 We started it like down here. So we had the same distances and so we had a pattern and then we made a random number generation on each of these tables and shuffled up the pattern. So now I've got two different sets of numbers that were created with this kind of distancing pattern. I didn't randomly pick between one and 100, but rather we basically had units of five that we put in place and then made a random number generator to kind of shuffle them up. And the reason I wanna just take a look at that is because if we then take these two and plot them, the numbers that were generated from that process, then we get something that looks like this. So you can see this is something that is not exactly as random, right? This has some randomness to it, but this is more like something that might happen if you were trying to create randomness yourself. So if someone said put a random set of numbers, but you had to think of them in your mind, you would probably space out the numbers so they look something like this. So when you see something like this versus something like this, this one is often gonna be the more random set of numbers because you end up with this clumpiness. Like when we think of randomness, we don't think that these clumpy things are gonna happen, but with randomness, it does happen. When we try to do something randomly, we tend to eliminate the clumpiness that happens in randomness. So that's just something to keep it to point out here. Now note that when we generated this number, whether they be randomly generated or not, the two data sets that we created are still not correlated because we used the same process again, but we didn't tie them together. So even though it's still not exactly as random as the other set, we still have a very low correlation as indicated by the trend line. Okay, so now let's do our calculation to see that mathematically. So this is our z score of the one times the z score of the other divided by the n minus one. So if we did that calculation, we can take then the, this is our first data set with our random numbers we take the z, which is gonna be each number minus the mean. So in this case, it would be 19 minus the mean, which is gonna be, in this case, minus 48.82 divided by the standard d, 29.730. And that's gonna be our negative about one. And we do that all the way down for all the related items. And we do that for the y as well. So here's our random data set two. And then here's all of our z's for it. So that first data point in the data set, 27 minus the mean for the second one, which is 49.51 divided by the standard d, 26.56 gives us 84. So we're gonna go back on over, say, okay, that's gonna give us about 85. And then we multiply the z's together so that we get about one times 0.85, gives us about 85, of course. And then this one's gonna be the 1.54 times the 1.89, gives us, hold on a sec, k-posso the 1.54 times the 1.22 is gonna give us the 1.89 about rounding is involved here. So if we do that all the way down, then that will give us our sum. We just need to sum them up to give us the numerator. So we do that over here. We sum them up, we create a little table. The numerator is just gonna be the sum of this outer column. The denominator, I'm gonna make a subcategory for it by putting n minus one, or you can call it the denominator, colon, do the subcalculation internally, like you might say, like a tax return kind of format, tabbing to show it's a subcalculation, indenting, in other words, 215 is the number of data points. So if we counted all the rows, minus one, and that's gonna give us the denominator, which I'm putting to the outside double indenting, 214. I now have the numerator and the denominator of our formula, numerator, denominator, on the outer columns of our worksheet, and we can divide it out 25.43 divided by the 214, is gonna give us the 0.11884 about. So that's gonna be our correlations, fairly low correlation. And we can check that with our data analysis, which is under the data tab. You have to turn it on. So if you don't have it on, you can go to the options and turn it on, which we show in the worksheet how to do that. When we do this in Excel, you can then pick the, well, you could pick the correlation, this, I picked the correlation right there, and that would give you an input field, and then it would spit out, if you pick the correlation, this number we're looking at, this one here, which is comparing the one, the RAND1 and the RAND2 negative 0.11884, which is what we got up top there. So this one is not dynamic though, and remember that you don't get all the benefit of kind of looking at these Z scores as you're generating your worksheet. So it's a great tool to have, but it's somewhat limited to just kind of spit out the correlation at the same time. Now, if I did the same thing with the second data set that we made, the second data set, which was less kind of random, and I calculated the Z score in a similar fashion, and I calculated the Z score for both of them in a similar fashion, I multiplied the two Z scores together to get our column over here, and then we did the same correlation calculation, summing up the column to the right, the right most column, N minus one or denominator is gonna be the number of items, 215 rows. They're not all here by the way, well, they might be all here, but I don't think I copied all the rows in our example in one note, but they would be in Excel, minus one, 214, dividing out the numerator and denominator gives us a lower correlation, 0.00815, and this is the mean and the standard deviation to help us with the calculations of the Z scores, and then if I did that in Excel to double check it, Excel's given us a similar, well, the same, obviously result here. So the general idea then would be that when we're looking at the correlation, you can, we here shows randomly generated numbers to get an idea of the correlation. We can come up to kind of a hypothesis of how they would be connected, noting that although we looked at some of the stats that have kind of a lot in common, you know, the mean and the standard deviation are close and they're both have a tendency that's gonna go towards a uniform type of distribution, doesn't necessarily mean that there's gonna be, you know, the high correlation between them, and then we mapped out that difference between randomness here and randomness here, which is a good concept to understand, but whether or not your data set is more or less random doesn't necessarily mean that it's gonna have an impact on whether or not the two data sets are correlated together, moving together in some way, shape or form.