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How Choosing the # Trials vs. # Experiments Changes Results (15 minutes)
- This activity is best done after students have found the probabilities of a spin-$\frac{1}{2}$ system.
- While the students are performing the probability activity, write on the board data from the Stern-Gerlach experiment for ten experiments where ten events occur in each experiment. For this set of data, calculate the mean value for each experiment, the probability for getting the measured result, and the experimental standard deviation.
- Now, take this data and treat it as if one experiment were performed with one hundred events in it. Repeat the same calculations.
- Perform the same operation treating the data as one hundred experiments with one event in each experiment.
- Write out a data table on the board that shows how the important calculated values differ. In particular, have students notice the standard deviation changes significantly.
Sample Data
| $x_{1}$ | $x_{2}$ | $x_{3}$ | $x_{4}$ | $x_{5}$ | $x_{6}$ | $x_{7}$ | $x_{8}$ | $x_{9}$ | $x_{10}$ | |
|---|---|---|---|---|---|---|---|---|---|---|
| # Spin-up particles | 7 | 5 | 5 | 8 | 5 | 7 | 8 | 2 | 7 | 8 |
M=Number of events in a single experiment
N=Number of experiments
| M=100,N=1 | M=10,N=10 | M=1,N=100 | |
|---|---|---|---|
| $P=\frac{\bar{x}}{N}$ | .59 | .59 | .59 |
| $\sum x_{i}^2$ | 3481 | 385 | 59 |
| $N \bar{X}$ | 3481 | 348.1 | 34.81 |
| $S^2$ | 0 | 4.1 | .244 |
| $S_{p}=\frac{S}{M}$ | 0 | .203 | .414 |
| $\sigma_{M}=\frac{S}{\sqrt{N}}$ | 0 | .64 | .049 |
| $\sigma_{p}=\frac{\sigma_{M}}{M}$ | 0 | .064 | .0049 |
