Standard Deviation of the Mean

It's intuitive that the more samples we take the more accurate we can get our mean $\mu$. What's not intuitive, however, is how to quantify by how much our accuracy improves as we increase the number of samples (N) we collect. In order to quantify this data, we examined how error on the mean changes as we increase the number of samples. For my experiment I decided to vary the number of samples taken by taking increasing exponentials of 2 (i.e. $2^{1}, 2^{2}, 2^{3}, 2^{4}, 2^{5} ... 2^{12}$). Firstly, we can see how accurate the calculated mean values are for each data set as we increase the number of samples collected by comparing the mean of the means for all of the different sample sizes. So after taking the mean of each individual data set, I took the mean of all 10 data sets with the same number of samples. This is plotted in figure 4. As you can see in the figure, the mean of the means seem to be approaching a constant as the number of samples increases. This makes sense because as you take more samples, the estimation of the mean approaches that of the parent population (equation 1).

Figure 4: This is a figure of the ``Mean of the Means''. As you can see here, the mean of the means are, on average, constant. However, you can see that for a lower number of samples, the mean of the means is not as consistent as the ones for higher numbers of samples. It appears that as the number of samples tends towards $\infty$, the means converge toward a specific number.

The accuracy of the data can be further examined by looking at the standard deviation of the means (SDOM). After calculating the mean of each set of data, the deviation between the means of the data can be calculated by taking the standard deviation of all the mean values. If the accuracy in measuring the counts per sample improves as we increase the number of samples for each data set, we would hope that the standard deviation of the means will decrease as we take more samples. I have plotted the standard deviation of the means in figure 5. We can see from the figure that the deviation does indeed decrease as we take more samples of data. The solid line plotted in figure 5 is the theoretical prediction of how the standard deviation of the means should behave based on the data. The theoretical prediction is based on equation 8⁷.

$$\sigma_{\bar{x}} = \frac {\sigma_{x}}{\sqrt{N}}$$ (8)

Equation 8 is the theoretical equation for the standard deviation of the mean. This, in effect, tells us how accurate our means are by giving us an error bound for the actual $\mu$ of the parent population. So for my data, the best estimate I have for the mean of the parent population is the mean of the data set with $2^{12}=4096$ number of samples. This has an SDOM of $0.0581$. Therefore, we can state the accuracy of this mean value to be $13.6832 \pm 0.0581$. This error goes to zero as N approaches $\infty$. Additionally, we can see from the equation that in order to improve the accuracy of our mean by, say, a factor of 2, we need to take 4 times more samples. This is due to the inverse square root dependency on N for the SDOM. Therefore, if we have $\eta$ more samples of data, our error (SDOM) decreases by a factor of $\sqrt \eta$.

Figure 5: This is a plot of the ``Standard Deviation of the Means''. The points are the standard deviation of the means based on the data. The line is the theoretical prediction of how the standard deviation of the means should behave based on the data.

This brings about the question of whether or not we can construct a light source that would not show any variations in the count rate. There is not any possible way to do this because the variations come from the quantum mechanical description of the system stating that there is a probability for whether or not a photon will be emitted from the atom. This is independent of how many samples we try to take. Therefore, even if we take an infinite amount of samples, the individual variations between the photon counts will not be zero. What will be zero, however, would be the deviation between the average means between each data set. If we can take an infinite amount of samples for each data set, each data set as a whole should give us the same number of photon counts as an average.