What does it mean for data to have variability as it prefers to statistical questions? Give an example

A statistical question is a question that can be answered by collecting data and where there will be variability in the data. This is different from a question that anticipates a deterministic answer. For example, "How many minutes do 6th grade students typically spend on homework each week?" is a statistical question. We expect that not all 6th grade students spend the same amount of time on homework, and the time spent on homework may also vary from week to week. On the other hand, this is not a statistical question: "How much time did Juana spend studying last night?" This question is based on a single number and has a deterministic answer.

The question "How many buttons are in the jar?" is answered by counting the buttons. This produces a single value--it is not answered by collecting data that vary.

The question "What is a typical number of holes for the buttons in the jar?" is a statistical question. To answer this question, students might compute the mean or the median (both measures of center that are used to describe a typical value). But in either case they would need to collect data on the number of holes by recording a value for each button. Because not all buttons have the same number of holes, there would be variability in the data that would be used to answer this question. That is what makes this a statistical question.

The question "If Zeke grabbed a handful of buttons, what are the chances that all of the buttons in his hand are round?" is a statistical question because this is asking for a probability that would be estimated by having Zeke grab many handfuls of buttons. For each handful grabbed, whether or not all of the buttons were round would be recorded. This would result in categorical data (with values of "all round" and "not all round"), but again there would be variability in this data. This data could then be used to estimate the probability of interest to provide an answer to the question posed.

Statistics is the study of variability. Students who understand statistics need to be able to identify and pose questions that can be answered by data that vary. The purpose of this task is to provide questions related to a particular context (a jar of buttons) so that students can identify which are statistical questions. The task also provides students with an opportunity to write a statistical question that pertains to the context.