Every inhabitant of the planet żMi is either a Yabby or a Zetapode. The two groups, collectively known as żMitians, contain roughly equal numbers.
The following two statements are simultaneously true:
(i) Zetapodes are, on average, much taller than Yabbies.
(ii) All the “very tall” żMitians are Yabbies. In fact, the tallest Yabbies, of which there are many, are much taller by far than the tallest Zetapode.
(Here’s an example in which both statements are true. There are 500 members of each group. All Zetapodes are 6-feet tall, making their average height 6 feet. Of the Yabbies, 400 are 3-feet tall, and 100 are 10 feet tall, making their average height 4.4 feet. The average żMitian is 5.2 feet tall. Keep these numbers in mind if you plan to market seats to their hover car manufacturers.)
Question: Which group is taller?
My Answer: If pressed, I’d say neither. But only trivially so, just as I’d say neither if asked, “Which is taller, milk or jealousy?”
I think a better response would be to ask for clarification, as the question in its current form is unintelligible. Importantly, it would also be unintelligible if (i) were all we knew about the two groups, or, likewise, if (ii) were all we knew. My bet, though, is that given just (i), many people would say (without clarification) that Zetapodes are taller; given just (ii), they’d say (without clarification) that Yabbies are taller.
(And some smart aleck, given (i) and/or (ii), would interpret the question as asking which group’s individual members’ heights sum to the bigger number. I’m fine with this, so long as the interpretation is made clear [which implies, for example, that a 200,001-member group of 3-inch tall people is 3 inches taller than a 500-member group of 100-feet tall people].)
Here’s another fun fact about the inhabitants of żMi:
(iii) Seven percent of żMitians reach the important developmental stage known as “shedding” at age 22.
Question: One żMitian is selected at random. What’s the probability the selected żMitian, who happens to be a Zetapode, sheds at 22?
My Answer: Zero. This is due to conditionalizing on a fact I forgot to mention:
(iv) Only Yabbies shed. (This, by the way, means that 7% of żMitians shed at 22, while 14% of Yabbies shed at 22.)
So, an answer of 7% would be acceptable given ignorance of (iv). But it’s important to keep in mind that lack of information is common, maybe even the norm (that’s kind of the whole point of probability); this includes ignorance about what sort of information one is lacking.
Concluding Question: Why did I take the time to share these examples that many people will find obvious, maybe condescendingly so?
My Answer: Because I very frequently run into such examples in the real world. They come from online user-commenters, journalist, academics, politicians, and, well, you get the idea. Sometimes it’s hard to tell whether folks are reasoning poorly; are being unintentionally sloppy with their words and concepts; have simply not read the study they’re citing and instead are repeating what they’ve heard others say (who have also misrepresented the study, for whatever reason); are making an intentional rhetorical move by exploiting the ambiguities of such talk (e.g., by leaving out phrases like “on average”; switching freely between “you” singular and “you” plural; switching freely between “we” and “they,” e.g., in reference to one’s home-team: “They lost last week’s game” but “We won last night’s game”); and so on.
The general, accumulative effects of all this, it seems to me, are confusion, anger, and an overall impediment to progress in areas that matter.
I also happen to find such examples fascinating in their own right, even when they’re simple—especially when they’re simple.
More importantly, this post is a kind of self-check, as I’m no innocent here, both as speaker and listener. Self-checking is crucial given the barrage of statistically grounded claims (made without clarification) coming in these days from all points along the political spectrum.
Finally, there are ways to gracefully, effectively, sincerely, and rightly traverse a sort of talk I might seem to be criticizing here; for example, a kind of socially conscientious talk that doesn’t claim to be—nor does it need whatsoever to be—grounded in statistics. This is a different sort of talk and not one I aim to address here, though I acknowledge that that sort of talk and the talk I am criticizing sometimes travel on the same breath, as it were, blending, each tainting the other.
-The Wikipedia entry “List of Cognitive Biases” features an extensive list of “systematic patterns of deviation from norm or rationality in judgment.”
-The 99% Invisible podcast episode “On Average” (#226, 8/23/2016) includes a segment on the U.S. Army’s early attempts at designing a universal airplane cockpit (hint: their [literally] fatal flaw was in relying on averages rather than ranges).
-For a fun and counterintuitive conditional probability question picked apart in detail, check out my post: “An Easier Counterintuitive Conditional Probability Problem (with and Without Bayes’ Theorem).”
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