Common Core Standards: Math
1. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
Students are in seventh grade now, which means they've moved up in the world of statistics. They're no longer answering the question, "What is the average height of the students in our class?" Now, they're answering a different question: "What is the average height of all seventh grade students in the U.S.?"
It probably took a little bit of time and effort to measure everyone's heights in their sixth grade class, but it's virtually impossible to measure the heights of every seventh grade student in the U.S. What ever will we do? Well, that's where this standard comes in.
Students should know that a population is a collection of all the individuals, objects, or events that we want to study. If the population is small and easily accessible (e.g., all the students in a single classroom), then studying everyone in the population is reasonable. When the population is large and not easily accessible (e.g., all the seventh grade students in the U.S.), then we study a sample of the population.
Of course, we can't just study any old sample. We need to find a sample that represents the population we're interested in. For instance, if we want to find the average height of all seventh grade students in the U.S., and we choose a group of fifty boys who play on their schools' basketball teams, then our sample is probably not likely to be representative of the population. Sure, we could measure the heights of these boys and calculate the average height, but we shouldn't assume that's a good estimate of the average height of all seventh grade students in the U.S.
So how exactly do students get a sample that is representative of the population? You could teach students all sorts of fancy sampling techniques, but at this point, we only care about one: random sampling. In a random sample, every person, object, or event in the population will have an equal and independent chance of being selected for the sample, and the sample is selected by some chance process, like drawing names out of a hat or rolling dice (affectionately called "number cubes" in Common Core lingo).
Students don't have to do much in this standard other than understand, so a good long lecture and an endless slew of homework problems could do the trick. If you're achin' for a good in-class activity, though, have students calculate the average population using data from all fifty states. Then, have students collect two samples, one random and one nonrandom, and find the average populations of those samples.
If students mark the random sample averages on one class dot plot and the nonrandom sample averages on the other, they should see that the dot plot of the average populations from the random samples much more closely estimates the average population of all fifty states than the dot plot from the nonrandom samples.
And if it doesn't, blame it on probability.