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I'm a private math tutor. I'm fairly new at this, and this semester is the first time I've been tutoring for a statistics class at a community college. I enjoy experimenting and learning about ways to explain math.

I'd like to get some feedback on explaining sampling and distributions to students who struggle, especially those who don't have a lot of cognitive power.

In any math topic, to help them understand math concepts I relate those to everyday experiences. Here is my idea for sampling and distributions.

I say imagine that you are a hairdresser (or substitute other profession they can relate to). I say, imagine it's the end of your work day and you're reflecting back on what happened. You think about the number of clients you had (an example of a discrete random variable), the amount of time spent cutting (a continuous random variable), or anything you like.

I say, imagine explaining your workday to a friend. I get them to imagine this hypothetical day and the language they would use. For instance, would they say to their friend, it was an unusual day with a lot of clients? Or a typical day with an average number, or typical number of minutes spent cutting? What are some ways in which days can be unusual?

Hopefully they start to form this mental picture of both the experiences of a single day (a sample) and how days vary over time, including what patterns are more frequent (a distribution).

We talk about some of the factors that might affect the distribution of clients. Are holidays very busy, while Fridays slow? We think about the population of all potential clients in this city and parameters of this population, and how they would affect what the hairdresser sees.

I haven't done this yet, but even the Central Limit Theorem could be explained by asking them to differentiate between day to day variation, and the variation of weekly averages.

So, just wondering if the statisticians here have similar or better ideas.

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  • $\begingroup$ I think comparing things to every day examples is helpful. Business/money is especially so as it involves something people care about and have experience with (have worked for instance as a waiter and seen tips). Other examples are sports (especially with males) or gambling. $\endgroup$ – guest Oct 14 '18 at 21:51
  • $\begingroup$ I think your hairdresser example is great. However, at the end of the day, don't think "the perfect explanation" is the key to unlocking good performance by slow students. They just need a lot of repetition/drill. But definitely mix in some explanations too. But don't think of it as the explanation winning the day. Mix in visual examples and just examples of different number lists (lots of different ways of looking at it). But also proceed with mechanical drill even if the "aha" light has not clicked. Some people will get it clicking later. Some never will (but can still learn methods). $\endgroup$ – guest Oct 14 '18 at 21:53
  • $\begingroup$ I teach physics (and rarely math) at a community college. When they write up labs, significance tests come up. Many of them have a tremendously hard time with this, no matter how many times I try to help them one on one. They get a 2 sigma result, which is a 5% probability. They want to say that .05 is the probability that they're wrong or right, or that the hypothesis is wrong or right, or that it's the probability of getting this result. They don't get that: (a) it's a probability conditioned on a certain hypothesis, and (b) it's the probability of being off by at least this much. $\endgroup$ – Ben Crowell Oct 15 '18 at 2:50
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    $\begingroup$ "variation of weekly averages" will be a very abstract construct for the students you're describing. $\endgroup$ – Jasper Oct 15 '18 at 20:23
  • $\begingroup$ @Jasper You could be right. What I'm thinking, though, is that people have a concept of a "good week" or a "bad week." They understand the difference between "a bad day" and "a bad week" and that one bad day does not make a bad week. So far I have only been working with students who are so far behind, that we are nowhere close to the CLT, so it's just a theory at this point. $\endgroup$ – composerMike Oct 15 '18 at 22:18
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Another way to make this idea concrete is to physically enact it. It requires some additional preparation, but you could do something like this:

  1. Print out many thousands of slips of paper, each with a single number printed on it. These numbers can be generated randomly using a spreadsheet (e.g. Excel or Google Sheets), and if you are clever you can get the numbers to be drawn from a normal distribution with a mean and standard deviation of your choice.
  2. Put all of the slips of paper in a box and ask the student: How could you figure out the mean of this data? Obviously actually adding them all up and dividing by the number of slips is impractical -- you need some way to estimate it, since finding it exactly is impossible.
  3. So you take a sample of, say, 40 slips and find the mean of the sample. That gives you an estimate -- but how reliable is it?
  4. If you take a different sample, you'd get a different sample mean. What if you took 50 different samples? You'd get 50 different sample means. How could you use those to estimate the population mean?

This kind of activity really clarifies the difference between the population mean (which is unknown, and in a sense unknowable) and a sample mean. Follow this out and you get to the idea of the distribution of sample means, and how it relates to the (unknown) background distribution, and then to the CLT.

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    $\begingroup$ This is a good experiment, but I'm also trying to connect the concept of a distribution with something that my students care about. Some "slow" students probably will not get the concept without a lot of mechanical drill of the sort you describe here, but it would be nice if the purpose could be motivated somehow. I can imagine many students looking at these slips of paper and saying: WHY? $\endgroup$ – composerMike Oct 15 '18 at 1:13
  • $\begingroup$ The example could be motivated with any number of real-world referents: people's heights, or annual incomes, or monthly rent in a particular city. Any time there's something in the real world you want to know about, but measuring it directly via census is essentially impossible, sampling arises naturally. $\endgroup$ – mweiss Oct 15 '18 at 1:28
  • $\begingroup$ I'm referring to the concept of a distribution. My students can certainly understand that people's heights differ, but the non-mathematical ones don't naturally wonder if there is a distribution with some inherent properties. Not saying I have the only answer, but I want them to imagine some personal, real-world scenario in which there is a meaningful distribution which they might find their own words to talk about, or their own questions to ask about. $\endgroup$ – composerMike Oct 15 '18 at 4:54
  • $\begingroup$ @composerMike A lot of examples exist of this type. A simple example to get you started thinking along these lines: you work for a company that has created a division that contains 35% females. Yet you know the company has a qualified staff which contains 55% females. How likely is this to happen by random chance. I use a study of empathy in rats (from U of C) where 24 of 30 rats freed trapped cagemates. If the rates were indifferent (p=0.5) how likely is this could happen (I have each student flip coin 30 times to simulate this). These problems all hinge on one question: is there an effect? $\endgroup$ – Matt Brenneman Oct 17 '18 at 18:26
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I agree with @mweiss that you need to physically enact this.

Luckily, your students have random number generators on them at all times. Statistics classes these days are set up with some kind of technology, either a TI-83 or Excel or StatCrunch or something similar. You can do the following experiment:

  • Explain that $X$ is the random variable of random numbers generated by (on the TI-83 or 84) $MATH \to PRB \to rand$. If you have covered uniform distributions in any detail, great -- you can ask them to remember that $X \sim U(0,1)$, $\mu_X = 0.5$, $\sigma_X = \sqrt{\frac{1}{12}} \approx 0.289$.

  • Sample from $X$ and get a picture of your data. On the calculator this is easy; press enter like 30 times. The data will be all over the place. Each time you do this, and this is crucial, make sure the student knows that these are values of $x$.

  • Decide on a value of $n$; 10 is fine. Explain that after you have chosen $n$, $\overline{X}$ is the random variable of means of $x$'s. If you have covered the Central Limit Theorem, you can write down the distribution but don't do that!!! If you do that, you will rob them of the surprise.

  • Sample from $\overline{X}$ and get a picture of your data. On the TI calculator you can actually type $(rand+rand+rand+rand+rand+rand+rand+rand+rand+rand)/10$ so that you can just press enter over and over. If you have a big class of students, I would instead recommend that you have each student actually get 10 values of $x$ and average them to get their own $\overline{x}$ and write it on the board. Either way, the point is to make sure the student realizes these new numbers are values of $\overline{x}$ and not $x$. The point here is to tell the difference between those things. Of course the data will not be all over the place.

Then you can ask: Cool; this looks like a normal distribution. I wonder which normal distribution it is?

The point is that by making the whole thing a concrete exercise, you can get them to tell the difference between the letters $X$, $\overline{X}$, $x$, and $\overline{x}$, which is the real hurdle of this topic.

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