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Main content
Current time:0:00Total duration:6:26

Impact of transforming (scaling and shifting) random variables

AP.STATS:
VAR‑5 (EU)
,
VAR‑5.F (LO)
,
VAR‑5.F.1 (EK)

Video transcript

let's say that we have a random variable X maybe it represents the height of a randomly selected person walking out of the mall or something like that and right over here we have its probability distribution and I've drawn it as a bell curve as a normal distribution right over here but I could have many other distributions but for the visualization sake it's a normal one in this example and I've also drawn the mean of this distribution right over here and I've also drawn one standard deviation above the mean and one standard deviation below the mean what we're going to do in this video is think about how does this distribution and in particular how does the mean and the standard deviation get affected if we were to add to this random variable or if we were to scale this random variable so let's first think about what would happen if we have another random variable which is equal to let's call this random variable Y which is equal to whatever the random variable X is and we're going to add a constant so let's say we add so we're going to add some constant here I'll do the lowercase K this is not a random variable this is a constant it could be the number 10 so these are random Heights of people walking out of the mall well you're just going to add 10 inches to the height for some reason if you want to figure out what the distribution of people's Heights with with helmet on or plumed hats or whatever it might be how would that affect why would the mean of Y and the standard deviation of Y relate to X so we could visualize that so what the distribution of Y would look like so instead of this instead of the center of the distribution instead of the mean here being right at this point it's going to be shifted up by K in fact we can shift the entire distribution would be shifted to the right by K in this example and maybe K is quite large maybe it looks something like that this is my distribution for my random variable Y here and you can see that the distribution has just shifted to the right by K so we have moved to the right by K we would have moved to the left if K was negative or if we were subtracting K and so this clearly changes the mean the mean is going to now be K larger so we can write that down we can say that the mean of our random variable Y is equal to the mean of X the mean of X of our random variable X plus K plus K you see that right over here but has the standard deviation changed well remember standard deviation is a way of measuring typical spread from the mean and that won't change so and for a random variable X this is this length right over here is one standard deviation well that's also going to be the same as one standard deviation here this is one standard deviation here this is going to be the same as our standard deviation for our random variable Y and so we can say the standard deviation of Y of our random variable Y is equal to the standard deviation of our random variable X so if you if you just add to a random variable it would change the mean but not the standard deviation you see it visually here now what if you were to scale a random variable so what if I have another random variable I don't know let's call it Z and let's say Z is equal to some constant some constant times X so remember this isn't the K is not a random variable it's just going to be a number it could be say the number two well think about what would happen so let me redraw the distribution for our random variable X so let's see if K were two what would happen is is with this distribution would be scaled out it would be stretched out by two and since the area always has to be 1 it would actually be flattened down by a scale of two as well so it still has the same area so I can do with my little drawing tool here let me try to first I'm going to stretch it out by oops first actually I'll make it shorter by a factor of two but more importantly it is going to be stretched out by a factor of two so let me align the axes here so that we can appreciate this so it's going to look something like this it's going to look something like this when you scale the random variable this is what the distribution of our random variable V is going to look like I'll do the disease color so that it's clear as you can see two things one the mean for sure shifted the mean the mean here for sure got pushed out it definitely got scaled up but also we see that the standard deviations got scaled that the standard deviation right over here of Z that this is a this has been scaled it actually turns out that it's been scaled by a factor of K so this is going to be equal to K times the standard deviation of our random variable X and it turns out that our mean right over here so let me write that too that our mean of our random variable Z is going to be equal to that's also going to be scaled up x or it's going to be K times the mean of our random variable X so the big takeaway is here if you have one random variable that's constructed by adding a constant to another random variable it's going to shift the mean by that constant but it's not going to affect the standard deviation if you try to scale if you multiply one random variable to get another one by some constant then that's going to affect both the standard deviation it's going to scale that and it's going to affect the mean