I don't know whether to laugh or cry when I see statistical
arguments like that.
Ignore the names on the various lines of data; this message
is 100% statistics, 0% politics. The data could equally well
be about frogs or asteroids; the statistical principles would
be the same.
Let's model the underlying reality as a Bernoulli process, i.e.
the toss of a weighted coin. Actually in this case we have a
/generalized/ Bernoulli process, i.e. the toss of a 16-sided
lopsided die.
Each /sample/ consists of N=679 tosses of the die. We want to
know how big the random statistical sample-to-sample random
variability will be, so we imagine collecting a huge number
of samples and doing the statistics. There are some surprisingly
simple formulas for the results:
expectation value = m = N p
variance = σ^2 = N p (1-p)
std deviation = σ = √(N p (1-p))
You can verify that the std deviation is half the square
root of the sample-size value *IF* the probability p is ½.
The percentage error is then half of 1/√N. This agrees
with everything we know about the toss of a fair coin,
unbiased random walks, central limit theorems, et cetera. https://www.av8n.com/physics/probability-intro.htm#sec-random-walk
OTOH let's be clear: the std deviation is ½√N *IF* the
probability p is ½ ... and *NOT* otherwise.
It turns out that rather commonly, pollsters take an
interest in situations where the probability is close
to 50/50 ... but that doesn't cover all situations, and
certainly not the present situation. The caption says
679 persons were polled, and claims the margin of error
is 3.8 percentage points. That's numerically equal to
1/√N (with no factor of ½). That could have been the
right answer if ...
1) if the probabilities were near 50% (which they aren't);
and
2) if statistical sampling error were dominant (which
it probably isn't, but let's not worry about that); and
3a) if we interpreted it as 1σ = 3.8% of 50% ... not
3.8% of full scale; or perhaps
3b) if we interpreted it as a two-sigma error bar instead
of the customary one-sigma error bar, so we have
2σ = 3.8% of full scale.
If you want, you can arbitrarily lengthen the error
bars in an attempt to account for methodological mistakes
or other systematic errors, but then please don't call
it "sampling error" and please don't throw around the
√N formula as if it were based on actual mathematics.
There are eleventeen reasons to think this polling data
is worthless, but sampling error is not high on the list.
You know without even thinking that the NPR graphic is
bogus. Given that the variables have to be positive,
you know it doesn't make sense to have 2% ± 3.8% let
alone 1% ± 3.8% or 0% ± 3.8%. You can see in the graphic
that the lower error bars have been artificially pruned.
Seriously, if you are rolling a die where a certain
number cannot possibly come up, it does not make sense
to say that the frequency is zero ± 3.8%.
The magenta error bars are 3.8% of full scale, as used by
NPR; meanwhile the black bars show the 1σ statistical
sampling error. The latter are significantly shorter,
partly because they represent 1σ error bars (which is
what physicists are accustomed to seeing) and because
they include the factor of √(p(1-p)). The better bars
do *not* support the conclusion that the article is
trying to reach. Even with this smallish sample size,
the weaker outcomes are distinguishable from the stronger
outcomes. Sure, the error bars overlap for some of
the nearest neighbors, but not for everybody.
In the real world, people make decisions all the time
based on imperfect data, often on data a lot dirtier
than this.
It's particularly ironic that the article makes a big
fuss about how "mathematical" it is. Jeepers, if you're
going to write a "math" article please make some effort
to use non-ridiculous math.
===================
Technicalities:
Note that for simplicity, I used an a_priori analysis.
That is, I assumed a model for the underlying probabilities
and explored what the observable consequences would be.
In reality, we need to do an a_posteriori analysis. That
is, we need to use the observations to estimate the
underlying uncertainties. That's doable, but it's more
work than I feel like doing at the moment.
For example, if you draw a sample of 679 readings,
and a particular value does not occur, you should not
infer that it's probability is zero. It could be just
a moderately low probability compounded by bad luck.
Hence the need for a_posteriori modeling.
Note that virtually all "least squares" curve fitting
is a_priori. It is virtually never the right thing
to be doing. If you're lucky, it's not wrong by much,
but it remains wrong in principle, and sometimes it
goes badly wrong in practice.
======================================
Technicalities aside, I leave this as a question and
a challenge for the folks on this list: how many of
your students would be able to look at this NPR
article and realize that it is completely bogus?
If not, why not? This isn't rocket science. This
is reeeeally simple statistics. At this level, it
is easier to explain than F=ma ... and more broadly
useful to ordinary non-physicist citizens.
Also a very great deal of modern physics depends on
probability and statistics, which is one more reason
why it's worth teaching.
===============
FWIW note that a Bernoulli process is named after
Jakob Bernoulli ... not to be confused with the
fluid-dynamical Bernoulli principle, named after
Daniel Bernoulli ... not to be confused with any
of the other Bernoullis.............