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I have another question concerning one of previous posts: Stem-Leaf Display

In the textbook, in a later paragraph, they remark about how "...there are no observations that are unusually far from the bulk of the data (no outliers), as would be the case if one of the 26% values had instead been 86%."

Isn't the $4\%$ value particularly distant from the bulk of the data? And how would having an additional $86\%$ value in place of one of the $26\%$ values affect anything? Also, is the representative value always found in the place where the bulk is, where most of the data is concentrated?

Another remark the author makes:

"The most surprising feature of this data is that, at most colleges in the sample, at least one-quarter of the students are binge drinkers. The problem of heavy drinking on campuses is much more pervasive than many had suspected."

Wouldn't it actually be nearly $50\%$ of students at universities be binge drinkers, because that is where the values are concentrated?

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up vote 1 down vote accepted

I’ll start with the last question. The statement is about the number of colleges with at least $25$ binge drinkers, not about the number of students who are binge drinkers. Remember that each entry gives the percentage of binge drinkers at one college. Thus, every entry that is $25$ or more is for a school where at least a quarter of the students are binge drinkers. If I counted correctly, only $16$ of the $150$ entries are below $25$: the one on the $0$ line, the $10$ on the $1$ line, and the first $5$ on the $2$ line. That means that at $134$ of the $150$ colleges in the survey, at least a quarter of the students were binge drinkers. I think that $134$ out of $150$ qualifies as ‘most’!

Most representative value really isn’t a technical term with a precise meaning. In some distributions there really isn’t any value that could be described as ‘most representative’ in any useful sense.

The $4$% value is fairly far from the large number of points in the $30$s and $40$s, but it’s not separated from the rest of the data by an inordinately large gap: it’s just $7$ percentage points from the next datum, and we expect some spreading in the tails of the distribution. An $86$% value, on the other hand, would be separated from the rest of the data by $18$ percentage points, a much bigger gap. The larger size of the gap would show up visually, too: the empty $70$ line would between $68$ and $86$ would stand out quite clearly. Changing one of the $26$% values to $76$%, on the other hand, would make the upper end of the distribution look a bit more like the lower end: I would not call that an outlier.

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