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We've spent a lot of time visualizing different types of data where there was one variable across a bunch of different categories. It could be like a histogram or a frequency distribution, a pie chart, etcetera. But now we're going to start taking a look at visualizing data sets in which there are multiple variables like two or more. And one of the most common situations of that is called a scatter plot. Alright?

Now that's something that you may have done in a previous math course, like an algebra course, so this may be a little bit of a refresher, but there's a couple of important conceptual things that we'll address and some terms that we'll need to know. So let's go ahead and get started here. All right. So basically, a scatterplot is just a graph on an x y coordinate system, and it's just a graph of paired numerical data. So this is basically just a bunch of pairs of numbers where there's one independent variable, which we call the x, and one dependent variable, which we call the y.

So when I whenever I think of a scatterplot, I basically just think of a bunch of x and y points on a grid. It looks exactly something like this. Alright. So we're going to talk about this second point in just a second here, but let's just jump right into our example. So a teacher is taking a survey of students to determine different factors that might affect test scores.

We've got these different tables that show these different sort of sets of data, and we're going to go ahead and plot the data if necessary, and then we'll go from there. Alright. So let's take a look at this first graph over here, this first table, which just shows test scores versus time spent studying in minutes for an exam. So we've got time on the x axis, and then we've got the scores on the y axis. So, basically, what happens here is you just pair up each one of these numbers, and they just form a bunch of x y points that you would just draw on a grid.

Alright? Now all of these points except for the first two are already drawn for us, so I'm just going to go ahead and just draw the first two just to just in case it's been some time since you’ve done this. All you do is you just grab two of these numbers over here in the same sort of column, and that's going to be your x y pair. So you're just going to go 50 on the x and then 86 on the y, and you're just going to drop a point right there. So this coordinate pair would be right over here, and this would be 50, 86.

Alright. So I'm going to do this one more time over here because all the rest of these are already drawn. Sixty ninety two would look something like this. You'd go to sixty and then all the way up to ninety two and drop a dot right there, and that would be back coordinates. All right?

So that's just how you sort of plot numbers on a scatterplot, but that's really all there is to it. All right? So now let's take a look at the second part of this question, which is and determine what type of correlation each graph has. One of the things you might notice about this graph over here is that generally, as the data points go to the right, the y values also increase. As students spend more time studying for that exam, they get higher test scores, which hopefully should be expected.

Right? So we actually say that these things are correlated, these x and y values. A correlation happens whenever the data points basically form some kind of a trend or some kind of a pattern. Now, there are different types of correlations that you might see, but the most common one that we work with in this course is called a linear correlation. This is basically where the general trend of the data is pretty much like a straight line, where you can draw a straight line that kind of cuts through most of those data points.

We say that those things are linearly correlated. Alright? So what we can see here is that as the x values increase to the right, the y values also increase. In other words, if you were to take the slope of this line here that you just drew, think back to algebra rise over run, we would say that that slope is positive. And because it's positive, we say that this correlation here is a positive correlation.

Right? That's all there is to it. Positive correlation just looks like this. Now let's take a look on the right over here because we have slightly different data. We have test scores versus the number of pins on a student's backpack.

And clearly, we can see a slightly different relationship that's going on here. So as we can see, the more pins you have on the backpack or more students, pins students have on their backpack, the worse they actually did on their test scores for some reason. In this case, what we actually see here is kind of the opposite. As the x values increase, the y values are actually decreasing. If you were to draw a line that kind of cuts through most of these things, the slope of that line would actually be negative, and therefore, we just say that this is negative correlation.

Alright? So positive correlation looks like this, and negative correlation kind of looks like that. That's really all there is to it. Alright? Now I want to point out one important conceptual thing between these two datasets.

Right? Here we saw that there was a relationship between test scores and time spent studying on the exam. And as you spend more time studying, this basically sort of gave you a higher score on your test. So this kind of makes sense here that these two things would be correlated because it could be something like a cause and effect relationship. If you spend more time studying, you're going to get a higher test score.

Over here, we saw that there is a relationship between test scores and number of pins in the backpack, even though that makes absolutely no sense. So I just want to point out here that two variables can look like they're related to each other without it actually being cause and effect. It's not like you're going to go to your next class and have more pins on your backpack, and you're going to do worse on that exam. Alright? It just looks like that way just sort of because of coincidence.

So one really important thing here that you may see in your course is that correlation does not imply causation. Just because two things show a sort of relationship like this doesn't mean that one causes another. Alright? Now let's take a look at our last two graphs over here. We're going to look at these two slightly different sort of datasets.

Here we have test scores versus the time spent sleeping in number of hours. So clearly, we can see here that all the data points are forming some kind of a trend. But instead of it being a straight line like this or straight line like that, it actually sort of forms a little bit of like a parabola or like a u shape or something like that. Whatever it is, it's definitely not a straight line that kind of represents the relationship between these variables. We see here that students who spend less time sleeping did worse on their exam, and also students who started sleeping more also did worse.

So we would just say here that this is a nonlinear relationship. That's really all you need to know about that. The final one that we'll see here is we have test scores versus number of siblings, versus the different students. And clearly, we can see here that this relationship isn't a straight line either upwards or downwards, and it's not even a curve or something else like that or a different kind of shape. So we would just say here that this data is just not correlated at all.

So you can actually have no correlation whatsoever. So you have positive, negative, nonlinear, and no correlation. Alright, folks. That's just the basics. Let's go ahead and take a look at some practice problems.

Back, everyone. Let's take a look at this example problem here. It's a little bit different than what we've done before. Instead of actually plotting specific points in a scatterplot, we actually have all of the three datasets we're going to be using, and we also have three scatterplots already with their points already drawn. What we're going to be doing is just matching each data table to its scatterplot.

So we're just going to match each table to the plot that's down below. Alright? And then we're just going to describe any correlations that we see. Okay? Let's go ahead and get started here.

We're going to take a look at the table A. So we've got numbers on the x-axis. Remember, it just tells us what the x and y are. We have no idea what these numbers represent, and that's perfectly fine. So we have numbers that go from zero all the way to five.

Alright. So if you go ahead and look at the graphs, if you just look at the x-axis, sometimes this can actually sort of give you a hint as to which graph it's going to be. So I've got numbers that go from basically like single-digit numbers on the x-axis on the first two graphs, and then I've got numbers that go from four all the way to 20 on the third graph. So it's probably not going to be that one. It's probably just going to be one of these two over here, but let's take a closer look.

So I've got the first data point over here that I'm just going to try and locate on the graph zero, ten. If you look at graph one, zero, ten would be over here somewhere, and that dot would probably be somewhere over there. There's no dot there. So it's probably going to be already this graph over here that we can pretty much tell. It's going to be graph two.

But let's just take a look at it and compare a couple more points just so we can be absolutely sure. The next point that I see over here would be one, four. Right? One on the x, four on the y. Do I actually find that in table A?

I can. So this x, this x one, four, is definitely going to be one of our points over here. Right. Just one more over here, which is going to be this. This one is going to be two, one, and that's going to be this ordered pair over here.

So it's definitely going to be that one. So table A is definitely going to be matched up with scatterplot number two. So let's take a look at the second one, which is table B. These values on the x-axis will go from a low of five all the way up to a high of 21. Alright.

So I've got numbers that go all the way to five, and I've also got numbers that go to 20 on the x-axis between actually, between these two graphs. Right? We've already taken care of this one, so we're just going to be looking at graphs one and three. So let's take a look at a couple of these data points. So we've already got in a, which is checked off.

So for b, the first data point that I see over here is five, ninety-one. So if I go ahead and try to locate this on graph number one, this is going to be somewhere over here. I've got five on the x-axis, and I go all the way up to ninety-one, and that's a point that's right over here. Now maybe somebody just drew the points wrong or something like that. It could be this point.

It's pretty close. Let me look at a couple more data points just to be absolutely sure. So if you look at the second pair, there should be twenty-one, fifty-eight. So if I go to twenty-one on the x-axis, clearly, we can see here that this doesn't even go to twenty-one. Therefore, it's probably not going to be graph one.

So it's definitely going to be graph number three. So for b, we're dealing with graph number three. And if you go ahead and look at a couple more of the points, you'll see that this lines up. We've got fourteen, sixty-eight, which is going to be this point right over here. We've got seventeen-eighty-one, which is going to be this point right over here.

And then we've got, you know, eight, ninety-four, which is going to be this point over here. So all these things definitely are lining up this way. Alright? So for b, that's going to be that third graph, which means that just by default, we have that c is just going to be graph number one. Alright?

So we have what all of these three, data tables are and where they match up to their respective graphs. So I'm just going to go ahead and sort of draw a little arrow like this just so you know which one's which. Now let's go ahead and just take a look and and just describe the relations that we see in each dataset. So over here, we can see that all the data points are actually kind of all over the place. Right?

There's no sort of, like, specific trend where I can kind of draw a line and say, yeah, most of the data points sort of seem to follow this trend. Got data points here lower on the right, lower on the left, sort of in the middle on the right, then lower on the right. It definitely doesn't seem like it's like a nonlinear thing because then I have this weird data point that's out here. So all I would say here is that there's really kind of no correlation here. Remember, those are my four possibilities.

So this is going to be no correlation. Alright? If you had to write this out as a complete sentence, you would just say there is no linear correlation between x and y. Alright? Now let's take a look at the second one over here, which is b.

We would just say here that, for example, b over here, we can see that the data points also don't really seem like there's a line that sort of fits with all the points over here, because if I draw this line, then I'm kind of leaving out these two points. If I draw this line, then I'm kind of leaving out these two points over here. But we can definitely see that there is a little bit of a curvature that's going on between these points. It almost kind of looks like a parabola. So it looks like what's going on with this dataset over here is that there is some kind of correlation, but it's just not linear.

So we would just say this is a nonlinear correlation. Now, finally, for the last one, we can see over here that these data points, as we go and look through them from left to right, they clearly seem to be sloping downwards like this. If you draw a line, it would be a downward sloping line like this. Remember that the slope we say is negative, therefore it is a negative correlation, and that's what we would say here. So this is just a negative correlation.

Alright. That's really all there is to it. That's how we sort of had go ahead and match up each data table to its scatterplot and then just briefly describe the correlations that we see. Let's take a look at the next video.

Most of those problems, the scatter plots you'll be using will already be given to you. But in some cases, you may have to make them, either drawing them by hand or, if you have access to technology, you can use graphing calculators to do this much more quickly. You can visualize the data and look for trends and correlations and things like that. I'm going to walk you through how to do that in this video. We're going to use our TI-eighty four, and I've got a list of steps here to help you along.

But the best way to get practice at this is to actually go ahead and do it with me. So let's go ahead and get to it. All right? So we're going to take a look at this example problem over here. We're going to use data that we've actually already looked at before.

This is test scores versus time spent studying on an exam. So we're going to create a scatterplot of this data using a graphing calculator, and we're told explicitly to use the time as the x-axis. This is very commonly going to happen in your problem so that there's no ambiguity as to how the scatterplot should look. All right? So basically here, the time axis is going to be the x data, and the y axis is going to be the scores.

All right? So what's the first step? We have to get all this stuff into our calculators. So we're going to have to go ahead and do that. So open up the stat menu.

You're going to go ahead to the edit button and you should have a blank screen that looks like this. If you already have numbers in L1 and L2, you'll have to clear those and start with a blank page. But let's go ahead and get to it. All right. So how do I get all these numbers in and which data goes into what column?

Basically, what we're going to do here is we're going to set the data in L one. That's going to be the X data and the data in L two to be the Y data. We'll see why that is in just a second here, but this is always good practice to do it this way because it's by default what the calculator is going to use it for the x and y data. Alright? So let's go ahead and get to this.

Basically, I'm just going to go ahead from left to right. So I'm just going to read off these numbers and plug them in. This is going to be 50, then sixty, zero, forty, forty-five, thirty, fifty, ten. Oops. So we got thirty, then fifty, then ten, and then twenty.

So it should have nine numbers over here, and then you just go ahead and do the exact same thing for the second column. Right? They don't necessarily have to be in order, but you should always keep them sort of paired up like this. So however you plugged it in for the x axis, you want to plug it in the exact same way for the y axis. Alright?

So this is going to be eighty-six. Moving it off from left to right. It's ninety-two, sixty-seven, seventy-nine, eighty-three, seventy-five, ninety-six, sixty-five, and then seventy-three. So you should have the same number of entries in both of the columns, which is exactly what I have over here. All right?

That's step one. Next, what we're going to have to do here is we're going to have to open up the stat plot menu and enable some features. You don't have to exit out of this window. Just hit the second button. And the button right above it is the y equals to open up the stat plot menu.

I'm going to go ahead and just go into that first one over here. Click on so you're enabling it. Now, as for the type of graph you should do, the first one by default is already the scatterplot, so you don't have to do anything there. And then basically, this next part here is the calculator trying to figure out which data to use for the x-axis and which for the y-axis. This is exactly why we use the x data for the L1 and we plugged in the y data into L2 is because that's what the calculator already uses by default.

So you don't have to mess around with any settings here. Okay? So then all we're going to do is we're just going to go ahead and, basically, just go ahead and hit the Graph button. Now, if you hit the Graph button and you don't see anything, you might be thinking, oh, no, I did something wrong. And don't worry, you didn't do anything wrong.

By default, your window size for the graphing calculator is going to be usually from negative to positive ten in the x and y axis. But that's not really where our data is, so we're gonna have to adjust the window a little bit, and that's the fourth step. Alright? So we're gonna go over here and open up the window menu, and we're gonna have to adjust a few numbers here. All we have to do is just set the x min, x max, y min, y max to the right numbers here based on what our data is.

Right? So in the x axis, in the x column over here, what I see is that I've got values that go from zero all the way up to 60. What I'm going to do is, for my x min, I'm just going to set it to a number that's really close, but a little bit less than that minimum number. So my minimum number is zero. I'm just going to go ahead and set this to negative one because I don't want the dots to be right on the edges of the window.

It would look a little bit messy that way. Now, for the x max, I'm just going to set this to my highest value is 60. I'm just going to go ahead and set it to 65 so I had a little bit of buffer room there. All right? Now, for the x scale, this essentially controls where the tick marks are going to be.

I don't want sixty tick marks on this graph, so I'm just going to go ahead and increment them by ten, just like they look here on this graph over here. So I'm going to set that to ten. Just do the same exact thing for the y values. My y values, my lowest one is sixty-five and my highest one is ninety-six. So I'm just going to set them to numbers that are kind of around those things.

So for my y min, sixty-five, I'm just going to bump that down to sixty. And for y max, that's ninety-six. I'm going to bump that up to one hundred. Now, for the y scale, I'm just going to do the same exact thing as I did for the x and increment it by ten. All right?

So once you're done with that, now you should be able to hit the graph button, and you should see a beautiful scatterplot that looks something like this. If you don't have the grid lines and the labels, you can go ahead and enable those really quickly by going to the second Zoom button, which opens up the formatting screen. And you can just enable the grid line and labels and things like that. And some of those things will change how the graph is going to look. But it should look something like this.

So basically, you should have dots that kind of go like this. And if you remember, this is we actually graphed something like this in a previous video. All right? That's really all there is to it, folks. So let me know if you have any questions, and let's get some practice.

Thanks for watching.