In this activity, students analyze data from samples of viewers for different TV shows. The data in this activity is used to begin the analysis as well as to get students thinking about the different shows the sample could represent. The purpose of the activity is to get students thinking about how measures of center from a sample might be used to make decisions about the population of a group.

Select students to share how they determined which shows matched with which data set. The purpose of the discussion is for students to notice that the shows are meant to appeal to different age groups.

This activity continues the work begun in the previous activity for this lesson. Students compute the means for sample ages to determine what shows might be associated with each sample (MP2). They also consider the variability to assess the accuracy of population estimates. A sample from a population with less variability should provide a more accurate estimate than a sample that came from a population with more spread in the data. In the discussion, students think about why a sample is used and why an estimate of the mean is helpful, even though it may miss some important aspects of the data. The discussion following the activity also asks students to think again about why different samples from the same population may produce different results.

Provide calculators to calculate statistics such as the mean and MAD.

The purpose of the discussion is to understand why it might be helpful to estimate the mean of a population based on a sample.

Some questions for discussion:

• “Why do you think a sample was used in this situation rather than data from the population?” (There are probably millions of people who watch these shows, and it would be difficult to collect data about their ages from all of them.)
• “How could we improve the estimate of the mean for the populations?” (If we include more viewers in the sample, then the estimate will probably improve with the additional information.)
• “If a sample has a large MAD, what does that imply about the population?” (It implies that the data in the population is very spread out.)
• “If a sample has a small MAD, what is the relationship between the data and the mean?” (Most of the data is close to the mean.)
• “Which estimate of the mean for the population do you expect to be more accurate: the mean from a sample with a large MAD or the mean from a sample with a small MAD? Explain your reasoning.” (The mean from a sample with a small MAD is probably more accurate. If the sample does not have much variation, then we might expect the population to not have much variation either. That means the data might all be close to the mean.)
• “What do you notice about the mean and MAD for different samples related to the same show?”  (The means are close, but the MADs can be different by a lot.)
• “Why are the answers for two samples different for the same show?” (Different people are included in the samples, so the numbers may change some, but if they are representative, they should be close. In these examples, even though the MADs may seem very different, the relative size compared to the other shows is similar.)

Some students may wonder why they need to calculate the mean when it might be obvious how to match the titles by just looking at the data. This example included 10 ages in each sample so that the important information could be calculated quickly. In a more realistic scenario, the sample may include hundreds of ages. A computer could still calculate the mean quickly, but scanning through all of the data may not make the connection to the correct show as obvious. 

• “Notice that there is a 56-year-old in sample 6. What are some reasons you think they might be watching this show?” (Maybe a grandparent is watching with their grandchild. Maybe an older person is interested in how science is shown on TV.)
• “The questions ask you to consider means, but are there any data sets for which median might be a better measure of center? Explain your reasoning.” (Yes, sample 6 has a wide range of data, ranging from a 1-year-old to a 56-year-old, but most of the data are around 10 years, so the median might be better to use for that sample.)
• “A lot of families might be watching Learning to Read with their children or older people may be using the show to learn English. How might this affect the mean? How could you recognize that there are two main age groups that watch this show?” (It would bring the mean up from what it would be if only kids were watching the show. The mean would not make the two age groups obvious, so looking at a dot plot or histogram might be more helpful with this group.)

In this activity, students use data from a sample of movie reviews to estimate information about all the reviews for the movie. Based on the distribution of the data, students are asked to choose an appropriate measure of center and measure of variation, then apply their calculations to the entire population. Finally, students gauge their trust in the measure of center they have chosen based on the associate measure of variation.

Students must make use of the structure of the distribution to determine the best measure of center and variability to use (MP7).

The purpose of the discussion is for students to review how to choose a measure of center and its associated measure of variation. Additionally, students use the measure of variation to help them think about how much to trust their population characteristic estimate.

Consider asking these discussion questions:

• “Which measure of center did you choose and why?” (I chose the median since the distribution is not symmetric.)
• “Based on the situation, do you think other movie reviews would have non-symmetric distributions as well?” (Yes, usually a lot of people will agree whether a movie is good or bad, but a few people will have strong opinions on the other end of the scale.)
• “A random sample of 20 reviews for another movie also has a median of 90, but its IQR is 30. Do you think this movie is more or less likely to be considered for the award?” (I think it is less likely to be considered since there is more variability in the sample, so it is harder to estimate the median for all of the reviews.)
• “A random sample of 20 reviews for a third movie has a median of 50 and an IQR of 20. Is it possible this third movie will be considered for an award?” (It seems unlikely, but it is possible. The random sample may have randomly selected the 20 worst reviews, and all the other reviews gave it a 100 rating.)