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韦伯定律及其应用

Weber's Law - Numberphile

Well, I thought we’d kick off
by doing a little experiment with you, Brady.

Er… I’ve got some weights here.

And I want to see how sensitive your hands are.

Do I mean sensitive hands?

I want you to lift some weights.

This is a workout session.

And I want to see if you can tell the difference

between how heavy different things are.

So I think, can I take the camera?
–你可以把这个当水平线–嗯 好的
– You’ll take it as horizon.– Yeah, ok.
It’s fine, Brady. You’re fine, you’re fine.

It’s not the most flattering angle though.

Which one of those is heavier?

That one.

All right, so that one is…
–我可以看一下么–当然
– Can I look?– Yeah.
120克
120 grams.
–好–你另一只手里 是100克
– Okay. – And in your other hand, you have 100 grams .
–好–好的 现在 我要加大难度了
– Okay.– Okay, now, I’m gonna make it slightly harder.

You could tell the difference between 120 and 100.

So you can sense 20-gram difference then.
(汉娜去拍婚礼 处理公司的事儿了)
Hannah is available to film weddings and corporate events.

Is what you’re shooting vaguely usable.

I’m sorry.

Okay.

Okay, and now, okay now.

I feel like that’s heavier.

Incorrect.

So that is 200 grams and 220.

And you can’t tell the difference.
–这是情有可原的–好吧
– And there is a reason for that.– Okay.

Okay, so what was going on there

is something that Ernst Weber discovered in 1834.

We want to do exactly this experiment.

He noticed that even though you can sense

a small change in between two weights,

when the initial weight is very small.

As that initial weight gets bigger,

you can no longer detect that change.

The just noticeable difference that you need changes,

depending on the size of the weights.

Now, this led him to come up with an equation

which is known as Weber’s law.

That small difference, that tiny change in the intensity,

if you like the feeling, depends…

It’s a ratio really that you’re sensing,

and that ratio is constant.

So although you can detect a 20-gram difference,

when you’ve got 100 grams in your hand,

you can detect it;

when you’re at 200 grams,

you need a much bigger difference

that’s more than its just noticeable difference

before you can detect it.

And the thing is… is that although in weights,

you know often in that situation,

where you’re testing different weights,

actually, I think that this Weber’s law

applies to a lot of different situations.

So, you know, if you’re in a really dark room

and someone maybe turns on their iPhone torch.

And you can see it kind of like lights up the entire room.

But when you’re in a really bright room,

if someone flicked on their torch wouldn’t make…

you wouldn’t notice the difference.

And this explains why, as you get older,

years seem as though they’re going faster.

Time kind of speeds up, as you get older, right?

Even though a year is the same length always,

the ratio of how long that year is to how long your life has been

ends up getting smaller and smaller and smaller.

So, how it feels changes over time.

So this here is Weber’s law,

but what it means is that the way

that we experience things in life actually follows a logarithm.

So if you think about how something feels, your response to something,

and then you compare that to the sort of intensity

of whatever it is your feelings.

So this could be light, could be sound, could be weights.

The way that this changes, it takes sort of this shape here.

It’s kind of a logarithmic shape.

So what that means is if you start off down here,

and you’re at a very low weight, okay.

And then you take a really tiny change in the weight,

so this 20 gram change.

So this is gonna be I, this would be I plus delta I.

The difference how that feels is gonna be quite big.

But then if you’re over here,

so now this one will be sort of 100 grams down here.

And let’s say when you’re over here,

you’re up at 200 grams.
–哦 可能是y轴画的太短了–是的
–Ah, maybe the y-axis is too small.–That’s right.

Even if you take the same change in weight,

so, I plus delta I.

Let’s make this I1, and I naught.

That same change in stimulus

is gonna feel like almost nothing in your response.

So essentially what this means is

the way that we feel stuff, the way that we perceive stuff in life

It actually follows a logarithmic relationship

which is what this curve here is.

I think that is quite interesting,

because when you come across logarithms first in school

which I think that, you know, when you’re sort of 16 or 17,

they feel really counterintuitive,

they feel like they don’t really make much sense.

But actually this is exactly how you perceive the world.

The exact parameters on this curve change

depending on what you’re talking about.

So there’ll be one curve of light,

there’ll be another curve of sound.

How quickly it goes up and how quickly it bends over.

But the basic mathematical structure of this,

which has been shown time and time again,

in all sorts of different experiments,

is that we perceive things logarithmically.

So there is an amount, a change in stimulus

that you can just about notice.

And we know exactly, because of this equation, where that is,

depending on what you’re talking about.

People who do marketing use this completely to their advantage.

So I swear the Cadbury Dairy Milk over time

has gradually got smaller and smaller and smaller,

and no doubt, it probably has.

But the people who does make these decisions know this equation,

know the way that we perceive things is logarithmic,

and know the exact amount

that they can shrink their chocolate bar by before you notice.

That’s what they’ve done.

Couldn’t I put the mass, the weight on the…

You have no one check those things.

Or they also… a really expensive items,

they know that they can creep up the price by a much bigger sum

before you notice that the price has changed.

Whereas things like… things that are really cheap, you know,

buying sort of pints of milk, or eggs, whatever,

you’ve got to be a lot more careful.

You can only eke out by a couple of pennies

before people start to notice.

Well, so the reason why I came across this was

because I was doing some research into how judges make sentences

and how they decide on sentences.

And actually this stuff becomes really important.

It’s not just sort of people trying to make a little bit more money

or noticing different size weights.

Three months in jail is three months in jail.

It doesn’t matter whether you have been in jail for,

or if you’ve been in jail for thirty years already.

Three months of jail cost the same amount of money through taxpayer,

still depriving someone of their freedom the same amount.

But the thing is that a six-month jail term

feels a lot longer than a three-month jail term.

But a twenty-year and three-month jail term

doesn’t really feel like that much more than twenty-year jail term.

So as a result, there’s this study which looks at

the sentences that people give out,

the judges around the world give out.

And there are these huge gaps in the timelines

that are available to them.

And it’s because of this, it’s because

it doesn’t… it just doesn’t feel like enough of a difference

to give someone a sentence

that’s not one of these kind of preferred numbers.

So you get lots and lots of fine granular sentences down low

and then you just get nice big round numbers like 20 years and 30 years.

Because people think logarithmically.

Thank you for watching now.

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jm