Fun, impressive, or compelling examples of scaling of the standard deviation like $1/\sqrt{n}$?

Question

I'm teaching a class where statistics is not the main topic, but I would like to introduce the idea that if you take $n$ measurements of a variable that are independent, identically distributed, and uncorrelated, then the mean has a standard deviation that scales down like $1/\sqrt{n}$. I thought it would be fun if there were some impressive or compelling real-world example of this.

Samples of things that I imagined along these lines: --

• Maybe someone has a webcam somewhere that shows their robot repeatedly flipping a coin, with running stats.

• Long-term observation of Brownian motion.

• Maybe there is a publicly accessible government database that has collected large amounts of some kind of relevant data over time.

• I would think that the examples with the largest $n$ and the best statistical properties would probably be from experiments in nuclear or particle physics, but this seems low in charisma and not as compelling and understandable.

I would be equally happy with something more of the flavor of the sum of random variables having a linearly increasing variance, which is mathematically the same thing, although not necessarily the same thing in terms of psychology/fun/impressiveness.

A vaguely similar type of statistical demonstration would be the famous University of Queensland pitch drop experiment.

Answer 1

If you have a bunch of identical dice, (I recommend non-standard dice; as of writing, mathsgear is a good source of interesting ones), you can just pass out dice to students -- then have them collect a lot of data.

I do this with my general education statistics students. My experience is it takes most of a class period, but you can get the class data up on the board as a histogram and they can see the standard deviation go down AND see how much it goes down by.

Prerequisites:

• The students need to know how to find the population mean and standard deviation for a die they are looking at.
• The students need to know that the inflection point on a normal distribution occurs one standard deviation away from the mean.

Outline:

• Pick a die and walk through calculating the mean and standard deviation for a single die roll. With a standard die you can write $\mu_x = 3.5$, $\sigma_x \approx 1.71$.
• Pass out a die of that type to each student.
• Task the class with calculating $\overline{x}$ with $n = 10$ several times. I have been asking groups of four students to find 10 values of $\overline{x}$, so each group must execute 100 rolls each. If your class has 8 groups, you can get 80 data points (You could try asking each individual student to find some values of $\overline{x}$ but my experience is that you will not get every student to actually figure out what is going on this way).
• Tell students to write all their values of $\overline{x}$ on the board. Because you have picked $n = 10$, you have nice round numbers for your $\overline{x}$'s (Note: you will be tempted to choose $n=9$ or $n=16$ but the payoff for $n=10$ is higher).
• Draw the bar graph of all the data points. Notice how almost-bell-shaped it is. Find the inflection points of the shape. Small but serious issue: $\overline{X}$ is not really normally distributed since $n = 10$. My experience is that things work out close enough.
• Note that the inflection points are $\frac{\sigma_\overline{x}}{\sqrt{n}}$ away from $\mu_\overline{x}$.

Key question to ask students:

• Are the $\overline{x}$'s more or less spread out than the $x$'s?