Experiment 1

Laboratory 1

Uncertainty and statistics

Max Woo
10/16/14
Lab 11, Michael Ip
Partners: Priyank Mehta

Introduction

The purpose of this lab was to investigate the standard deviation for increasingly large data sets, and to verify the method for calculating the propagation of error. For the experiment, we sampled the fluctuating voltage output of a circuit board using a digital voltmeter.

Experimental Results

Part 1

For the first part, we connected the circuit board to a 30V power source, and then connected the first voltage output of the circuit board $V_1$ to the first voltage input of the voltmeter (the positive terminal of $CH0$), making sure to connect the negative terminal of $CH0$ to ground. We then connected the voltmeter to the computer, and using the “4BL Software Program”, we configured the voltmeter to sample at a rate of 1000 points/second. For the sample size, we switched between 6 different values: 10, 100, 500, 1000, 5000, and 10000, measuring the mean and standard deviation 10 times for each, giving a total of 60 data sets.

We then calculated the standard deviation of the mean using two methods. For the first method, we simply took the standard deviation of the measured means for each sample size. The results are graphed below:

Graph 1: A logarithmic graph of the standard deviation of the mean voltage measurements for sample sizes 10, 100, 500, 1000, 5000, and 10000

For our second method, we used an equation from statistical theory that gives the relation between standard deviation of mean $\sigma_{\overline{V}_N}$, the standard deviation of one ensemble $\sigma_N$, and the ensemble size $N$:

$\sigma_{\overline{V}_N} = \frac{1}{\sqrt{N}} \sigma_N$ (Eq. 1)

This method is advantageous because it only requires one measurement, whereas for our first method we had to make 10 measurements. Thus, we used this equation to come up with the calculated standard deviation of mean, using the measured standard deviation of the first ensemble for each sample size. The results are graphed below:

Graph 2: A logarithmic graph of the calculated standard deviation of the mean, calculated using Eq.1 and the standard deviation for sample sizes 10, 100, 500, 1000, 5000, and 10000

Part 2

For the second part, we kept the first setup and then connected the second voltage output of the circuit board (terminal $V_2$) to the second voltage input of the voltmeter (positive terminal of $CH1$), making sure to connect the negative terminal of $CH1$ to ground. We also connected $V1$ to the first input of the adder circuit $V_{in2}$, and $V2$ to the second input $V_{in2}$. Then, using a sample rate of 1000 points/second and sample size 10,000, we measured the mean and standard deviation of $V_1$ and $V_2$. Afterwards, we disconnected $V1$ from $CH0$ and connected the output of the adder circuit $V_{sum}$ to $CH0$. Using the same settings for the voltmeter, we measured the mean and standard deviation of $V_{sum}$. The results are shown below:

Table 1: The measured mean and standard deviation of the two inputs and one output of the voltage adder

Analysis

Part 1

We know that the standard deviation of the mean should decrease as the ensemble size increases, which is apparent from the overall downward trend of the both graphs. However, for the measured standard deviation of mean in Graph 1, the surprisingly low values for ensemble sizes 500 and 1000 show that clearly there was some experimental error. The calculated standard deviation of mean in Graph 2, on the other hand, has a much more gradual decrease. The large discrepancies between the values of Graph 1 and Graph 2 are most likely due to experimental error. However, because the slope of Graph 2 is more gradual, it is clear that method 2 more accurate.

Part 2

From the measured standard deviations of $V_1$ and $V_2$, we can calculate the standard deviation of mean for each using Eq. 1. We get $\sigma_{\overline{V}_1} = 0.006797 \text{ V}$ and $\sigma_{\overline{V}_2} = 0.003396 \text{ V}$. We know $V_{sum}$ is just the sum of $V_1$ and $V_2$:

$V_{sum} = V_1 + V_2$ (Eq. 2)

So using the equation for propagation of uncertainty for $f(x,y)$:

$\sigma_{\overline{F}} = \sqrt{(\frac{\delta f}{\delta x} \sigma_{\overline{x}})^2 + (\frac{\delta f}{\delta y} \sigma_{\overline{y}})^2}$ (Eq. 3)

and plugging in Eq. 2, we can derive the equation for the uncertainty of $V_{sum}$:

$\sigma_{\overline{V}_{sum}} = \sqrt{(\sigma_{\overline{V_1}})^2 + (\sigma_{\overline{V_2}})^2}$ (Eq. 4)

This is the theoretical uncertainty for $V_{sum}$, which we calculated to be $\sigma_{\overline{V}_{sum}} = 0.007598 \text{ V}$.

We can also find the measured uncertainty $\sigma_{\overline{V}_{sum}}$ using the measured standard deviation and Eq. 1, which gives us $\sigma_{\overline{V}_{sum}} = 0.007120 \text{ V}$. This is extremely close to our theoretical uncertainty, verifying the accuracy of the propagation of error equation.

However, if we calculate $\overline{V}_{sum}$ using Eq. 2 and our measured values of ${\overline{V}_1}$ and $\overline{V}_2$, we get $\overline{V}_{sum} = 1.0193 \text{ V}$, which does not fall within both our theoretical uncertainty and measured uncertainty of $\overline{V}_{sum}$. This indicates the presence of experimental error in part 2 of our experiment.

Part 3

Part 3 of the experiment was merely an equation derivation, and thus had no experimental setup or results. We started with a hypothetical quantity $F$ that was calculated from two quantities $x$ and $y$ using the equation:

$F = xy$ (Eq. 5)

Using the propagation of error equation (Eq. 3) we can find the theoretical uncertainty of $F$:

$\sigma_{\overline{F}} = \sqrt{(y \sigma_{\overline{x}})^2 + (x \sigma_{\overline{y}})^2}$ (Eq. 6)

Using the given measurements $\overline{x} = 1 \pm 0.1$ and $\overline{y} = 1 \pm 0.001$, we use Eq. 5 and 6 to calculate $\overline{F} = 1 \pm 0.1000$. However, if we instead assume $\sigma_{\overline{y}} = 0$ to ignore the uncertainty of $\overline{y}$, we still get $\overline{F} = 1 \pm 0.1000$. This shows that the uncertainty of $\overline{x}$ dominates the uncertainty of $\overline{y}$.

Conclusion

For part 1, we were able to observe the overall downward trend of both the measured (Graph 1) and theoretical (Graph 2) values of $\sigma_{\overline{V}_1}$ as the sample size increased. In addition, the smoother downward trend of Graph 2 indicated that using Eq. 1 to calculate the theoretical uncertainty was more effective than measuring $\overline{V}_1$ multiple times and finding the standard deviation. However, the large discrepancies between the theoretical and measured values of $\sigma_{\overline{V}_1}$ showed that there was a lot of experimental error affecting our results. This was most likely just because we were sampling random voltages and our results just happened to deviate from the theoretical value. The reduced discrepancy for higher sample sizes seems to confirm this conclusion.

For part 2, our theoretical and measured values $\sigma_{\overline{V}_{sum}}$ were very close, verifying the validity of the propagation of errors equation. However, our measured $\overline{V}_{sum}$ does not fall within either our theoretical and measured uncertainty. This is probably because $\overline{V}_{sum}$ was measured separately from $\overline{V}_1$ and $\overline{V}_2$, so $\overline{V}_{sum}$ was not actually the direct output of $\overline{V}_1$ and $\overline{V}_2$ as we had assumed.