In order to standardize a set of numbers, I first take their average. I then calculate their variance. And you may remember that the formula for the variance of a set of numbers is 1 over n times the summation i equals 1 to n of each individual number, minus the mean of the set, for which we'll use the Greek letter mu. Squared. So, it's the average squared distance from the mean. And for this particular set, the variance would be 665.3. And then I'm going to calculate the standard deviation. And by the way, these are all population measures, population variance, population standard deviation. So this would be square root of 1 over n times summation the from i equals 1 to n x minus mu x squared. So it would be square root of 665.3, which would be equal to 25.79. Okay, so now I'm going to record these two values that I need the most, the mean. And the standard deviation, which we write with the Greek letter sigma. And now I can get rid of my calculations, and I'm going to go onto step number two. So, this was step one. Step two. Is to subtract. The mean from each individual value. And that is going to give me -34.67. Then I'm going to go on to step three. I am going to divide by the standard deviation of x. So now I'm going to take each of these values, and I'm going to divide by 25.79. So, step three, divide by standard deviation, and I'm going to get -1.34. I'm going to refer to this new set as my Z-scores for x sub i. And I'm going to write them like this. I'm going to designate them x sub Z sub i. So their mean would be the mean of x sub Z, and the mean is equal to 0. And the standard deviation, that's a Z, x sub Z is going to be equal to 1. So you can think about the formula as follows. You have each individual x sub i minus the mean of the set of x, divided by the standard deviation of x. And this will always produce a set that has mean 0 and standard deviation 1. What I've done is calculate the Z-scores for a second set of values, which we're calling the y values. And what you'll notice is that although the x values have a very large range, with a standard deviation of 25 and a mean of 44, and the y values have a much smaller range and a much smaller standard deviation. When they've been standardized, both the x and y axis values have the same mean and standard deviation. So this is what we mean by standardizing them. So, now we're going to look at some relationships of ordered pairs, and our question is this. If we look at certain standard relationships like covariance, correlation, the slope of the best fit line that plots y against x, the y intercept, will they be same for the original values and the standardized values, or will they be different? And if they're different, how will be they different and why? First, we'll look at the covariance of x and y. You take each individual x sub i and subtract the mean of all x's from it. Take each individual y sub i that goes together with it in an ordered pair, subtract the mean of all y's from it. Multiply them together. Then you do that n times. You add them all up, and you divide by n. And our covariance for our original set, using this formula, would be 67.89. And our covariance for our new set would be 0.9. Next, we're going to look at beta. This is the Greek letter beta. It stands for the slope of the best set line. So, if you picture our. Data with a line, delta y over delta x equals the slope of that line equals beta. And you can think about our best fit model as being, for each y sub i, we're estimating that y sub i equals beta times x sub i plus alpha. That is our formula that has the smallest sum of squared errors. So we want to know beta, the slope of this line. And for our original values, beta is equal to 0.1. While beta for our standardized values is equal to 0.9. Similarly, we know that for our standardized values, alpha must always be equal to 0. The reason why we know this is because the mean, the mean is always on our best fit line. The mean of x and the mean of y, ordered pair. And we know that for standardized values, this is 0,0. So that we know that the best fit line must go through the origin 0,0. But that, of course, is not true in general. And in fact, our value for alpha is -0.54, whereas here, our value for alpha is 0. And then finally, we're going to look at the correlation. Correlation is equal to the covariance of x and y divided by the standard deviation of x times the standard deviation of y. So our values for R are 0.9 and 0.9. So what conclusions can we draw from this? Well, the only thing that remains constant over our transformation to standardized values or Z-scores is the correlation. The correlation is unchanged. Okay? For every standardized set, alpha is equal to 0, so that is simple. And for every standardized set, both beta and the covariance are equal to the correlation R. So we've made things very simple, indeed.