There's a pervasive idea in (at least) western culture that most people
are normal
. The grain of truth in this gets lost in a misinterpretation
that leads to labelling folk as weird
in so far as they don't match the
perceived standard for what's normal
. That perceived standard is an
illusion; in fact, the thing that's normal is that, while most of us are in many
regards fairly close to most of us, each of us is in some regards significantly
different from most of us.
(This subnormality strip
illustrates the point quite nicely.) That might conflict with some folk's
intuitions; but it's actually an inevitable consequence of statistical
variability.
The popular intuition on this is reinforced by something commonly taught in
statistics classes: the so-called normal distribution
, also known as
the Gaussian distribution
, which shows a nice spread about a central
value, with most of a population within a moderate range of that centre. The
probability density at distance x standard deviations from the mean of a
normally distributed variate is proportional to exp(−x.x/2), which dies
off robustly away from the mean as x increases. Only 27 in ten thousand of a
population falls more than three standard deviations from the mean. This fits
nicely with the conventional idea that being like everyone else is the usual
case. So far so good, but that's for just one measurable property of a
population. However, life is not so one-dimensional; there are many properties
of a population that we can measure.
I should at least mention, in passing, that not every attribute of a
population is even approximately normally distributed: that familiar bell-shaped
curve isn't the only game in town. For some of the things we can measure
of a population, the distribution isn't even continuous (how many cars does your
household own ? I'm guessing it's a whole number…) nor need it,
even when it is continuous, be unimodal
– that is, there may be
more than one centre
at which the distribution's density is highest. (At
a guess, and without even looking to see if I can find data, I imagine the
floor-areas of homes have several peaks, at sizes that developers think various
common types of household shall find comfortable – the single flat, the
couple flat, the family home and so on – while relatively few homes have
areas in between these sizes. Even when a continuum variate does have only one
peak (mode), it isn't always the average (mean) or the point with half the
population on each side (median), as in the case of the normal distribution.
Still, when a distribution is messier than normal, it'll only make my case
stronger; and it's strong enough even with the normal distribution, so I'll
stick with that in what follows.
Before moving on to that, though, I should also note that variates that are commonly described as normally-distributed are often only approximately so; and, in many cases, conceptually can't be normally-distributed. A common reason for this last is that the normal distribution has non-zero density along the whole number line, from minus infinity to plus infinity; the density is crazy small outside the vicinity of the centre, so this doesn't tend to matter in practice: but there is something preposterous about assigning a non-zero probability (no matter how small) to selecting a random living human adult and finding they have a negative body-mass or height. That's not just improbable, it's conceptually broken: the probability of selecting a human adult with negative height or body-mass is zero and shall remain zero even if humanity spreads out through the galaxy and populates ten milliards of planets with ten milliards of people on each; a distribution that allows a one in power(20, 10) probability of picking a random person and finding their height is negative would then be claiming to expect that one of those humans does indeed have negative height. Such a model would be Wrong. Fortunately, there are some perfectly good distributions, such as the gamma distribution, that assign zero probability to negative outcomes yet approximate the gaussian for suitable values of their mean and standard deviations, so we can conceptually use these to describe such variates even when we do in practice analyze the situation using their approximation by the Gaussian distribution.
So suppose we have diverse properties of a population that we can measure,
and that each of these properties is normally distributed. There may be some
correlations among them (e.g. taller people tend to have more mass than shorter
folk, on average) but we can typically quantify those correlations and
re-express our set of measured quantities in terms of some independent
variables, each of which shall typically be (tolerably well modelled as if it
were) normally distributed. So, rather than a line to represent the one
quantity we've measured for our population, we'll get a space with as many
degrees of freedom to it as we have independent variates expressed by the
quantities we've measured. Rather than a density per unit length along the
line, we'll have a density per unit volume
of that space;
that volume
shall be area if we have just two variates, the usual spatial
volume if we have three and some higher-dimensional generalisation of this if we
have more.
So, to a reasonable approximation, we can assume (some suitable parameterisation of) our multi-dimensional data are normally distributed and independent. In that case, if we have co-ordinates x, y, … representing the values of the different variates. As the're independent, the density in our higher-dimensional space shall be proportional to the product of their individual densities, exp(−x.x/2).exp(−y.y/2)…; and exp is a homomorphism from addition to multiplication, so this is just exp(−(x.x +y.y +…)/2), which we can write as exp(−r.r/2) with r as the usual distance of the position described by co-ordinates x, y, … from the central point. So, just as for the one-dimensional gaussian, this multi-dimensional gaussian's distribution is highest at the centre and drops off fast as we move away from it. Indeed, the probability density at a given distance r from the centre is just exp(−r.r/2) times the density at the centre, regardless of dimension.
And yet something crucial has changed: if we want to know what proportion of our population lies in some region of our co-ordinate space, that encodes some range of variation in the values of our originally-measured properties of the population, we integrate over a region of that co-ordinate space. This integration combines our density with volume; and the volume that's close to the centre point is small, and gets smaller the more co-ordinates we have.
In one dimension, the length of our co-ordinate line that's within a half of the centre is one; the length between half and one away from the centre is likewise one (half on each side of the centre) and likewise as we move outwards. However, in just two dimensions, the area within half of the centre is π/4 while the area between half and one of the middle is three times as large and the area between 1 and 1.5 is 5.π/4; each ring, going outwards, in which radius increses by half, has a larger area than the ones inwards from it. In three dimensions, the volume within half of centre is π/6, that between half and one is seven times as big, that between 1 and 1.5 is 19.π/6 and, again, each band gets bigger as we move outwards. The higher the dimension, the faster this grows with distance from the centre.
Although the density further out is lower, the density takes that lower value over a larger volume than at lower dimension; and that lets the lower density add up to a larger proportion of the population. In dimension n, we end up integrating exp(−r.r/2).power(n−1, r) over a range of r values to find out what proportion's distance from the centre lies in that range. Although the density, exp(−r.r/2), is highest at the centre, the power(n−1, r) is zero there, for n > 1, and grows as we move away; for higher n, it grows faster. So the typical distance from the centre grows with dimension – as, indeed, does the degree of variability about that typical value.
So what does this mean in practice ? If you look at the heights of
human adults, or any other single parameter of a population, you'll
usually find relatively few who lie outside some common range and more lie near
the middle of that range than near its ends, just as the conventional wisdom
about normality
says.
However, once you consider folk in the round
, taking into account
everything about them, rather than just one aspect of them at a time, any
attempt to describe a typical
person is doomed. In so far as you can
devise a chart of all the possible ways of variation among folk, that you need
to take into account to understand them holistically
(while still having
all the usual single-parameter measures of interest represented linearly within
your chart), folk are going to be spread out across that chart and, even where
there is clustering around some centre, only a small proportion of the
population shall be near any one such centre, with most folk significantly
distant from it.
We are used to people being described as weird
if they deviate
significantly from what we imagine to be typical. The above analysis says that
most of the population is weird in one sense or another. Furthermore, once you
consider how tiny a proportion of the population are close to any given
cluster-point, that could be considered a candidate to be described
as typical
, it turns out that being close to typical
is in fact,
itself, also weird
, in the sense of being somthing that very few folk
are. Consequently, everyone is weird, in one sense or another.
All of the above is true when folk are fully forthright and honest about how
they are, so that we can all look about ourselves and see the real variability
among folk, and our place within it. The real world is, however, complicated by
the fact that people act out rôles, perform personas, try to fit in
and in diverse other ways conceal our true natures from those around us. When
folk do this to appear to conform to some culturally-communicated conception of
what is normal
, they create the impression that this normality
is
more common-place than it really is.
Indeed, the one way one can rescue the conventional notion
of normality
is to say that being normal
is all about keeping up
the pretence of conforming to certain culturally-identified norms. One defence
of that is that civilisation is built on such pretences: where I've seen this
case made, it has hinged on a tacit presumption that folk are nasty and brutish
unless conditioned to pretend otherwise. This seems, to me, an unduly
pessimistic view of humanity. Perhaps those who make this case secretly fear
that they themselves would be nasty and brutish without the cultural
straightjacket of normality
(albeit possibly projecting this fear onto
others); but my general impression of humans I have dealt with is that a large
enough proportion of us are sufficiently decent that we could do with less
pretence and more acceptance of how varied we really all are. While I can
believe there are some who are nasty and brutish, my suspicion is that not only
are they few and far between but also most of them got that way as a rebellion
against being coerced into being something they were not. If we spent less
effort on crushing folk into rigid boxes and devoted more to understanding folk
the way they are, I'm fairly sure we'd all be better off.
In particular, coercing folk to shoe-horn themselves into rôles that
don't fit them is bad for their mental health, both directly by the distress
they experience in forever trying to keep up unnatural pretences and indirectly
by hiding their real nature from those who might help them to adapt better to
the world around them. I have seen this repeatedly from folk talking about
their experience of autistic tendencies: the social pressure to conform leads
them to hide behind a mask of normality
, but doing that is stressful for
them, leading to stress-induced behaviours that seem odd to others, and keeps
them from getting the help of those who understand the condition they're
enduring. I lived with a kindred condition
for three decades and likewise masked my distress and confusion about the world,
initially because I knew I would be teased about them if I did not hide them,
eventually just out of deeply-ingrained habit and the presumption that this
was the human condition
that I should endure with as much grace as others
seemed to. (I did not realise their condition was quite different from mine,
and easier to endure as a result.) The consequent failure to talk to others
about a permanent nightmare left me trapped in it – until, overwhelmed, I
went to talk to a doctor who recognised the condition and fixed what was wrong.
Without the social pressure to pretend, I might plausibly have talked to a
doctor about all of that a whole lot sooner.
Meanwhile, of course, the prevalence of acting makes it easy for malicious
players to hide their true stripes. Folk may recognise that their public
façade isn't real, but think nothing of it because everyone acts
to some greater or lesser degree. If more folk were routinely straightforward
in their dealings with others and sincerely sought to be true to themselves,
those recognised as acting would be subject to more scrutiny and suspicion as to
why they're doing that. That wouldn't, of course, stop the worst of the
psychopaths, as they would just learn how to pretend so convincingly it seems
sincere, but it would make that harder for them (so more likely folk would spot
the mask slipping) and the lack of others looking for artificial rôles to
play would make it harder for them to recruit accomplices.
Humanity's minds would be healthier if more of us were sincerely ourselves,
fewer of us pretended to be anything we aren't and our culture reflexively
opposed any attempt to coerce folk into being, or pretending to be, what they
are not. It would be easier for human cultures to embrace that healthier path
if we all learned that the notion of normality
on which it is founded is
a myth and the reality of a world full of wonderfully weird folk – which
is what we've always really lived with, however much we may have pretended
otherwise – is a glorious feast we can all rejoice in.
Written by Eddy.