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Weber's Law
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0:00
Well, I thought we'd kick off by doing a little experiment with you Brady. 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 I'm gonna need to take the camera. (Brady: You're taking over? Alright.) It's fine Brady, you're fine. You're fine. It's not the most flattering angle though.
0:31
Which one of those is heavier? Brady: That one. Hannah: Correct! Correct answer. Right, so that one is- - Can I look? - Yeah - a hundred and twenty grams. Okay, and then in your other hand you have a hundred grams? - 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.
1:03
Brady: Is what you're shooting vaguely usable? - I hope so.
1:09
Okay, and now, have a go now.
1:14
I feel like that's heavier. Hannah: Incorrect! So that is 200 grams and 220, and you can't tell the difference. And there is a reason for that. - Okay, alright. Okay, so what was going on there is something that Ernst Weber discovered in 1834 when he did 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
1:46
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, of the feeling
2:10
depends - it's a ratio really that you're 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't detect it when you're at 200 grams. You need a much bigger difference, that smallest, just noticeable difference before you can detect it. And the thing is, is that although in weights you're not often in that situation, where you're testing different ways, actually
2:36
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, you wouldn't mak- you wouldn't notice the difference.
2:58
And this explains why, as you get older, years seem as though they're, 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, in life, actually follows a logarithm.
3:30
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 you're feeling. So this could be light, could be sound, it 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 a very low weight, okay? And then you take a really tiny change in the weight,
3:54
so this 20 gram change, so this is gonna be i, this would be i plus Δi. 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.
4:14
Aww I've made my y-axis too small!
4:17
Even if you take the same change in weight,
4:21
so i plus Δi. Let's make this i1 and i0. That same change in stimulus,
4:28
is gonna feel like almost nothing in your response.
4:33
So essentially what this means is the way that we feel stuff, the way that we perceive stuff in life, doesn't follow a linear relationship, it actually follows a logarithmic relationship, which is what this curve here is. And I think that is quite interesting because when you come across logarithms first, in school, which I think they're, you know, when you're sort of sixteen or seventeen; they feel really counterintuitive. They feel like they don't really make much sense.
4:55
But actually, it's exactly how you perceive the world. The exact parameters on this curve change depending on what you're talking about. So they'll be one curve for light, there'll be another curve for sound - sort of 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. If there is an amount, a
5:23
change in stimulus that you can just about notice; and we know exactly because of this equation where that it is depending on, on what you're talking about. People who do marketing use this, completely to their advantage. So I swear that Cadbury's Dairy Milk, over time, has gradually got smaller and smaller and smaller. And, no doubt, it probably has. But, the people who make these decisions know this equation,
5:51
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 that's what they've done. (Brady: Don't they have to put the mass, the weight, on the packet?) - Yeah but no one checks those things! Or, so also, really expensive items they know that they can creep up the price by a much bigger sum
6:13
before you'll notice that the price has changed. Whereas things like, things that are really cheap, you know, buying sort of pints of milk or eggs or whatever. You've got to be a lot more careful, you can only eke it up 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. How they decide on sentences. And actually, this stuff becomes really important.
6:38
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, you know, three months already or if you've been in jail for thirty years already. Three months in jail cost the same amount of money to the taxpayer, you're still depriving someone of their freedom the same amount. But, the thing is, is that a six-month jail term
7:02
feels a lot longer than a three-month jail term. But a twenty year and three months jail term doesn't really feel like that much more than twenty year jail term. And so, as a result, there's this study which looks at the, the sentences that people give out; that 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
7:32
that's not one of these kind of preferred numbers. (Brady: So you get lots and lots of fine) (granular sentences down low, and then you'll just get nice big round numbers like 20 years and 30 years.) Because people think logarithmically.
7:46
Thank you for watching. Now, in the past you've probably heard me talk about Brilliant, the site full of puzzles and quizzes and lessons all about mathematics and science. The people who make it are guided by eight principles of learning, you can read all about them. Cultivating curiosity is one I relate to. For example, here's a problem from Brilliant: if the earth suddenly stopped spinning mayhem would ensue. I think that is perhaps a slight understatement,
8:14
but let's continue. What would happen to objects on the surface? Now quite aside from answering this question what I like is clicking here to discuss solutions. All sorts of people are explaining how they worked it out, and others are commenting on it. It's not just about the answer, I really like seeing how its explained. Its cultivating my curiosity. Now if you'd like to check out Brilliant go to brilliant.org/numberphile, so they know you came from here.
8:41
There's loads of free stuff on the site, but the URL below will give you 20% off a premium membership with access to even more stuff. Our thanks to Brilliant for supporting this episode, and we'll see you again soon.
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