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.

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 point value of your hand.
  4. Write the value of the mean on a sticky note and add to whiteboard.
  5. Repeat steps 1 - 4 four additional times, per person.

Discussion

Answer the following questions in your group:

  • What appears to be the average value of the sample mean?
  • How does this compare to the average point value in the deck of cards?
  • Which distribution appears to have more variability: the distribution of sample means or the distribution of point values in the deck?
  • How do the shapes of the two distributions compare?
  • What are the approximate 1st and 3rd quartiles for the distribution of sample means?
  • What does this suggest about the value of most sample means?

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 this year, and 80 samples from a previous year’s class.

Note that side-by-side histograms are somewhat misleading, since the number of samples in each class are different (so in particularly, the 2023 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: