Chasing Their Own Tails: When Central Tendency Junkies Attack, Part 1

“The whole of philosophy is like a tree, whose roots are metaphysics, whose trunk is physics and whose branches are the other sciences.”

– Renee Descartes, The Principles of Philosophy, 1644

This is the first of a series called When Central Tendency Junkies Attack


My third grade class was so large that we had to be divided into two rooms connected by a sliding door.  Each class had its own teacher – I was in Mr. Osborne’s room; my friend Abe was in Mrs. Ragaluth’s.  During free time kids from each class were permitted to move freely between the two rooms, and you would have thought I would have visited Abe as often as possible.  But I didn’t.  Though I was not a bossy kid, I told Abe that he’d have to come see me.

The reason was Zach.

Zach was the meanest kid any of us had ever met.  He was always throwing something, or stealing something, or punching someone, or ruining a game during recess. And when he got in trouble, he seemed incapable of remorse.  No sooner was his punishment over than he’d be back at it.  Zach’s toxic personality infected the room, and in response I – and many kids from Mr. O’s class – responded by imposing a social quarantine.

I had forgotten about Zach until last week, when I posted an article on this site with the milquetoast title “Jonah Lehrer is Not a Neuroscientist.”  In it I sought to illustrate the perils of pop neuroscience, and the risk that when boiling down a complex article into a sentence or two a pop science writer might, for tactical reasons, misrepresent the gist of the findings.  Along the way I suggested that mean values better represent what “crowds” are doing than medians do, and then made this comment: “median guesses are not guesses by a crowd, as Lehrer states.  They are guesses by a single person….” (or the mean of two, in even-numbered populations).

Many readers went bazonkers at this – to the point, for many of them, of ignoring the general thrust of the article.  There were times, after reading yet another outraged comment, that even I was left with the impression that my post had been titled “Medians: The Biggest Lie Since Roswell.”  Here’s a selection of criticisms:

David F:“…as many others have pointed out, you seem to be lacking even a rudimentary understanding of statistics.”

Anonymous Coward: “…surely you should try to understand what a median is before saying ridiculous things like ‘median guesses are not guesses by a crowd’… [you are] borderline innumerate.”

Tim: “The point made in this blog post was that median is not representative of the group as a whole… which is basically stats 101.”

Cathy Kerr: “It is troubling that non-scientists coming to your blog are cheering on your critique of Lehrer based on your credentials as a scientist when in fact (based on the “peer-review” of your commenters) what you have taught them is a marginal cranky view of statistics (eg., median is not a group statistic).”

MRW: “I’m not so sure Lehrer was confused about what the median is, but I’m having trouble believing that *you* are clear about it. Medians are not guesses by a single person by any reasonable understanding…. The median isn’t the guess of “one or two” people. It’s often a better representation of the overall guesses of the group because it depends less, not more, on what any one person guesses.”

Melanie M: In this scenario, the *only* way to get an accurate picture of central tendency would be to exclude the outliers entirely, log-transform data, or use the median as the measure of central tendency. Using the arithmetic mean, which is clearly greatly swayed by extreme datapoints, would actually (in my opinion, and the opinion of many others) be unethical.

Chad Orzel: “It’s particularly bad… [Freed] blasts Lehrer for using a median value as his example, without noting that the median values are generally pretty close to the geometric means…. He derides the median as the guess of a “single person,” which completely misrepresents the nature of that measure…. Median values are often the more appropriate choice for data that are unbounded on the high end, and thus tend to be skewed by outliers.” (Orzel was my favorite, because he made my ignorance official, titling his blog post “Neither a Neuroscientist Nor a Statistician.)

In this post, I don’t want to talk about the substance of my interlocutors’ points – I’ve addressed those already, and will again in the third post in this series, though in a novel “outside the box” way that we’ll be leading up to.  Instead, for now, I want to talk about the affect my comments stirred up in them, because their anger is where the philosophical action is.  In doing this – focusing on feeling, not cognition – I am pulling a typical therapy maneuver.  We call it listening to the music, not the words.  But I think over the next three posts you may see that even in intellectual debate, following the music can be a good strategy if you want to understand what’s going on.

In this case, I think what’s going on is fear.

Let me describe the subjectivist methodology I used to reach this conclusion – lest it seem to be pure assertion.  In screw-up therapy, whenever a patient gets angry at one of my “screw-ups” – as some of those quoted here have done – I try not to get defensive. Instead, I try to ask myself whether I am witnessing a defense mechanism in operation. I asked not “what are they so mad about?” but “what are they so scared of?”

The hypothesis here is that angry people are unconsciously worried that they are weak, and further that their weakness is on the verge of being discovered and exploited.  In this view, the dynamic reason an angry person is angry – regardless of the merits of their point – is that they are mounting a burst of anger the way a wounded animal, surrounded by lions, might make one last snarl.  They are trying to ward off attack. This is why the deep structure of anger is so close to that of paranoia.

The next step is to not reflexively do what a lion would do, which is attack – though sometimes, when someone is shouting at you, this is a difficult impulse to suppress (projective identification, for all you psychodynamic psychotherapy fans).  As a therapist (rather than a predator) the ultimate goal is to help the person heal. Over time one learns to essentially “look through” (not past, or away from, but through) the anger, and talk directly to the hurt and fear. It sounds corny until you actually try it, when it feels amazing, and quite scientific – to hypothesize a hurt person in there, and by speaking to it correctly, hear him or her talk back.

So that’s what I did here.  I wanted to make things better – not psychologically, but conceptually.  So I looked at Orzel’s post, and Anonymous Crowd’s comment, and all the other critiques, and asked “what are they so afraid of?  Why are my beliefs about medians making them mad to not be scared?”  I wasn’t trying to psychoanalyze them.  I was trying to understand the intellectual point that they were, I suspected, afraid might enter into play if they didn’t fight me off. But like a psychotherapist, I assumed that their fear might be unconscious. For you see, we can have unconscious intellectual fears; we can be scared of ideas.

And that’s when I got it.

They were afraid of Zach.


Of course, this was at the time merely a hunch.  I had to prove it, and so I called up my old school and asked the Secretary – who had been there back when I was in third grade – if she remembered Zach. I wanted to make sure I had the history right.  “Remember him?” she asked, “I was the one who typed up Principal Perrine’s Central Tendencies, Inc. letters.”

“What?” I asked.

“What what?” she asked back.

“What report? What tendencies?”

“Honey, Principal Perrine was so worried about Zach, and getting so many threats from parents in Mrs. R’s class to withdraw their kids from school, that he decided to get scientific proof that Zach was a threat to class cohesion, and then use it to kick him out of school.  He hired this sociology outfit called Central Tendencies, Inc. to do an investigation.  But they did such a bad job he ended up firing them and doing it himself.  That’s why Zach wasn’t in your class second semester.  Don’t you remember he left?”

“Oh yeah,” I said, “that’s right.”

“Here hon, I’ll send you the report.  It’s good reading,” she said.

Three days later, in the mail, I received five documents paper-clipped together.

The first was a report dated November 1, 1976. It began: “Central Tendencies, Inc. was retained to evaluate whether, as indicated by several parents in this year’s third-grade class, Mrs. R’s room is “much more aggressive” than Mr. O’s.  We interviewed each child in the third grade separately, and asked them to rate the aggressiveness of the kids in their class. To avoid having them rate each child individually we asked them how aggressive the average kid in their class was, using child-appropriate terms (see Appendix A).  Each child chose an actual classmate who they felt was average, and then rated his or her perceived aggressiveness. The results are shown below.”

They then included this graph.

“Based on these results,” the report went on, “we report no statistical difference, by independent samples (Student’s) t-test, in the aggressiveness of the two classrooms. We conclude that the distress of students in Mrs. R’s class is either imaginary or caused by another source. For a fee we are willing to explore what this might be.”

The next document was dated November 2, 1976.  It was an angry letter from Mr. Perrine to Central Tendencies, Inc. saying their results were “plain wrong” because “that kid is as mean a kid as I’ve seen in my life; he is ruining the vibe in Mrs. R’s class.” (Yes, it was the 70’s – even Principals said things like “vibe.”)

The third document was a second report, this one dated November 8th, 1976.  “In response to last week’s correspondence, Central Tendencies, Inc. administered a new questionaire to the third graders, in which we objectively measured the aggressiveness of each student in the class using a validated measure. See the graph below for results.”  The school secretary who sent me the report was kind enough to indicate my, Abe’s and Zach’s aggressiveness ratings on the graph:

“As you can see,” the report went on, “the mean aggressiveness rating of Mrs. R’s class is high, relative to Mr. O’s, but this is due to an outlier that – in a large sample – would be a right tail.  For skewed data with long right tails the median is a better and more appropriate – some might even say more ethical – measure of central tendency.  Indeed, if we were to report the mean we might be derided by colleagues as borderline innumerate or even cranks. We therefore used the median, and the median confirms the results of our initial survey.  The average aggressiveness of the two classes is similar – our two surveys show this both subjectively and objectively.  Thus again we conclude that the problems in the two classrooms cannot be due high levels of aggression in Mrs. R’s class. For a fee, we are willing to explore what the true problems might be.”

A second letter from Mr. Perrine followed, dated November 9th, with the simple handwritten sentence: “you may know a lot about statistics, but you don’t know anything about third graders.  You get an F, for Fired.”

The final document, which I found moving in its simple clarity, was a hand-drawn graph in Mr. Perrine’s inimitable style.  It was a bit wrinkled and seemed to have an awful lot of food residue on it. Like Proust with his madeleine, I found memories of him come flooding back.  When I awoke from my reveries, I saw that on the x-axis was the text of a question I now recall him casually asking me one day while I ate lunch: “hey Pete – how much of your Halloween candy would you give me to move to Mrs. Ragaluth’s class?” To which I had, I remember, replied “no way Jose.”  I know because that night I became quite anxious when my mother told me that Mr. Perrine’s first name was, as a simple point of fact, Frank. I now realized he had been using Halloween candy as a rating instrument.  He always was a clever guy – and he sure knew third graders.

After gazing at this final graph for a time, I picked up my laptop and began typing. You may find my thoughts in the next post, which should come out sometime later this week.  For now, however, I encourage you to honor your own thoughts.  Let them linger on the difference between these three graphs, on the three questions they asked, on the differences in their respective means and medians and modes.  Let them linger as well on me and Abe, and our two classes, and how so many friendships that year were altered by one angry little boy.  And most of all, let them linger on my interlocutors’ outrage. They like chasing tails, sucking outliers into the mainstream, turning skewed curves into normal ones, and they turned angrily upon me when I said I did not think such manipulation – at least in my view – represented what Lehrer called a crowd.  Let your thoughts linger – for to linger, to tarry, to daydream is the essence of psychotherapy and, I sometimes think, of life.  Might there be hidden in all of this tumult and confusion some idea, some deep idea, that they might fear, some idea in reaction to which they have intuitively manifested the instinct to moralize and ostracize and appeal to authority?

I’ll try my hand at answering this question in the next installment of this unexpected and admittedly campy trilogy, When Central Tendency Junkies Attack.  For my own part, I will be circling back to the quotation from Descartes, with which I began. But for now, linger and mull on the meaning of Zach.

And on the statisticans’ meaning – arithmetic and otherwise – of our lives.


  1. It’s probably worthwhile to look at the fetishization of central tendency measures as a symptom rather than cause or driving problem.

    Suppose a question is: What are the stats to be used for?

    Measures of dispersion and clustering seem most accurate/predictable. However, central tendency measures have rhetorical advantages and are predictable as well — we assume.

    However, the social problem seems to be that as more information and policy/professional behavior alludes (at least) to measurements the general literacy needs to rise. But our brains have limited capacities so we need to choose our stats wisely and critically.

    We would propose that the function of this blog and community it forms, is to “teach the teachers” about these matters. So a critical reading of J Lehrer is only secondarily about him or his article and primarily a matter of the meta questions being explored in the comments and these follow-up posts.

    It’s a shame, but predictable, that Lehrer didn’t get that wider issue which he could have both contributed to and learned from — but we are defensive creatures.

    However, meta is a dirty job and journalists are not paid to get anywhere near it — well, unless it’s meta-physical.

      1. Yeah, let’s take a wack at it. Easy target. The British government has handed out copies to all ministers. That is worrisome. Although the Brits did can their .5mm lb investment in behavioral economics, thank buddha. Course, I don’t want to have to buy the bloody thing!

  2. “The hypothesis here is that angry people are unconsciously worried that they are weak”

    Is that a statement about the mean or the median of the unconcious worry of angry people?

  3. This is the very question you should ask when choosing a measurement of central tendency: how much attention do the outliers need in your analysis? That’s why philosophy, then statistics, are critical to science. Your first analysis indicated a poor understanding of the philosophy of statistics, and a pack of ravening wolves jumped on you, many of them no better informed on philosophy, but having some experience with statistics.

      1. I haven’t ever thought about any general questions in philosophy of stats, I’m only looking at questions I’m using for my research or ways I or others are using stats in an argument. If we can come up with a specific question, I’d love to. :)

  4. It’s very strange that both “anonymous coward”, in response to your previous post, and the report you were sent from your 1976 3rd grade class, contain the term “borderline innumerate.” I did a google search, this doesn’t seem to be an extremely common phrase. I like your blog and agree with your analysis of Lehrer’s article, but I’d like a comment on the probability of this occurrence.

    1. Oh my friend – my “report” from my third grade class is supposed to be a self-evident joke – it’s made up data, to make a rhetorical point. But man, many thanks for the honor of the close read!

  5. With all due respect, I think your story actually supports the argument in the original literature (about crowd estimates) with regard to using median or geometric means rather than arithmetic means.

    When trying to show whether a disruptive student is a problem for a class, any class average is irrelevant, since the average student is not the problem, the problem is the classroom is unpleasant because of one (or a few) students.

    However, the median might sometimes be quite appropriate. For example, if the school was wondering if Mrs. R is a bad teacher, because the average student in her class is more aggressive (perhaps thinking that the teacher is somehow encouraging bad behavior), the fact that the data is heavily skewed by one exceptionally disruptive student is highly relevant. The fact that the median aggressiveness of the students is pretty much the same as in Mr. O’s class suggests that the problem is not the class as a whole, but only a few bad apples.

    So, absolutely, it is important that any choice of statistics actually reflects what people are interested in—the most common way of lying with statistics is to cite the irrelevant.

    I did not read the Lehrer article, don’t plan to, so I am not defending his writing as such. But the table (now much discussed) suggests three interesting findings:
    1. The median response to questions of the type asked in the study appear to be within a factor of 2 of the real answer, which counts as middling bad in my book.
    2. The mean response is sometimes off by a factor of 10, which is much worse.
    3. And judging from the difference between the results, the variance and skew of the median is dramatically lower than the variance and skew of the individual respondents answers, which is a fact of great value.

  6. While I disagree with your characterizations of me as weak and scared (and I’d classify myself as annoyed rather than angry), I don’t see anything to be gained in debating that point.

    Even in your example, if you want to know what’s typical of the group, the median still looks like the better choice.

    If you want to know whether Zach is a bad seed, neither the median nor the mean seems like a very good choice. Personally, I’d take one of two approaches:
    1) If you want to know if Zach is mean, compare him specifically to the group statistics in graph 2.
    2) If you want to know if Zach’s meanness is affecting the classroom atmosphere, use a measure of classroom atmosphere rather than “average meanness” (e.g., something like graph 3).

    Really, I’d probably use both approaches. Then there would be evidence that Zach is unusually mean and that this meanness is detrimental to the classroom atmosphere.

    The mean in your hypothetical happens to show your point, but it’s not just because it’s a mean. What if there was an unusually nice kid in the class as well? The classroom atmosphere could still be bad and Zach would still be mean.

    1. Incidentally, I’m puzzled by your first graph. Why do the students rate the average (arithmetic mean?) meanness as being nearly identical – it doesn’t seem to support your argument about means and medians or to agree with graph 2.

  7. Your post was an extremely effective pedant attractor. I was probably one of the less polite ones; I apologize for my coarsely worded criticism (which was at least mixed with praise for what I liked in the original article). I admire your response.

    What motivates our (my) pedantry? Maybe it’s fear, as you suggest. Maybe it’s frustration with misinformation (especially among those of us who have a hard enough time already teaching this kind of stuff).

    But I think one thing that REALLY attracts pedants is the opportunity to out-pedant (if I can coin that word) another pedant. If person B is calling out and correcting person A, the ne-plus-ultra of pedantry is being able to point out that B is mistaken. I can’t tell you the motivation behind that (it’s probably not anything good), but I suspect that was a large part of what was going on here.

    As to the graphs in this post, I’m not sure what point you were making. If you’re pointing out that it’s impossible to characterize a distribution with a single number, I wholeheartedly agree. Pretty much any time you attempt to describe a multi-dimensional vector (i.e. any continuous or discrete function) with a one-dimensional vector (i.e. a single number) you will lose information. But I don’t see how this is related to the previous post, or to the motivation of us rude pedants.

    1. Ignore the personalize and ad hominem attacks — just delete ’em. The distract the discussion, waste everyone’s energy and are just venting. There are lot of sites for hostile venting — very few for civil conversations.

      Plus, if you don’t bock and delete them they scare away the shy and polite people who really have the most to contribute. Polite doesn’t mean you don’t disagree — to the contrary, it would be impolite not to! lol

      1. Well, I’m not into censoring or even judging; that’s my nature, to let it all air out. But why do you think shy polite people will scare off? They don’t like controversy?

        1. It’s all about managing emotions, of course. The hostile-aggressive venters do it by externalizing and bullying, and shy people withdraw. “People who can’t control their own emotions have to control those of others around them.”

          The majority of blog visitor see the risk, which is real, that anyone who disagrees with a venter will be attacked and usually they are. Most people have thin skins so instinctively avoid forums where 1, 2 hostile folks hold sway.

          The other problems are:
          – We want as many different voices, especially conflicting, as possible – pluralism is best for problem-solving. Hostile folks want to silence all other voices.
          – Bullying and rhetorical tricks work – once they enter a group discussion. Some hostile folks will be civil when warned, some won’t
          – Instead of standing up to bullies on line, people just start to ignore the posts and fade away from the blog

          If you have a loud hostile person at a party — pretty soon everyone leaves. Only rarely is the person driving others away asked to leave.

          We moderate some professional groups and this is our experience. We have had to set a zero tolerance policy for off-topic or hostility to make the group feel “safe” for everyone.

          This is an endemic problem — most look away and hope it will solve itself. It won’t. From the Harvard Business Review blog policies: “No ad hominem attacks. These are conversations in which we debate ideas. Criticize ideas, not the people behind them”

          Also from a different source: “The point of ad hominem, personalized attacks is to take a swipe at someone’s character, to undermine their personhood, identity and integrity and thus the point and matter of fact being argued.

          The attacker’s brain has confused an idea they disagree with as a personal attack – so they attack back. Since their brain feels ideas hyper-personally, this is the domain they are stuck in and must drag others into.

          However, it is actually a deflection away from the topic in dispute. But, by playing on our instinctive reflexes to either defend ourselves or devolve into personalization, these tactics inevitably work and an effective distraction to information sharing, civil exchange and problem-solving.”

          All social groups need policing — even monkey groups it appears.

  8. Superb post however , I was wondering if you could write a litte more on this subject?
    I’d be very thankful if you could elaborate a little bit more.
    Bless you!

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