The ever-helpful Mathematics Education Research Blog posted about a new journal called Education Designer which includes a mathematics article.

The problem? One of the examples seems wrong to me. Take a look at this worksheet (which goes with this one). Specifically it makes statements like

*In this exam, the mean mark was greater than the modal mark.*

and expects the students to match them to a box-and-whisker plot. Except — how can you tell anything about mode from a box-and-whisker plot? Is this a UK-to-US translation issue or is the worksheet wrong? (EDIT: See the comments. It’s more like a cultural assumption issue.)

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Mr. K, on November 17, 2008 at 3:14 pm said:Mode is a particularly useless statistic for smallish or uneven data sets. It drives me nuts when we have to teach it using small sample sizes, and the give it equal weight to the others.

Much like histograms, it is susceptible to aliasing error based on bin selection.

Given all that, if you presume a somewhat even distribution, or you allow your bins to be the size of the narrowest quartile, it becomes evident that the mode will be in that narrowest quartile.

I’m not sure where the fault lies in this – there are enough unspecified assumptions to give everyone a share of the blame.

Alex, on November 18, 2008 at 1:28 pm said:“how can you tell anything about mode from a box-and-whisker plot?”

Short answer: by looking at the skew.

Long answer: most distributions resemble a bell curve (e.g. the normal distribution). If this is symmetrical, then the mean, median and mode are all the same. But if the data is clumped to the left (like in example H), then the mode is in the left clump; the median is marked, of course; and the mean is on the right.

http://en.wikipedia.org/wiki/Skewness for pictures of skewed distributions.

In the UK, this is part of the A-level maths syllabus (Stats 1).

Jason Dyer, on November 18, 2008 at 3:03 pm said:That is, of course, unless you don’t have a number repeated at all. I was thinking (taking something that would give example H) of a set like 10, 30, 32, 35, 40, 45, and 90. But I am understanding the main assumption here now (it’s a large number of people taking the test). Thank you!

I also see the cultural difference now — on US tests I have seen box-and-whisker plots tend to be on smaller data sets that can be tabulated reasonably by hand. Apparently on UK tests they focus more on the en masse aspect.