Let's do a couple of examples looking at standard deviation. I showed you the calculation for standard deviation for the attendance for the Cubs baseball games, and used the Excel Standard deviation for population. Then I showed you how to do it using a long hand, long version here. On this example, I've got the runs that the Cubs scored and the runs that were scored against the Cubs and the total score of the baseball game for each 162 baseball games from last year. We're going to focus on those Cubs for awhile. But anyway, let's do a standard deviation of the population and show you a standard deviation for the sample and see how they differ. So now, the population standard deviation, I can calculate using this little standard deviation formula STDEV.P and I'm going to scroll over the entire population of [inaudible] and I get my standard deviation, 3.05. I'm going to look at what the mean of these scores are. Average of this population. Here it is and what I've got here is I've got my mean and my standard deviation. So, the Cubs on average scored a little less than five runs, 4.98 runs and the standard deviation was three runs. I can tell already this data is a little bit skewed, and so we think about 68% of the observations are going to be on either side, three standard deviations or this one standard deviation away from my mean. So, if I'm scoring about five runs here, then about 68% of my sample is going to be between two runs and between eight runs. I can already tell it's going to be a little bit skewed because my standard deviation is pretty big. So, my guess is that the Cubs had some games where they scored say 11,12 and 14 runs or something like this, and then there's going to be games where they didn't score anything, they had zero runs. So, since the population is skewed on one end, the standard deviation isn't really describing a normal distribution. But I want to show you the difference between let's say, population standard deviation and a standard deviation for a sample. Here's my population standard deviation here, but my sample standard deviation uses, equals STDEV.S. Then I can take a standard deviation for a subset let's say, games 100 through 162, here we go, right there. Now I've got a sample standard deviation right here. So, you notice that my sample standard deviation is a little bit different than my population standard deviation. We saw the same thing when looking at the means. Let's take the sample standard deviation equals STDEV.S for this last set of games here. I've got a sample standard deviation of 2.94, it's almost three, so it's very close. I want to show you how I calculate my standard deviation doing a long version, if you will. To calculate the sample standard deviation, what I'm going to do, is I'm going to grab a subset of the whole population and I'm going to do it longhand, if you will. So what I've done is I've collected here the last 14 games or so of the Cubs and the number of runs that they scored. Now, just to double check and make sure that we're on the right page, I'm going to calculate the average for this group and here's the average right here, 5.285. Now, I've already placed this average into this column here. So, this is the number of runs the Cubs scored. This is the average score of the Cubs during this period of game. You'll notice that this game here with this 12 runs is fairly high. Now, I've got a difference and the difference is the number of runs minus my average, gives me a difference. Then in this column, what I've done is I've taken the square of that difference. Now, down here what I'm going to do is, I'm going to sum all of these squared differences, there they are. Now, I'm going to double check to make sure that I actually have 14 games. So, let's do the count, I'll do the count of these right here. So yeah, I have 14 games. Now to calculate my sample standard deviation, I need to take the sum of the squared differences and divide it by my sample minus one. So, I'm going to do that right here. I'm going to take the sum of my squared differences divided by here's my sample minus one, gives me this number right here. Then what I'm going to do, is I'm going to take the square root of this. So, I'm going to insert here a little note for myself of what I've done right here. This number right here is the sum of squared difference divided by my population, my sample size, minus one. So, here it is. This is my sum of my squared differences. Here's my sample size. So, right here is the sum of squared differences divided by my sample size, minus one. Now right here what I'm going to do, is I'm going to take this amount and take the square root of that to the power of 1.5 and I get 2.94648. So, this right here and I'll just insert a little note for myself, ''This right here is my sample standard deviation.'' You'll notice that this sample standard deviation is exactly as this one right here. I used STDEV.S for these same observations. I'm going to do it again right here, equals STDEV.S for these observations right here and paw. You'll notice that I'm getting the same number. Of course here I'm rounding up by a little bit, but it's the same number. So, we can do it using a couple of keystrokes in Excel. We can also do it doing long hand. Then what we say is, this is the spread around the mean for this data, which is for this example the number of runs that the Cubs had, and this is the sample of games, 14 games. This is the average number of runs they had in that game, and this is the standard deviation or the spread of those runs.
