Nov 17

How good are you, or any of us, at estimating numbers? Sometimes I'll be in
a crowd of people and I'll try to make a quick estimate of the numbers
around me. I usually don't have a way to verify my guess, but I feel pretty
confident in distinguishing between dozens, hundreds and thousands. More
precision? Only possible with much smaller crowds; or by counting.

That last is an interesting point: at what apparent size of a gathering of
people, or a set of things, do you turn to counting rather than estimating?
At what apparent size does precision start to fall off?

The number 4 pops up in answers to such questions. But why can't we
accurately estimate 7, or 10? Can we train ourselves to do so?

I don't know, but I had fun writing this column for Mint (today, Nov 17):

Less than four at one blow,
https://www.livemint.com/opinion/columns/less-than-four-at-one-blow-11700159353369.html

Let me know what number you can estimate accurately.

cheers,
dilip

---

Less than four at one blow


"The mind is unable," wrote W Stanley Jevons, "to estimate any large number
of objects without counting them successively." He was right. Think, for
example, of what you'd do if I asked you to look out of the window at a
traffic jam on the road below and tell me how many cars are involved.

You'd start counting.

But Jevons went on: "A small number, for instance three or four, [the mind]
can certainly comprehend and count by an instantaneous and apparently
single act of mental attention." Right again. Think of reporting how many
fingers of one hand I hold up. Definitely five or less, of course. One
quick glance and you'll be able to tell me "three", or "four". You
certainly won't count the fingers.

Jevons wrote these lines in a scientific paper reporting on a protracted
experiment he conducted. First, he would grab a number of beans in his
fist. Second, he'd throw them into a box. Third, he'd take a quick glance
at the box and write down a guess at how many beans were in there. Fourth,
he'd actually count the beans in the box and write down that number too.

Jevons was a patient, diligent experimenter: he did this bean and box
exercise 1027 times. He put the results into a table where the rows
represented his estimates and the columns the actual numbers. According to
the table, the smallest number of beans he threw into the box on a given
trial was 3; the largest 15. Presumably he didn't want to waste his time on
a grab of just 1 or 2 beans, and he never grabbed more than 15.

But Jevons found something fascinating with his trials. When there were
only 3 or 4 beans in the box, he guessed right every time - all 23 and 65
trials, respectively. But with 5 and more, he started making mistakes. The
more numerous the set of beans, the more his guesses were off the mark.

Thus with 5 beans, he guessed right 102 times, but also guessed "6" four
times and "7" once. With 6 beans, he guessed "6" 120 times, but also "7" 20
times and "5" 7 times. With 10, it was like this: "8", 6; "9", 37; "10",
46; "11", 16; and "12", 2. With 15, like this: "12", 2; "13", 1; "14", 6;
"15", 2.

Partly, what this table suggests is that though he did make mistakes,
Jevons's guesses even with larger numbers were mostly correct, or close.
I'm not sure that faced with 10 beans, I'd guess correctly 46 times out of
107; or with an error of just 1 bean 99 times out of 107. Or maybe I would,
and maybe that's one point Jevons's data makes - that we can make pretty
good estimates of smallish sets of objects.

But the more intriguing point his data makes is about the number 4. It
looks like a limit on our ability to quickly estimate the number of objects
in a set. Four or less, and we get it right every time. More than four, our
accuracy falls off.

This may seem obvious and unremarkable to you. Of course we can guess small
numbers precisely, and larger numbers not so precisely. Yet think about
this: why is the threshold 4? Why not 3, or 6, or 10? Does it have
something to do with our five fingers? Something else?

Jevons published his paper all the way back in 1871 ("The Power of
Numerical Discrimination", Nature, 9 February 1871,
https://www.nature.com/articles/003281a0) - and ever since, there are
scientists who have wondered about this threshold of 4. If Jevons wondered
about it too, he didn't have access to the tools of modern neuroscience to
help him find an explanation. But a recent study does have a possible
explanation ("Distinct neuronal representation of small and large numbers
in the human medial temporal lobe", Esther F Kutter et al, Nature Human
Behaviour, 2 October 2023,
https://www.nature.com/articles/s41562-023-01709-3).

The idea was to monitor how the brain behaves when faced with such an
estimation task. To do this, naturally we cannot simply implant electrodes
into the brains of living humans. Instead, these scientists found a set of
people who already have such implants - epileptics who have the electrodes
for medical reasons - and were willing to be studied.

These patients were asked to do simple mental arithmetic while their brains
were monitored. What they found is, to me, simply startling. For each
number, individual neurons lit up, or became active. That is, a "3"-tuned
neuron fires up only when offered the number 3; a "6"-tuned one, only the
number 6. Such "number neurons" had been identified in animals before, but
never in humans.

But after she analyzed 801 of these neuron firings, Kutter was able to
tease out two different patterns. As you might guess, one was for small
numbers, the other for large ones. The neurons tuned for 4 and below were
nearly error-free. That is, they lit up only for their preferred numbers
and not for others. In contrast, the neurons tuned to steadily higher
numbers than 4 were steadily less precise in their firing. Sometimes they
didn't fire when expected; sometimes they fired when fed a slightly
different number.

"We found a boundary in neuronal coding," the paper remarks, "around number
4 that correlates with the behavioural transition from subitizing
[perceiving a number correctly at a glance] to estimation."

There was one more facet to this. The neurological mechanism at work here
isn't just that an individual neuron responds to individual numbers.
There's also evidence that the number neurons for 4 and below are actually
prevented from firing when offered the wrong number, thus improving their
accuracy. These neurons, Kutter and colleagues explained, "showed superior
tuning selectivity accompanied by suppression effects suggestive of
surround inhibition as a selectivity-increasing mechanism."

So yes, this is the possible explanation for Jevons's 150 year-old
findings. "Possible", because remember that this is based on work with
patients who already have electrodes in their brains. They are implanted in
the part of the brain that deals more with memory, not so much with
numbers. Still, there are these number neurons there too. That is why the
speculation that this small-number neuron mechanism may underlie the way we
estimate numbers.

All this makes me wonder if I can train myself to accurately guess larger
sets of objects. I'm aiming for seven.


-- 
My book with Joy Ma: "The Deoliwallahs"
Twitter: @DeathEndsFun
Death Ends Fun: http://dcubed.blogspot.com

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