Activity

Objectives

In this activity, you will describe the distribution of a statistic by observing its average value and variability between samples. Compare to the distribution of the variable in the population.

Assignment

Arrange yourselves into groups of 3 or 4. on the following activity. Write your answers in your own notes and add your data to the histogram at the front of the class.

Background

A standard deck of 52 playing cards consists of four copies each of the numbers 2 through 10, along with Jacks, Queens, Kings and Aces. In many games, it is important to track the point value of a hand of cards. Face cards (Jacks, Queens, and Kings) are usually treated as 10 points, while Aces are treated as 1 point. Number cards (2 - 10) are treated as the number shown.

Below is a distribution of the point values for a standard deck of cards:

Activity

Each person should do the following:

  1. Thoroughly shuffle one of your group’s deck of cards.
  2. Draw 10 cards from the deck (without replacement) to form a sample.
  3. Compute the mean value of your cards (counting Aces as 1 and Faces as 10)
  4. Write the value of the mean on a sticky note and add it to the appropriate interval on the front chalkboard.
  5. Repeat steps 1 - 4 an additional four times. After you’ve used physical cards at least once, you may complete remaining shuffles using the random card shuffler at https://www.random.org/playing-cards/

Discussion

As a group, answer the following questions:

  • What is the theoretical mean value for the data set of card values?
  • How does the distribution of sample means compare to the distribution of card values?
  • What is the relationship between the centers of the two distributions?
  • Which distribution appears to have more variability?
  • How do the shapes of the two distributions compare?
  • What does the variability of sample means suggest about the means in repeated samples?

Results

The distribution of card values in the population is given below, with the population mean given by the purple line:

Based on the 200 trials conducted in class on Monday, the distribution of average point values in hands of size 10 is:

We can also consider two smaller sets of samples, 120 samples from the 9am section and 80 samples from the 10 am section:

Note that side-by-side histograms are somewhat misleading, since the number of samples in each class are different (so in particularly, the 9am section will have higher peaks).

We can fix this by instead indicating the proportion, rather than count, of observations in each bin:

Finally, suppose we had a class of 20000 students and each student performed the experiment 5 times:

The results are summarized below: