help desk - 2nd grade mode question

A ktm reader sends this question from her daughter's 2nd grade math book.

flaw of averages, illustrated

 

The Flaw of Averages: Why We Underestimate Risk in the Face of UncertaintyThe Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty

flaw of averages

the book

the online animations & simulations

and the music! playing The Big Enchilada now -- love it (directions for clicking through to the music here)

 

The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty 

what is an average, anyway - part 2

re: what is an average, anyway? anonymous wrote:

One thing I never realized until I started teaching is why we calculate the sum, then divide by the number of numbers to find mean. We're actually finding how much each person would get if the items were distributed equally. So if there were 3 apples, 5 apples, and 10 apples, if we combined them we would have 18 apples. Dividing by 3 gives 6 apples per person if the apples were shared equally. This also shows why we use median rather than mean for items like income where there can be outliers that skew the results when the items are shared equally.

This reminds me of Ron Aharoni saying that when he taught arithmetic he realized there were subtleties to elementary mathematics that he hadn't thought about.

Must find that passage & post!

gasstationwithoutpumps on mean, median, & mode

gasstationwithoutpumps on mean, median, & mode

re: what is an average, anyway?

Mean and median are both "measures of central tendency". They are ways of summarizing a large set of data in a quick way that simplifies the data and removes noise.

If you know that the average rent for office space is $4 a square foot, and you need 1000 square feet per person, then you can quickly judge whether you can afford an office for your staff, without having to go through hundreds of real-estate listings. Of course, you also need to know something about the variance, and not just the mean, as there may be a few not-so-nice places that are much cheaper than average.

 gastationwithoutpumps

what is an average, anyway?

A friend of mine asked me to walk her through mean, median, and mode --- and it came to me, thinking about it, that I don't exactly know what an average is beyond the obvious.

Another issue: I'm so sick of my own child (& everyone else's) being lost in group means that I've come to feel some real antipathy towards the very concept of a group average. Meanwhile the concept of a personal average makes sense and seems obviously useful ----

Here are my questions.

What is useful about averages? 

What do averages tell us?

And why did the calculation of averages come to be so important culturally?