 which we're not as able to do in this data set. The nice thing on this side of being able to take each calorie count in Excel, even though it's quite a long set of data, is that when you add this up, it adds up to basically pretty close to 100%, which is kind of a nice double check. So what we would like to do, though, is we'd like to say, okay, well, I'd like to kind of group this stuff together so that these p of x's would be for the range of like anything less than zero and then everything from zero to 400, for example, I would like to kind of sum that up into a group. And then when I do my frequency count for my actual counts, we can put them into bins or buckets. There's a couple of different ways we can do that when we're looking at our actual data that we could just use this to kind of sum up into our buckets, sum these up. And remember that normally we don't like to do that here because we're talking about the area under the curve, but because we're using such fine data, we get to a pretty close number or we can use a formula, which will be the between formula so that we can use the norm.dist in the upper range minus the lower range. So let me show you what I mean. We'll do a couple of these over here. We're going to say that we have the x's and the x's are going to go up by 400. So these are going to be basically our buckets. So we got it going from zero up to 400 and then 400 to 800 for and then 800 to 1200 so that we have our buckets instead of just one calorie at a time. So then the actual frequency, if I was to do my frequency calculation, then now I can do my frequency and have buckets that are much larger. So for example, this one would be saying everything that's greater than zero up to and including 400 in our actual count over here of our actual data. And we had five of those. This one would be saying everything over 400 up to and including 800 of the actual count, we got 14 of those. We're doing this with the frequency calculation, the data array being our table on the left and then the bends being these bends. And it gives us our buckets, which is nice. So now we have actual numbers in here as opposed to if we did it one calorie at a time, when we did our frequency, we had almost zero numbers with one, like a one showing up every once in a while, although I don't have a lot of the data in here because I didn't want to copy it all over. But we do this in Excel if you want to check it out. The sum adds up to 157, which is a double check that these bends are adding up because that's what our actual count was of our actual number of data. Now we can take the percentage of total and it gives us something that may actually be relevant. Meaning I could say, okay, this is going to be five divided by the total divided by the 457 and that gives us 1.09. And this is the same thing, 14 divided by the total, 14 divided by 457 is 3.06 if I move the decimal two places over. So now the question is, well, if I can do the same thing for my P of X information, then I have something that's kind of comparable. And we're going to do that with a formula that looks something like this and then we'll do it another way as well. This is a sum if formula and it's saying sum if and we're picking up the sum range, which in this case we're looking at the P of X is over here. So we want the P of X is comma the criteria. So we want, we're going to pick up the criteria range, which is going to be this information. And then we're picking up the criteria. It's got to be less than or equal to this number over here. It's got to be less than or equal to, in this case, the zero. So that's so I'm summing that up. So it's basically saying, pick this up. If the X is less than zero, sum this up. So it's summing everything up down to when the X is zero. And then this one is summing everything up. If I did a similar formula, it would look a little bit different. But this one would be summing everything up. If X is greater than zero up to and including 400, and we would be picking up this column if this area, the X, was between zero and 400. Now, remember, usually that's just an approximation. But because we have such fine units, we come up with a pretty close approximation. So that's one method that we can use. And now we have comparable numbers. I'm like, OK, there's 800. Actual data came out to 14.06 versus the 3% that if we use this technique of summing it up. And I can see if this adds up to 100. It adds up pretty close to 100% on this data, which might give another indication that that's given me a pretty accurate calculation. And then I can also do it this way. I can take my P of X count times the count, meaning now that I have these numbers, I can multiply them times the number of counts that we had. So I can take this number of 0.036, 0.0036, 0.0036 times the count of 457, and then we get around 2. And now I can compare this 2 to this 5. So these are the two ways that we can kind of compare our data sets. I can convert the norm.dist to a count by multiplying it times the count and comparing it to our actual frequency count. Or I can convert our actual frequency count to a percent to compare that to the percent. The only difference here now being that we put these items into buckets so that we're not talking about one calorie at a time. The other way you can do this buckets thing, which is more accurate, would be to say, I'm going to have a lower buck, a lower bin, and an upper bin from 0 up by 400, 400 to 800, 800 to 1,200, and so on and so forth. And then use our norm.dist formula to calculate it. This one's calculating the first one, which is just norm.dist for the 0. But for the second one, you'd have to have norm.dist of the upper part minus the norm.dist for the lower part. So it would look like this. You'd have norm.dist of the higher x, and then the mean, the standard deviation, it needs to be cumulative. That's the 1 minus the norm.dist for the lower part, the 0, if you're talking about the second one, and that'll give you that in between. This should be more accurate than what we did up top summing. But because we have such a fine unit of measure, they're pretty close either way that you do it. We get the 0.36 to 1.04, the 3, the 6.82, the 0.36 to 1.04, the 3, the 6.82. So they're pretty close either way. This one comes out. You could see if I add them up to exactly 100%. This one came up to 99.99. OK, let's go on to our graphs then. So once we have this, then we can kind of grab them together. We can put both of these on the same graph. This is graphing, I believe, the actual percent column and the p of x column. So we're looking at our actual data and the function data on one graph. And you could see they line up fairly close, giving us further indication that a bell curve might be an appropriate tool to use to approximate this data. This one is a, I believe this is a bar graph of this data over here. Probably should have labeled it better. This data over here, where we picked up all of this. Now this was a long column because remember, if we did it one calorie at a time, that we had a whole lot of, we had 106,517 calories in total. So if I graph that even with a line graph, then you get something that looks almost like a smooth curve. It doesn't look like this jagged thing over here or it doesn't look like this jagged thing here because we have such fine data that we're using. This isn't the actual data. This is the curve that we're plotting. But we're plotting it on a calorie by calorie basis. And that's one of the reasons, again, that when you think about the area under the curve and you think about the integral calculus of it, because you have such the fine, so fine of the lines, could be the reason why that if we sum up, we sum everything up this way, that we come up to something very close to if we did it with a calculation of the norm dot dist. So there is that one.