Why do variances add

Video transcript - [Narrator] So in previous videos we talked about the claim that if I have two random variables, x and y, that are independent, then the variance of the sum of those two random variables or the difference of those two random variables is going to be equal to the sum of the variances. So that if you have independent random variables, your variation is going to increase when you take a sum or a difference.

And we've built a little bit of intuition there. What I wanna talk about in this video, it's really about building even more intuition, is get a gut feeling for why this independence is important for making this claim.

And to get that intuition, let's look at two random variables that are definitely random variables but that are definitely not independent. So let's let x is equal to the number of hours that the next person you meet, so I'll say random person, random person slept yesterday. And let's say that y is equal to the number of hours that same person was awake yesterday.

And appreciate why these are not independent random variables. One of them is gonna completely determine the other. If I slept eight hours yesterday then I would have been awake for 16 hours. Or if I slept for 16 hours then I would have been awake for eight hours. We know that x plus y, even though they're random variables, and there could be variation in x and there could be variation in y.

But for any given person, remember, these are still based on that same person. X plus y is always going to be equal to 24 hours. So these are not independent, not independent. If you're given one of the variables it would completely determine what the other variable is. The probability of getting a certain value for one variable is going to be very different, given what value you got for the other variable.

So they're not independent at all. How can I calculate the variance? What Excel formula do I use to calculate the variance between two cells? How do you find the sample variance in 89, 57, , 73, 26, , 81? How do you find the mean of the random variable x?

He following table shows the probability distribution for a discrete random variable. What is the variance of X? How do I find the variance in 40; 56; 59; 60; 60; 62; 65; 69; 75; 84 statistics? What is the difference between variance and variability? How do you find the variance of the data 2, 4, 6, 8, 10?

Random Variables View all chapters. Mean and Standard Deviation of a Probability Distribution. Addition Rules for Variances. In R , a fancy piped way to compute the variance is this:. Then subtract the mean of this column. Then take it to the 2nd power square it, that is. Then compute the mean of the numbers.

Which is exactly the same as the R code above; slightly more complicated, I feel, because the steps are nested because of the brackets. Note that the R function var does not divide by n but by n However, for larger samples, the error is negligible. Thus, depending on whether you are interested in a guess of the population variance or just the variance of your data at hand, the one or the other is slightly more appopriate.

Here, we just divide by n and by happy. In general, my opinion is not to worry too much about tiny details for the purpose given , but rather to try to grasp the big pictures. Now, we are concerned with the additivity of the variance.

So we are supposed to some stuff up! Suppose we repeat our dice throwing experiment, but now with 2 dices instead of 1. After each throw, we sum up the score. After some hard thinking we feel reassured that this should yield a number between 2 and It appears that the means are adding up, which makes sense, if you think about it.