 that data mirrors what's in the actual total, right? So now I've got the averages for each of those 75 and I can compare that to the actual middle number and you can see some of them are higher, some of them are lower. So that's kind of what we would expect. If I take an average of the averages, I come out to the 2210, which is pretty close to the 2189. And then of course we can build our histograms. So you can start to play with the pictorial representations. Now this one is actually a histogram of the sample of data. So in this case, we took a histogram of this sample 75, where we took 20 of them. So this is plotting out those 75 calorie counts in the buckets of from 538 to 1155 and so on and so forth. The middle point is here at 1773 to 2390. We had eight that landed in here. And again, we know the actual for the entire population is at the 2189. So that's for just one sample. And this one is just for one sample. This is 74. So in this case, we took sample 74. All of these numbers for 20 tests. And it looks like this. Obviously they're not gonna look exactly the same because these are two different sets of 20, which were randomly selected from the entire population. And then we took another one. This is number 73. So if we take each one of these samples that were randomly selected, we get different histograms. And if you take larger samples, then you'll get different shapes of the histogram rather than taking 20. You can take as many as you want. 105,000, you can test it out and excel and play with larger samples and see what the adjustments to these histograms would be. And then this one is taking a histogram of the averages. So in this case, remember that this column is the average of all of the 75 samples of 20 that we took. And so if we take a histogram of the averages, then here's what that looks like. So here's from 1879 to 2039. And again, we would expect the middle point here to be at the 2189 because that's the actual amount. And so you see it's starting to look something more like that. And then it's tapering off and we've got it kind of skewed to the right for this histogram. And we could see that when I take the average again of all the averages, that middle point is at 2210. So this is just another tool. We do this in Excel as well if you wanna see how to do this in Excel, but you can start to play with these data sets and use your pictorial representations and get to fairly large data sets so you can get an idea of what happens when you're using small numbers versus larger numbers and then you can actually build your histograms and charts and whatnot based on the results and you're gonna get a better intuitive fuel of what is actually happening. In future presentations, of course, we'll get into more specifics on how we can kind of describe some of these things mathematically, but it's quite useful to play with this stuff and Excel, which allows you to do fairly large amounts of data to see what the impact would be when you make stay histograms based on those results and also it's great practice for using Excel.