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so imagine that we've got your arm here and got these nice big strong muscles and you're holding in your hand oh wait so let's say that you're holding a two pound weight okay so here's a two pound weight and let's say that you're holding it in your hand and you want to lift it up so as you lift this two-pound weight you kind of notice that there's some resistance and you you definitely are able to notice that there's a weight in your hand so 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 two point zero five pounds so I've got this two point zero five pound weight and it's the same exact shape and everything and is just 0.005 pounds heavier and let's say that we replaced this guy with with the two point zero five pound weight and let's say that I asked you to lift this new two point zero five 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 so basically what I'm trying to say is that this point zero five pound increase in the weight for most people probably would not be noticeable so let's say that instead of giving you a two point zero five 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 so and then I asked you to lift the new weight you most people would probably notice this new increase this new weight and so basically what I'm trying to say is that a addition of point zero five pounds probably wouldn't get noticed whereas in addition of 0.2 pounds would get noticed so the threshold at which you're able to notice an increase or a change and 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 so we can abbreviate this as J and D so in this case the just noticeable difference let's say for just sake of argument is 0.2 pounds 0.2 pounds okay so let's imagine that instead of starting with a 2 pound weight instead we started with a 5 pound weight so 5 pound weight so the 5 pound weight is much heavier than the 2 pound weight and in this case if I gave if I replace the 5 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 5 pound weight you may not notice the point 2 pound increase so whereas if I replace the 5 pound weight with a 5 and a half 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 point 2 pound increase so 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 point 2 pound increase they need a bigger just noticeable difference in order to actually be mentally aware of the change in weight so basically when you're holding 5 pounds let's say for sake argument the just noticeable difference is a little bit higher than when you're holding 2 pounds now these numbers were just kind of thrown around and if you actually did this experimentally the actual numbers might be different but the concept generally would remain the same so in the 5 pound weight category that just noticeable difference is 0.5 pounds so from this we can kind of come up with in equations so let's just define some variables so I or the intensity of the stimulus is equal to 2 pounds in this case and 5 pounds in this case and Delta I so - I would be the just total difference so it would be five pounds five oh sorry five point five pounds - five pounds equals half a pound so for the five pound example I would be five and Delta I would be 0.5 and then in the two pound example I would be two and Delta I would be point two so basically there is actually a guy back in the day named Webber so Webber noticed in 1834 that the ratio of the increment threshold so 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 so if we were to take 0.2 divided by two and if we were to take 0.5 divided by five this ratio is actually equal na equals 0.1 and this ratio would be more or less fairly constant for a bunch of different weights and so that's what Weber's law is so Weber in 1834 it realized that there was this relationship so we can write this as an equation so Delta I over I equals K and so K is a constant for each individual person there's this particular threshold and the ratio the background intensity to the incremental threshold is relatively constant and that constant is this K value so this part of the equation over here is known as the Weber fraction so this works for sensory tactile stimuli like lifting a weight but you can it also works for auditory stimuli so 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 cure 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 and 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 and so that's what the Weber Weber's law is basically saying so if we take this equation over here let me just give myself a little bit of space so if we take this equation Weber's Weber's law and rearrange it so we have Delta I equals the background intensity times this constant rearranging 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 so if we were to draw a little graph for the draw graph work 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 so 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 are in a quiet room so quiet quiet room and so this law would generally hold true for almost any type of stimulus and it's a good rule of thumb it's not exactly you know set in stone but it is kind of how most of your different sensations operate where if there's a bigger background intensity you need a bigger difference threshold in order to actually perceive the sensation and in the real world sometimes people add a different value so they you've 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 the real-world situations so this equation can be modified in order to more accurately represent what goes on in the world world