Weber's law and thresholds

Voiceover: Imagine that we've got your arm here, and you got these nice big strong muscles, and you're holding in your hand a weight. Let's say that you're holding a two-pound weight. Here is a two-pound weight. Let's say that you are holding it in your hand and you want to lift it up. As you lift this two-pound weight, you notice that there is some resistance, and you definitely are able to notice that there is a weight in your hand. Let's say that we took this weight and you didn't know, let's say I grabbed the weight out of your hand and I replaced it with another weight that was 2.05 pounds. I've got this 2.05-pound weight, and it's the same exact shape and everything, and it's just .05 pounds heavier. Let's say that we replaced this guy with the 2.05-pound weight. Let's say that I asked you to lift this new 2.05-pound weight, and you might lift it, and you may notice that it's different, but most people in general would not notice any difference. They would just think it's the same exact two-pound weight. Basically, what I'm trying to say is that this .05-pound increase in the weight, for most people probably, would not be noticeable. Let's say that instead of giving you a 2.05-pound weight let's say that I gave you a weight that was 2.2 pounds, and let's say you close your eyes and I took this two-pound weight out of your hand, and I replaced it with this new 2.2-pound weight, and then I ask you to lift the new weight. Most people would probably notice this new increase, this new weight. Basically, what I'm trying to say is that a addition of .05 pounds probably wouldn't get noticed, whereas an addition of 0.2 pounds would get noticed. The threshold at which you're able to notice an increase or a change in weight or really any sensation so that threshold, where you go from not noticing a tiny little change to actually noticing a tiny little change is known as the just noticeable difference, noticeable difference. We can abbreviate this as JNT. In this case, the just noticeable difference, let's say for just sake of argument, is .2 pounds, .2 pounds. OK. Let's imagine that instead of starting with a two-pound weight, instead, we started with a five-pound weight, so five-pound weight. The five-pound weight is much heavier than the two-pound weight. In this case, if I replace the five-pound weight with a 5.2-pound weight, because it's a lot heavier and you're using a lot more muscle fibers in order to lift the five-pound weight, you may not notice the .2-pound increase. Whereas if I replaced the five-pound weight with a 5-1/2-pound weight, so 5.5-pound weight, and I asked you to lift it, you might notice the half a pound increase, but you might not notice the .2-pound increase. Basically, what's going on here is that since you're using more muscle fibers, you're using more sensory neurons, they're not as sensitive to small increases. They're not as sensitive to the 0.2-pound increase. You need a bigger just noticeable difference in order to actually be mentally aware of the change in weight. Basically, when you're holding five pounds, let's say for sake of argument, the just noticeable difference is a little bit higher than when you're holding two pounds. Now, these numbers we're just throwing around, and if you actually did this experimentally, the actual numbers might be different, but the concept generally would remain the same. In the five-pound weight category, the just noticeable difference is .5 pounds. From this, we can come up with an equation. Let's just define some variables. I, or the intensity of the stimulus, is equal to two pounds in this case and five pounds in this case. And delta I, so delta I, would be the just noticeable difference. It would be 5.5 pounds minus five pounds equals half a pound. For the five-pound example, I would be five, and delta I would be .5, and then in the two-pound example, I would be two, and delta I would be .2. Basically, there is actually a guy back in the day named Weber. Weber noticed in 1834 that the ratio of the increment threshold, the ratio of the increment threshold, which is this over here, to the background intensity, which is this over here, so this is the background intensity, is constant. If we were to take .2 divided by two, and if we were to take .5 divided by five, this ratio is actually equal, and it equals .1, and this ratio would be more or less fairly constant for a bunch of different weights. That's what Weber's law is. Weber, in 1834, realized that there is this relationship. We can write this as an equation. Delta I over I equals K. K is a constant for each individual person. There is this particular threshold, and the ratio, the background intensity to the incremental threshold is relatively constant, and that constant is this K value. This part of the equation over here is known as the Weber fraction. This works for sensory tactile stimuli like lifting a weight but it also works for auditory stimuli. Imagine you're in a quiet room. If you're in a quiet room with someone else, you can whisper. You can talk really, really softly and the person can hear you. But if you're at a rock concert, you have to be yelling at the top of your lungs in order for someone next to you to hear you. That's because the background intensity in a quiet room versus a concert is different, and so the delta I, which is whether you're just whispering or whether you're yelling, is different in accordance with the background intensity. That's what the Weber's law is basically saying. If we take this equation over here, let me just give myself a little bit of space. If we take this equation, Weber's law, and rearrange it, so we have delta I equals the background intensity times this constant, if we arrange it, we can see that this Weber's law predicts a linear relationship between the incremental threshold, which is this value over here, and the background intensity. In other words, as the background intensity gets bigger, the incremental threshold gets bigger. If we were to draw a little graph, if we were to draw a graph where the x-axis is the background intensity and the y-axis was the incremental, the difference threshold, what we would see would be this linear relationship. Using the concert example over here, a really, really big background intensity would result in a delta I, so this would be ... Let's say delta I in this case was how loud you were talking. The delta I would have to be a lot bigger than if you were in a quiet room, so quiet room. This law would generally hold true for almost any type of stimulus. It's a good rule of thumb. It's not exactly set in stone, but it is kind of how most of your different sensations operate, where if there is a bigger background intensity, you need a bigger difference threshold in order to actually perceive the sensation. In the real world, sometimes people add a different value. You got delta I over I equals K. This is the normal Weber's law. Some people will even add in another constant over here in order to take into account the baseline level of activity that needs to be surpassed in real-world situations. So this equation can be modified in order to more accurately represent what goes on in the real world.