Recall in our discussion of mean, median, mode, we used some of the statistics from the Chicago Cubs. We used game duration. Let's just switch a little bit. We're going to look at game time attendance, and so here's the data for the attendance of each of those games at Wrigley Field, games one through a 162. Now, if I go to the bottom of this and I look at the average, I'm going to type in equals average here. I highlight the entire population here. I'm going to get an average of 37,479 people went to each of these cubs games. That's my average. I'm going to insert a little note to myself that we know that we're looking at the average here, so we can recall this. That's my mean. Okay, and that's my population mean. Okay. So there's my population mean is 37,479. Now, if I type in stdev that gives me some different options here. stdev.p and stdev.s. The stdev.p gives me the population standard deviation. My stdev.s gives me my sample standard deviation. Because we're looking at all of the games for the Chicago Cubs, we have the entire population. This would be my stdev.p. This would be my population standard deviation. So I'm going to type that in here, and now what I'm going to do is I'm going to highlight all of the games, games one through 162. Here we go, and when I hit "Enter" here, I get this number, 654.3, and then some change. So what does this mean? This means that on average, there were 37,479 people who attended these games. But there wasn't always that number of people who attend to the games. Sometimes there were fewer people who attended the games. Other times, there were more people who attended the games. If we think about the number of people who attended the games as being a partially or normal distribution, then what happens is we've got a percentage of the people who attended the game on either side that is when we observe all the people in all of the attendancies of the games. On either side of this mean 37,479, one standard deviation away from this mean on either side a certain portion of the population will be within 6,543, either plus or minus of 37,479. So this is telling me the spread, right? This tells me how far away people are from that mean. A smaller standard deviation means that there's less variance, less change away from that mean. More of the observations occur within a smaller range of that mean. How much larger standard deviation means. A greater percentage of the population of what we're looking at falls a little bit further than that mean. So let me show you how to calculate the standard deviation. I'm going to show you how to calculate this by hand. Going to our Excel spreadsheet, notice that we have our population mean right here, 37,479. I'm going to copy this into this column that says mean and scroll all the way down. I just pasted the value here, and then I hit this scroll down all the way down. So you'll notice that every single one of these has 37,479. Now, remember the calculation for the population standard deviation is the difference between the observation and the actual mean, and so here's what we got. Let's take that difference. So I'm going to take this difference right here. Here's this minus my mean. Now, in this case, it gives me a negative number, and that's okay. I'm going to copy that little calculation all the way down. Now, what I'm going to do is I'm going to square this guy right here. So now what I've done is I've taken the square difference. So here's my mean. Here's my observation. My difference is the observation minus the mean. X minus the mu what we call. Now, I'm taking the square of that. So x minus mu squared. What I'm going to do is, I'm going to scroll that down for the whole group right here, and now what I'm going to do is, I'm going to take the sum of the squared differences right here. So right down here, I'm going to equals the sum of all of these guys. So this is the sum of the squared differences. It gives me a very large number. Now, if I have the number of observations, which is 162. So I'm going to put a little note here by the way. Sum of squared differences. That's what that value is right here. This whole column gives me the square differences in everyone individual. This right here is my differences in column G. In Column H is the sum of the square of the differences. Then down here, this gives me the sum of my squared differences. This is the number of games that I have, is 162. That's what n is. I'm going to insert a little shape to remind me what this thing is right here. This is my n. That's my n, and so when I take this as my sum of my squared differences divided by the number of my population, I get this number right here. Very large number, and if I then take the square root of that number right here, equals this guy to the power of 0.5, I get this. You will notice something, and I highlight this, so you can see it. This guy, the square root of the ratio of the sum of squared differences to the population. You'll notice look, this number right here is exactly equal to that number right there. This is the standard deviation as calculated by Excel. This is the standard deviation as calculated using a long handed version. So we get the individual observation, the attendance, we subtract it from the average, the mean, and we get our difference, and then we square that difference. Then we take the sum of that squared difference, and then we divide the sum of the squared differences by n, which is right here, and then we take the square root of that and we get the square of the ratio of the sum of squared differences to the number in the population, and that becomes our population standard deviation. So when we are looking at our standard deviation and we say what does this mean? Our x bar in this situation is 37,400 in change. Our standard deviation is this 6,500 in change, and so what this says is that 68.26 percent of the time that the Cubs are playing, you've got a stadium that's filled with 37,000 plus or minus 6,500 in attendance. That's how you use the standard deviation, and that's what it means.
