# Sample Variability Lab (Roulette)

Copyright © 2007–2020 by Stan Brown

Copyright © 2007–2020 by Stan Brown

**Summary:**
Having studied individual behavior in a normal distribution,
now you turn to the **distribution of sample means**.
Compared to individual bets at roulette, this lab illustrates the
behavior of samples of 30 bets. You and your classmates will explore
the center, spread, and shape of the distribution of sample means.

Color | $win, x |
P(x) |
---|---|---|

Red | ||

Black/ Green | ||

∑ | — |

Let’s take a highly non-normal population: the discrete probability distribution of wins and losses on $10 bets at roulette.

In US roulette (see above), there are
38 numbers: 18 red, 18 black, and 2
green. The ball is equally likely to land on all of them. In this lab,
you’ll simulate **$10 bets on red**: if a red number comes up, you
win $10, and if a black or green number comes up, you lose $10.

Construct the discrete probability distribution by filling in the table. Sketch the histogram to see why this distribution is called “highly non-normal”. Then compute the mean and standard deviation of the discrete PD:

μ = __________ σ = __________

What do μ and σ mean? Interpret them in English:

Color | $win, x | freq., f |
---|---|---|

Red | ||

Black/ Green | ||

∑ | — |

Now take 30 numbered slips from the supply. These are the outcomes of your 30 bets. Sort them by color, and enter the values in the table at right. Be sure to use the correct symbols for the statistics of this sample:

*n* = __________
*x̅* = __________
*s* = __________

In terms of this gambling situation, how do you interpret
*x̅* in English?

What do you see by comparing your *s* to your *x̅*? Your *s*
to σ?

Now consider the distribution of the means of all possible samples of 30 bets from this population. What would be the mean and standard deviation of that distribution?

We can’t actually construct all possible samples, but we can use the samples gathered by your classmates to give some idea. Write down your classmates’ sample means here, as well as your own:

**Center**: What’s the mean of those sample means? How does it
compare to the mean of the population?

**Spread**: What’s the standard deviation of those means? How does it
compare to the variability of the population?

The mean you constructed in this section isn’t
μ-sub-*x̅*, but it’s an approximation to it. If we had
hundreds of samples of 30 each, instead of just a few, it would be a
better approximation.

Similarly, the standard deviation of the sample means from
your classmates isn’t the Standard Error of the Mean (SEM or
σ_{x̅}), but it’s an approximation.
The approximation would be better if we had hundreds of samples of 30.
Compute the standard error as follows:

σ_{x̅} = σ / √*n* =
__________ / √__________ = ___________

How does this compare to the standard deviation of the sample means in this class?

**Shape**: Your instructor will show you a histogram of
the class’s sample means. Compare its shape to the histogram
of the population (individual bets).
If we had a very big class, and plotted a histogram of
everyone’s sample means, the Central Limit Theorem says it
would be roughly normal, even though the original population was highly
non-normal.

What is the main idea you should carry with you from this
lab? It is how to
**describe the sampling distribution of the mean**:

**center**: μ_{x̅}= μ**spread**: σ_{x̅}= σ/√*n***shape**: Even if the population is non-normal, the larger the sample size, the closer the sampling distribution is to normal.

Sample means vary less than the
individuals in the population do, by a factor of √*n*, the square root of
sample size. The value σ/√*n* is called the
**standard error**.
By the 68-95-99.7 rule, a sample mean is 95% likely to be
within two standard errors of the population mean, even though the
individual members of the population may be quite far from the
mean.
This is why you can
**use a sample mean to approximate a population mean** —
but that’s a story for next week.

Updates and new info: https://BrownMath.com/stat/