In this activity, students begin to see numerical evidence that different samples can produce different results and thus different estimates for population characteristics (MP2). Students look at a small population and some different collections of samples from this population. Although the data for this population is small enough that it is not necessary to use a sample, it is helpful to get an idea of how data from a sample compares to the population data.

The purpose of this discussion is to show that different samples can result in different estimates for a population characteristic, and its purpose is to act as a reminder of reasons we might choose one measure of center over another.

Some questions for discussion:

• “What is the population for this situation?” (all of the paintings that were sold) 
• “What are the samples used in the calculations?” (the first two paintings that were sold, those sold at a gallery show, and the oil paintings)
• “Why did the different samples have different means?” (because they used different paintings)
• “Why were the means for the first two paintings sold and those sold at the gallery show so far off from the mean of all the paintings?” (because they contained the cheapest one and most expensive one, respectively, with only a few other numbers to balance it out)
• “Based on the numbers in the population, does it make more sense to use median or mean?” (Median. The $1,200 painting is much greater than the rest of the values, so the measure of center is affected much more by the one painting when using mean.)

In this activity, students begin to see that some samples represent the population better than others. Students compare the dot plot of a population of data with the dot plots of several samples and discuss some aspects that would make some samples better than others (MP7). In the discussion, the phrase “representative sample” is defined.

Direct students' attention to the reference created using Collect and Display. Ask students to share how they determined which samples fit with the population better. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond.

Consider asking these discussion questions:

• “What are some aspects that make for a good sample? a bad onead?” (A sample is “good” if it has a similar distribution to the population data. A sample is “bad” if the data does not have a similar distribution to the population data. For example, Sample 3 is bad because it is not centered in the same place.)
• “If you were to find a measure of center to represent a typical value for the population, would you use mean or median?” (I’d use the median because the data is not approximately symmetric.)
• “The population in this example has a mean of 1.76 hours and a median of 1.5 hours. Sample 1 has a mean of 0.33 hours and median of 0.5 hours. Sample 2 has a mean of 1.73 hours and a median of 1.75 hours. Sample 3 has a mean of 3.33 hours and a median of 4 hours. Based on this information, which seems to represent the population the best?” (Sample 2)
• "Why might someone purposefully use Sample 1?" (They may want to promote the idea that these kids do not sit in front of a computer very much every day and selectively choose the smallest numbers from the population.)
• "One of these samples was selected by randomly picking 20 students from the group. Which one do you think it was? Explain your reasoning." (Sample 2. The other samples seem almost deliberately chosen to use mostly low or high values, so Sample 2 seems the most likely to be selected randomly.)

Define representative sample. A representative sample is a sample that has a distribution that closely resembles the population distribution in center, shape, and spread.

Explain that a sample with the same mean as the population is not necessarily representative because it may miss other important aspects of the population. 

• Example 1: If the population for a question is all of the humans in the world, and we use one person from each country as our sample, it may not actually be representative of the population. A larger country such as China is under-represented because there are actually many people living there, but only one is included in our sample. Similarly, a smaller country like Cuba might be over-represented because it has fewer people living there, but it is represented in the sample exactly the same way as all of the larger countries.
• Example 2: The average height of people in the world is approximately 66 inches. Two people may be found: one who is 87 inches (7 feet 3 inches) tall and one who is 45 inches (3 feet 9 inches) tall. Their mean height may be the same as the world’s, but these two certainly do not represent the heights of most people.

Explain that a representative sample is the ideal type of sample we would like to collect, but if we do not know the data for the population, it will be hard to know if a sample we collect is representative or not. If we do know the population data, then a sample is probably unnecessary. There are ways to select samples that are more likely to produce representative samples that will be explored later.

This activity is additional practice for students to understand the relationship between a sample and population. It may take additional time, and so is included as an optional activity.

In this activity, students attempt to recreate the data from the population data using three given samples (MP2). It is important for students to recognize that this is difficult to do and that some samples are more representative than others. Without knowing the population data, though, it can be difficult to know which samples will be representative. Methods for selecting samples in an unbiased way are explored in future lessons.

Students may consider that each of the auditors’ samples should be added together to create one larger sample rather than considering that the auditors may have chosen the same data point in their separate samples.

Therefore, each auditor having a data point at $41,000 may mean that there is only one data point there, and each auditor included it in the sample, or it may mean that there are actually three data points there, and each auditor included a different point from the population. 

The purpose of the discussion is for students to understand that getting an understanding of the population data from a sample can be very difficult, especially when it is not known whether samples are representative of the population or not.

Display the population dot plots for all to see.

For furniture sales, the samples came from data represented in this dot plot.

For electronics sales, the samples came from data represented in this dot plot.

Ask students:

• “How close was your estimate to the actual dot plot? Consider the shape, center, and spread of the data in your answer.”
• “Are any samples better at mimicking the population than others?”
• “What could the auditors have done to make their samples more representative of the population data without knowing what the population would be?” (They could have included more information in their samples. They should have also thought about how the samples were selected. For example, if the auditors only came on months when there were large sales happening, they may have missed important data.)