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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.

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    $\begingroup$ " 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. $\endgroup$
    – guest
    Jan 4 '20 at 23:49
  • $\begingroup$ 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. $\endgroup$
    – Jessica B
    Jan 5 '20 at 9:12
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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?
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