 mean, we get to the negative 2.25. And then I take the five minus the 5.25. And I get to the point 25. I take the six minus the 5.25. And I get to the point 75. And I take the seven minus the 5.25. And I get to the 1.75. So that's what we have here. There's those three numbers. And then to get to the Z score, we take those numbers and divide them by the standard deviation. So all we're doing now is the next step. We would say, okay, what do we do again? We took the three minus the 5.25 divided by the standard D divided by 1.71. And then the next one would be five minus the 5.25 divided by the standard D 1.71 and so on and so forth. So we have those. Here's the second one approximately 1.15. And then we do the same with the Ys. Here's all the Ys minus the mean of the Y. So we'd say, okay, the Ys over here would be, for example, 105 minus the 209. Boom. And then we would take the next one, 185 minus the 209 and so on and so forth. So if I go over here, we're going to say there we have it negative 24 negative eight. And then we take each of those and divided by the standard D. So we're just would do then same thing. If I took this first one 105 minus the 209 divided by the standard D 99.92 about 1.04 on the negative. So there we have that. And then we just multiply the Zs together. So these two together. And that will give us then if I take this first one 1.32 times the 1.04, we get the 1.37 and so on. So if I sum up this last column, I get the numerator. So I can then use my little table here and sum that up. I'm going to put it in a table format. Here's the sum of this column. I can actually do it in the calculator. Why not? Because there's only four numbers. 1.34 plus 0.04 minus 0.04 plus 1.39 gives us about 2.77 rounding is involved. Then the denominator is just n minus one n is the number of items. There's rows one row two row three row four row and minus one is going to give us three. And then we have the numerator and denominator in the outer columns 2.77 divided by three is going to give us the 0.92. Notice again the format that I have here of this formula. Kind of useful to put it in a table when you're working like an Excel spreadsheet or something like that. It's useful to see it this way. You can build your worksheets this Wednesday. This is the numerator, which is this bit. And then the denominator I'm going to do a sub calculation. And I'm going to break that out as many sub calculations as I need and pull them into the inner column indicating it's a sub calculation with the colon with the indentation n minus one. The result then bouncing back out into the outer column which I can call n minus one or simply in this case the denominator and then I'm dividing out just the outer columns 2.77 divided by 3.92. Now I can see this in Excel and use Excel to do this with the analysis tool which isn't in Excel by it's in Excel but it's not turned on by default. You can find that in the options. We do that in the Excel problem. If you want to look at that in more detail but then in there I can do the correlation and just pick up this data set. You have to have the data set next to each other. So I just highlight that data set in Excel and Excel will then give me this prompt and I'll have to populate. Here's where the data set goes. I'd have to check off the range or that I had the labels involved if I clicked on the labels and then tell me where I want to put it if I was to put it in Excel and it'll give me something like this and I'm focused in on the X and the Y which are intersecting here. There's the 0.9 to 1.9 and so on that we got to here although we rounded it. So this isn't dynamic however so if I change the data set this isn't going to change with it so it's not a great tool for your worksheet if you're making a dynamic worksheet but it's a great tool to analyze the data upfront or to check your data kind of as we are doing here. You can also use the the same data analysis tool and look at this descriptive data and I just want to point that out even though it's not our main point of focus here to give you this kind of descriptive information for the X and the Y. This is our general kind of statistics info. You've got the mean you got the standard error, the median, the mode, the standard deviation, the sample variance, the minimum, the maximum, the sum, the count and so on. And this again is not dynamic. It doesn't change as your data changes so it's a good tool to use as a preliminary analysis. It might be the first thing you do before you build something out of your data set to get a feeling or an idea of what's going on with them and you can highlight multiple data sets and have it spit out or you can use it as a check figure for your data sets. So just a quick recap here. We're now looked at a perfect positive correlation, a perfect negative correlation. Now we're looking at more of a realistic example where it's not perfectly correlated, but you have a general trend. This one being one where in advance you would expect to see some kind of general trend. And by plotting out that trend, you can get more understanding about the data sets and possibly giving you predictive power into the future, such as how many hens would I would I need to buy by, you know, using the mathematical formula? Obviously that, you know, these hens, we're doing, we're doing great. And then these purchases of hens were kind of slacker hens and they weren't up to, you know, the production line that that we were expecting from them. But again, laying eggs is I'm not laying eggs is difficult, I would I would assume. So I'm not complaining. I'm not like, you know, it's tough work. But you would think like the other hens were doing, you know, did a little bit better some, you know, than these hens. But then this one is outside. But then you have the trend line, and the trend line can help you to predict to predict, of course. And and then, of course, we can see what the exact correlation is with our calculation here, mathematically, which will give you an understanding of how good that relationship is, how reliable you can kind of be on using that, you know, basically the trend line possibly to make predictions, you can do that with a formula calculation, which is useful sometimes, because as we'll see in future examples, breaking this information out like this, looking at the Z scores will often give you more information or could quite likely give you more information than simply using Excel to spit out the Z the correlation. So but either method would be good. And then, of course, graphing it when you graph it out, you give that pictorial representation. So we can look at the correlation conceptually, we can might have an idea about what the correlation might be. And then, of course, we can plot out it on a graph and see it pictorially and pick up the formula of the trend line, which could be useful. And then we can do a mathematical calculation of the correlation, in this case, having, of course, a positive correlation, but not perfectly positive.