 The standard deviation is a central concept in statistics and in probability. So let's take a closer look at it. So we saw that we have many choices for the center. We can use the median, the mean, or any other measure of center. And we have many ways to measure the deviation. We can use the absolute deviation, the square deviation, or we can be inventive and create another function of the deviation. And once we have those deviations, we have many choices for the representative deviation. We can find the mean or any other measure of center. But people don't like having choices. It's why when you go into a fast food restaurant, you only have one item on the menu, and you only have one type of drink you can order. And in some places, lawmakers are trying to make it easier for people to choose who to vote for by not letting them vote. And where was I? Oh yeah, we were talking about choosing measures of deviation. While we do have a lot of choices, it's actually easier if we all agree on measuring deviation in the same way. And so it's useful to introduce the population standard deviation. Let D be a set of data values for a population. The population standard deviation is the square root of the mean of the squared variations from the mean. So for example, let's find the standard deviation of the set of data values, five, eight, ten, five, and six, and assume that these data values form the population. That's an important distinction that we'll talk about more later. So definitions are the whole of mathematics. All else is commentary, so let's pull in our definition for population standard deviation. So here we want to find the square root of the mean of the squared variations from the mean. So that suggests we need to start out by finding the mean. Next, we want to find the sum of the squared deviations from the mean. So those deviations will be the difference between each data value and the mean, 6.8, and we want to square those values. Then add them to get 18.8 the sum of the squared deviations. Now we'll find the mean of the squared deviations. That's the sum divided by the number one, two, three, four, five data values. And that's our mean squared deviation. And so the population standard deviation is going to be the square root of the mean of the squared variations. So we'll take our square root. And properly speaking, since our data values only have one significant figure, we should round this number to two. Now there's one important change we have to make. For some rather technical reasons, we need to make one modification when dealing with a sample instead of a population. Unfortunately, there's no good way of writing this without introducing some notation. So here it goes. If our set D is a sample drawn from a larger population, the sample standard deviation is still going to be the square root of the sum of the squared deviations, but we're not going to be taking the mean. We'll be dividing by n minus one. And the easiest way of remembering this rather daunting formula is just to remember that the sample standard deviation uses n minus one to compute the mean standard deviation instead of n. So let's say we have our same set, but this time let's assume these data values form a sample from a larger population. While we're still going to find the mean, we're still going to find the sum of the squared deviations from the mean, but this time instead of finding the true mean of these deviations, we'll compute what you can think of as a pseudo mean using five minus one for instead of five, and so that gives us 4.7. And then our sample standard deviation is still going to be the square root of this amount. And again, properly speaking, since our data values only give us one significant figure, we should round the sample standard deviation to one significant figure, so we'll round this off to two.