Interpreting Categorical and Quantitative Data Statistics

Mean (Number of) Girls (and Boys)

“Find the mean number of siblings.”
Student 1: (0+1+2+3+4+5)/6 = 2.5
Student 2: (1+4+6+4+2+1)/18=1
Student 3: (1+4+6+4+2+1)/6=3
Student 4: (0+1+2+3+4+5)/18 = 0.83
What sort of misconceptions can you dig out of these student responses? If this were your class, how would you respond?

3 replies on “Mean (Number of) Girls (and Boys)”

I know we are supposed to respond to the student work but I find it very difficult not to critique the question itself. This table is very difficult for me to interpret. The question is ambiguous. No wonder students find it difficult. (Or I have many misconceptions and would not do well in this class, a possibility I am a little worried about!)

The right-hand column is labeled “frequency”. But if this column can be “totaled” to 18 then I suggest it is not a frequency at all. I guess it is “number of people who have this number of siblings” but it could be “number of families that have this number of siblings” and that would have a different meaning. (Example, first row: Is there 1 person in the sample who is an only child or is there 1 family in the sample that is childless? If a brother and sister are in the sample, is that recorded as 1 instance of 2 siblings or 2 instances of 1 sibling each?) So is the question asking for the “mean number of siblings” per family or per person? Those would have different answers, right?

If the goal is to have students question and debate all these issues (which would seem valuable to me!) that’s great. But it seems to me that the goal was probably to have students multiply column 1 by column 2 and divide by 18, and even I am not that motivated to do that because the information given to me is so ambiguous.

I disagree a bit with secretseasons. I think the question is not ambiguous, but the meaning of the answer is. We should certainly do the computation and then ask “the mean number of siblings for whom?” — though I’d guess that if this came up in class, it would be the mean number of siblings that the 18 kids in this class have. And I agree that it would be interesting to have the conversation about the difference between average number of siblings that a kid has and the average number of kids in a family!

Also in statistics, “frequency” is commonly used for a count of this sort; we might use “relative frequency” if we wanted the column to add to 1.

For the student work, I think the fundamental question to be asking is “what does frequency mean?” I don’t think that any of the student responses show that they know that this table is shorthand for 0,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,5. Part of the problem (particularly in these days of technology!) is that we leap far too soon to these kinds of summary tables instead of working directly with data. The kids should have a lot of experience making tables like this for themselves before leaping toward problems like computing the mean of the data summarized here.

I think I would first ask how many children there are to begin with, and then ask how many siblings there are. Maybe draw them out, give them names. If we can find those two, we can then find the mean.

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