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

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.

• " 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/n−−√." While this is an important intuitive concept in measurements, I question your introducing it into a class where stats is not the main topic, and especially with juco students. Concentrate on helping them learn the core material in the subject you are teaching. Commented Jan 4, 2020 at 23:49
• I would think, if you are going to do this, the subject needs to be relevant to the main topic of the course, which you haven't stated. Commented Jan 5, 2020 at 9:12

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 $$\mu_\overline{x}$$.

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