in the writing reports section. (That is: write out the equation, show substitutions, then show the answer). I strongly recommend using Excel in this report to find the numerical values.

Show:

• how you got your average, SD, and SDM.
• how you got the class SD and SDM. (Be careful! Don't average the SD's of the groups!!)
• how you got the average of the averages and the SD of the averages. These numbers should NOT be the same as the previous ones
• how you got the theoretical data in Table III
• a graph of BOTH the actual data and the theoretical data. The graph should be Number of Darts vs. Bin Number. (Number of Darts, n_x, is on the y axis. Bin Number, x, is on the x axis.)

Normally, SD and SDM go in the error analysis section of the Discussion; they will revert to there in the next lab.

Conclusion: Make sure you tell me about how statistics and uncertainty are related to probability. I am looking to see if you have a clear idea of what the SD and SDM mean; make sure you demonstrate this.