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Buttons: Statistical Questions


Alignments to Content Standards: 6.SP.A.1

Task

Zeke likes to collect buttons and he keeps them in a jar. Zeke can empty the buttons out of the jar, so he can see all of his buttons at once.

Task_1_295826ebc1b74f53d11f89e10b14ad1b

  1. Which of the following are statistical questions that someone could ask Zeke about his buttons? (A statistical question is one that anticipates an answer based on data that vary.) For each question, explain why it is or is not a statistical question.

    1. What is a typical number of holes for the buttons in the jar?
    2. How many buttons are in the jar?
    3. How large is the largest button in the jar?
    4. If Zeke grabbed a handful of buttons, what are the chances that all of the buttons in his hand are round?
    5. What is a typical size for the buttons in the jar?
    6. How are these buttons distributed according to color?
  2. Write another statistical question related to Zeke’s button collection.

IM Commentary

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.

Solution

    1. Statistical question
    2. Not a statistical question
    3. Statistical question
    4. Statistical question
    5. Statistical question
    6. Statistical question

    Each of the statistical questions would be answered by collecting data and there would be variability in the data.

    Questions identified as not statistical questions are not answered based on 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 "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. It is not a statistical question.

    The question "How large is the largest button in the jar?" is a statistical question. The size of the largest button is a population characteristic and this question would be answered by collecting data on the sizes of all the buttons in the population. In this way, the question would be answered in a way that takes variability in the population into consideration. 

    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. These data could then be used to estimate the probability of interest to provide an answer to the question posed.

    Like the first question, the last two questions (v and vi) are statistical questions because they would be answered by collecting data that vary. To answer the question about the typical number of holes, data on number of holes would be collected for each button in the jar. The question about how the buttons are distributed according to color would be answered by recording the color of each button in the jar and then summarizing these data in a table or a graph.

  1. Some possible statistical questions are:

    • What is a typical shape for buttons in the jar?
    • What is the distribution of the diameters of the round buttons in this jar?

Michael says:

over 1 year

I don't think that question iii is correct. It seems the largest button in the jar sort of varies the same way the number of buttons in the jar varies. The fact that the administrator changed the answer after a complaint goes to show that the answer could be argued.

cameron says:

over 2 years

Why would there be a typical shape in a button jar?

Cam says:

over 2 years

Good question, and I see you're not the first to bring this up. I personally view the word "typical" here as a placeholder for the (somewhat loaded, given the context) word "average." So the answer of 3 would fill in the blank in the sentence "Typically, a button in the jar has around 3 holes", or something to that effect. I agree that having it fill in the phrase "A typical button in the jar has 3 holes" is a much stronger statement.

I'd be interested in hearing suggestions for better wording.

roxypeck says:

about 2 years

I do think that wording questions like these is sometimes awkward. I think of "typical" as one of those weasel words that is trying to indicate some sort of summary measure for a population--like average or median for numerical populations or mode for categorical populations. I try to stay away from "a typical button" because that sounds more like it is talking about one specific button. It always bothers me when people talk about something like the "typical student" because if you put a bunch of the summary measures together to get this typical student (brown eyes, 5 feet tall, weight of 120 lbs etc.) you might be describing a student that doesn't even exist in the population. The description of typical refers to the population as a whole and I don't really think there is such a thing as "the typical student". But that is just a personal pet peeve, so feel free to just ignore me!

roxypeck says:

about 3 years

I agree. The size of the largest button in the population is a population characteristic and this question would be answered in a way that takes variability in the population into consideration. I would consider it a statistical question. I have edited the solution to reflect this. Thanks for pointing this out.

Lauren Glessner says:

about 4 years

When looking at question iii, I was under the impression that, since the largest button is being compared to the remainder of the set, that it was a statistical question, since the size of the other buttons in the set must be known, as well. Size varies amongst a group, so isn't it statistical? I see that the one button's size is only one answer, but the data must be collected on all the buttons in order to determine the size if it's being compared to others. Where am I going wrong? I found other information out there that supports my claim.

roxypeck says:

over 4 years

Yes, iv is a bit tricky. It is considered a statistical questions because it couldn't be ansered by just taking one handful. To estimate the chance that all of the buttons in a handful are round, you would need to repeat the process many times and in each case note the outcome. There would be variability in those outcomes. After loking at a number of repetitions, it would be possible to use the proportion of the time that all buttons were round as an estimate of the chance (probability) of getting all round buttons.

Robert W. Hayden says:

over 4 years

Not a disagreement but another way to look at it. You could also answer the question by examining ALL the buttons and their shapes COULD vary. ( I see only round ones in the photo but I can't see all the buttons in the jar.) My way or Roxy's, it is a question about the group of buttons as a whole, and the buttons can vary, rather than a question about a single button that has but one shape. It is a question about the collection as a whole that can only be answered by looking at individual buttons and cataloging their varied shapes.

A possible source of confusion could be over the unit of analysis. If all the students in the class have button collections then we might be interested in how many collections are more than 98% round. Now we do treat the proportion for each collection as a single number and the variability is among collections rather than among individual buttons in the same collection.

Rachel Aponso says:

over 4 years

Can you clarify why iv is a statistical question? The way I read it, I understand the question to be asking for the experimental or theoretical probability of that one handful. Where is the variability in that one sample?

roxypeck says:

almost 5 years

Thanks for your comment. I agree that the wording of the question and the commentary could be improved. I have edited both the question and the commentary to try to clarify. The main distinction between a statistical question and one that is not a statistical question is that a statistical question is one that anticipates an answer based on data that vary. This wasn't clear in the origiginal question wording. I hope you find the revised wording to be an improvement.

St. Christopher School says:

almost 5 years

I'm a bit confused by the use of the word "typical." Doesn't that indicate a single answer and not a range of answers? I'm trying to figure out how I would explain the difference between questions containing the word(s) "typical" and "average" or "measure of center."