Another measure of my data, another descriptive statistic for my data is something called the standard deviation. Now, we're going to talk about standard deviation later on in this program when it comes to regressions and standard errors. But the standard deviation is something that a lot of people are aware of and you might not know how to calculate it. Likely, you've seen something like this. Here is a very classic bell curve and this is what we also call a normal distribution. Now, what I've shown here is I've shown a normal distribution and the percentage of the population that falls within a particular range from the mean based on what we call the standard deviation. So, you've probably seen this bell curve or this normal distribution. What does it mean? Why do we care about it? How do we calculate the standard deviation? That's what we're going to do next. The standard deviation is a measure of the difference away from the mean that certain proportions of your data fall. On this screen, I have the formula for the standard deviation. Now, I've described the standard deviation with this small letter s. Sometimes it is referred to as, using the Greek letter s, that's the small Sigma right here. So, people will sometimes use this Greek little Sigma or they might use the s to describe your standard deviation, and either is okay. What we have here is, the numerator here, we've got the square difference of each observation from the mean. Recall that the mean, right this x bar here, is the summation of all the observations divided by the total number of observations. That's my x bar, x here is the actual observation that we're looking at. So, you say, I'm looking at observation one, I'm looking at observation two, I'm looking at observation three, and what have you. That's x. When we take the x minus x bar, we get a difference, an error, some amount that were off from what the average is. So, if we square that difference and then sum up the entire differences, we will get a sum of the squared differences. Now in this case, this is what we call a sample standard deviation here. I'm going to divide by the number of data points n and subtract one, and then I take the square root to get the standard deviation. If I do not take the square root of this, I get something called the variance and I usually will describe the variance using this symbol with a square, right there. So, the variance is the sum of the square differences divided by n minus one, and that gives me my sample variance. If I then take the square root of the sum of the differences, the sum of the squared differences divided by n minus one, then I get the standard deviation. So, now you'll notice that my standard deviation gives me this guy, and my variance is this guy with a square, and so my standard deviation is in fact the square root of the variance. People might be talking about these things in your MBA programs and so you might want to make sure that you're comfortable and familiar with Sigma, Sigma squared, the variance, the standard deviation. Let's calculate some of these things. Here we have the formula for what we call the population standard deviation. Now, the symbol over here is the Greek letter Sigma and it's the small Sigma and what we have here in the numerator is some of the things that we've already discussed. Now mu, that's the population mean, x is the observation of any one individual observation in a population. N is the number of observations that we have. So, in this case, what we have is, we've got x, which is the individual observation minus the population mean mu and we square that difference, and then we sum up that difference for the entire population and divided by the number of observations that we have in that population. If we take the square root of that calculation, we get what's called the population standard deviation. Here, I've got the formula for the sample standard deviation. In this case, the only thing that is different is some of the symbols and what we're describing and the fact that we hadn't have to subtract one from the number of observations. So, here it's x bar is the mean of the sample, x is the individual observation within that sample and so we take the observation minus the sample mean, square that and we sum it for each observation in our sample, divided by the number of data points in our sample minus one and take the square root of that and we call that our population standard deviation. Let's do some examples and I'll show you how to calculate the population and a sample standard deviation and we'll do it using some shorthand in Excel and then we'll also show you how to do it longhand that is. Here's how you do each one of those steps in the calculation.
