In this activity, a dot plot is created by the class that includes all of the calculated sample means. Students then compare that display to the data from the entire population to better understand the information that can be gained from a sample mean (MP7). In the discussion that follows the activity, students look at similar displays of sample means for samples of different sizes (MP8). Students should see that larger samples tend to more accurately estimate the population mean than smaller samples.

The center of the dot plot created by the class can be thought of as a mean of sample means. The phrase can be difficult for students to think through, so remind them what the dot they added to the dot plot represents and how they calculated that value. Consider displaying the class dot plot for the rest of the unit to help students remember this example to understand similar dot plots.

Display the dot plot of sample means for all to see throughout the unit. It may be helpful to refer to this display when viewing dot plots of sample means in later lessons. 

The purpose of the discussion is for students to understand that the centers of the two dot plots in the activity are close, if not the same, but the sample means have less variability, and the shapes of the population distribution and the sample mean distribution are probably different. Additionally, students should use the dot plots shown here to see that larger samples have sample means that are still centered around the population mean, but have less variability than the means from smaller samples. Therefore, larger samples should provide better estimates of the population mean than smaller samples do.

Items for further discussion:

• “If the sample size were increased to 30 instead of 20, what do you think the distribution might look like?” (There would be less variability in the same means, but they should be centered around the same value.)

For a different set of data representing the wingspan in centimeters of a certain species of bird, compare these dot plots.

Here is a dot plot of the population data.

Here is a dot plot of the sample means for 100 different random samples each of size 10.

Here is a dot plot of the sample means for 100 different random samples each of size 30.

• “What do you notice about the distribution of the sample means as the sample size increases?” (The variability decreases, but the center stays in about the same place.)

• “Use the dot plots to explain why the sample mean from a random sample of size 30 would be a better estimate of the population mean than the sample mean from a random sample of size 10.” (Since the variability is less, the sample mean is probably closer to the true mean with size 30 than with size 10.)

In this activity, students continue to look at how variability in the sample means can be used to think about the accuracy of an estimate of a population mean. If the means from samples tend to be very spread out, then there is reason to believe that the mean from a single sample may not be especially close to the mean for the population. If the means from samples tend to be tightly grouped, then there is reason to believe that the mean from a single sample is a good estimate of the mean for the population (MP7).

Students may get confused about what the dot plots represent. Refer to the class dot plot of sample means from the reaction time activity to help them understand that a single dot on the dot plot represents a mean from a single sample. Each dot plot shows means from several different samples.

The purpose of the discussion is to think about what the information in the given dot plots tells us about the accuracy of an estimate of a population mean based on the sample mean from a single sample.

Consider asking these questions for discussion:

• “In the first dot plot for ‘Trivia the Game Show,’ what does the dot at 68 represent?” (There is a sample of 10 viewers that has a mean age of 68 years.)
• “Based on the dot plot shown for ‘Learning to Read,’ do you think any 7 year olds are included in the data?” (Yes. Because one of the sample means is 6.8, there are probably a few 7 year olds and some 6 year olds in that sample.)
• “Why is this kind of dot plot useful?” (It shows how variable the means from samples can be, so it gives a good idea of how accurate a sample mean might be as an estimate of a population mean.)
• “Why might it be difficult to obtain a dot plot like this?” (Each dot represents a sample of 10. It might be hard to get 100 different samples of size 10.)