Missing value given the mean

- [Voiceover] Let's say you're in the babysitting business and you like to keep a log of whom you are babysitting. So in the last month you babysat six children and you wrote the ages of all six children in your log. But then when you go back to your log you notice that some blue ink spilled over one of the ages and you forgot how old that child is. And at first you're really worried, your whole system of keeping records seems to... you know, you've lost information. But then you remember that every time you wrote down a new age that month, you recalculated the mean. And so you have the mean here of being four, the mean age is four for the six children. So given that, given that you know the mean, and that you know five out of six of the ages, can you figure out what the sixth age is? And I encourage you to pause the video and try to figure it out on your own. So assuming you've had a shot at it. So let's just call this missing age, let's call that question mark. So let's just think about how do we calculate, how would we calculate a mean if we knew what question mark is? Well, we would take the total. We would take the total of ages, of ages, we would then divide that by the number of children. We then divide that by the number of ages that we had, and then that would be equal to, that would be equal to the mean. Or another way to think about it, if you multiply both sides times the number of ages, the number of ages on that side and the number of ages on that side, then this is gonna cancel with that, and we're gonna be left with the total The total is going to be equal to, is going to be equal to the mean times the number of ages. Mean times, and I'll just write times the number, times number of data points, or number of ages. So maybe we can use this information, 'cause we're just going to have this missing question mark here and we know the mean and we know the number of ages. So we just have to solve for the question mark. So let's do that. So let's go back to the beginning here, just so this makes sense with some numbers. The total of ages, that's going to be five plus two plus question mark, plus question mark, plus two, this two, plus two plus four plus eight. We're gonna divide by the number of ages. We're gonna divide it by the number of ages. Well we have six ages here. One, two, three, four, five, six. Six ages. And that's going to be equal to the mean. This is going to be equal to the mean. The mean here is four. So let's see, and this is just how you calculate the mean. So let's see if we can simplify this. So five plus two is seven. Let me do this, that's the wrong color. Five plus two, five plus two is seven. Two plus four is six plus eight is 14. 14. And then seven plus 14 is 21. So we're left with 21 plus question mark over six is equal to four. Now we can do what we did when we just wrote it all out. We can multiply both sides times the number of ages, the number of data points we have. So we can multiply both sides times six. We can multiply both sides, both sides times six. So six on that side, six on this side. Six in the numerator, six in the denominator, those cancel. So all we're left is, on the left-hand side we're left with 21 plus question mark. Alright. All of these other green numbers, those just simplified, five plus two plus two plus four plus eight is 21 and we still have the question mark. So we get 21 plus question mark, I want to do that green color, 21 plus this question mark, the thing that we're trying to solve for. The missing number is going to be equal to, is going to be equal to four times six. Well what's four times six? That's 24. And so what's the question mark? 21 plus what is equal to 24? And we could, of course, you might just say, well it's gonna be three. Or, if you want to, you could say well, question mark is going to be, question mark is going to be equal to, is going to be equal to 24 minus 21. Which is, of course, three. Which of course, so let me just write this down, so the question mark is equal to three. So the missing age, you were able to figure it out based on the information you had, because you had the mean, you were able to figure out that behind this blotch, that behind this blotch you had a three. It's exciting.