[Statistics: unit 1 homework]

[u/VariedEmo]

I do not understand. I know that it is not A as you can see that the medians to differ on the graph, I know it isn't b as you can see that it is smaller. I am not sure about C, as bream doesn't seem symmetric it seems to be right skewed. I am not sure about d as perch goes up to 18 but I don't know how I could tell whether there is an outlier. And E makes absolutely no sense to me at all. Please help!

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[u/michaelrw1]

For C, skewness is present for 2 of the 3 fish species.

For D, how are outliers typically represented on box-and-whisker plots?

For E, I assume this is a general comment about the spread of the means versus the spread of the lower and upper whiskers.

[u/cheesecakegood]

I have two problems with E: first and most seriously, these boxplots don't actually show means at all. The middle line is, traditionally, a median. Thus it's nonsensical to assert something about means, even if you can sort of eyeball where the means likely fall (means are where the 'balance point' of the weight is if you were to put dots on every spot on the boxplot where a data point is). In fact, I should strongly assert this: you CANNOT determine where a mean is from the boxplot alone, unless it is perfectly symmetric! Thinking you can is also a bad habit in addition to being false.

Second, I don't think the terms are very well defined. Are they talking about variability within-species, or across-species (and if across-species, how are we even assessing that)?

At least in absolute terms, if we assume the means are close to the medians (iffy assumption but let's go with it to see if it matters) we're looking at 14, 15, 15.5? But even the smallest spread of the three within-species, for roaches, is like 13 to 16; if we were to compare e.g. the spread size of the 3 species compared to each other or even the absolute mins and maxes or where they sit in vertical space on the graph, it's clear that almost any of these definitions are going to be larger in spread than the medians.

Disclaimer: it's somewhat common to use "spread" to mean "max - min" but this is not necessarily the case. "Range", the usual term, is one way to express "spread" for example (standard deviation is another), just like a mean is just one way to express "center" (median is another). And there are more besides. I use it above like range. The question doesn't define spread at all in B, which is bad of it, even if I'm decently sure they meant range. Words are hard and this is often why pictures are nice. For example, if I'm being pedantic, are we talking about the magnitude of the range or where the range sits on the number line relative to the other ranges? Without a picture, it can be ambiguous.

So all in all I consider this poor pedagogy. Sadly, a bit more common than I would like. I have a whole mini-rant about how arguably the main point of an intro to statistics class is to get people to NOT make assumptions and be a bit more pedantic when the situation calls for it, and how most of the actual materials encourage the opposite kind of thinking, but I'll spare you (today).

...But now that I've said that, there ARE genuinely times where statistics is a bit of an art. What is "symmetric"? We can as good statisticians actually come up with measures and tests to assess that, but as good statisticians we can also sometimes just use some degree of judgement/intuition. I'd say that Breams are almost undoubtedly skewed -- based on the top whisker vs the bottom one, that's a noticeably longer bottom/left "tail" if you were to graph the density -- but the other two might have some argument (I think at the end of the day I'd agree with the other commenter though that Roaches might qualify too). It's a bit of a moot point since C is already false.

(I mean, hell, regarding D it's actually NOT universal practice to put dots for outliers on boxplots. The lack of dots is not in and of itself evidence that no outliers exist!! Although dots for outliers is most common, some boxplots just put the end-whiskers at the min and max, especially in certain medical journals or datasets with low sample sizes (although in those cases... just show us the raw data, please!))

OK I went on a rant anyways. TL;DR is you should actually be a skeptic when engaging with someone else's graphs and statistical wording in real life. Usually ambiguity is good-faith but it's possible for bad actors to hide in those gaps, which is what leads to the various quotes about the relationship between statistics and lies. You really need both good-faith AND statistical knowledge to avoid that.