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Welcome back, everybody. So in recent videos, we've taken a look at x and y scatter plots, and we've been able to sort of visually tell just by looking at the data whether there were trends or correlations in them. But sometimes you may have to more specifically quantify how related two variables are. Fortunately, there is a word for that, and it's called correlation coefficients. Now I know that sounds kind of scary, but all I'm going to show you in this video is that it's just a number.

It's a simple number that represents how related two variables are. So let's take a look at a couple of examples. We're going to jump right in. Let's get started. Alright?

So basically, this correlation coefficient, sometimes called the linear correlation coefficient, is given by the letter "r", and basically, it is just a number that is between negative one and positive one. So you can kind of imagine it on a number line like this. And what it does is it measures two things: it measures the direction and the strength of correlation between two variables.

In fact, let's just jump right into our example so I can show you how this all works. So we have these three correlation coefficients that are given to us over here: 0.13, 0.64, and negative 0.96. They're all numbers between negative one and positive one and they always will be. All we have to do is we're given these three numbers and we just have to match them up to their appropriate graphs.

So let's go ahead and get started here. I mentioned that r measures the direction and strength. So let's talk about the direction. The direction of correlation is always going to match the sign of that r value. So, for example, in a couple of videos, we talked about how positive correlation shows an upward trend like this.

Those are going to be positive r values. So basically, whenever you have positively trending upwards like this, your r value is going to be positive and vice versa. Whenever you have something that sort of slopes downward like this, that's negative correlation. Those r values will be negative. All right.

So let's take a look at our three numbers. You'll notice that these two are positive and this one is negative. So we just have to find which one of these graphs shows a downward sloping trend in data. And if you take a look, we should see it here that there's basically only one, and it's going to be the left one. So automatically, just by looking at the graphs, we can figure out that this one is negative 0.96.

So sometimes you can tell just right off the bat just by looking at the signs of the r values. Alright? Okay. So that's a little bit more about the direction, but let's talk about the strength of that r value.

So we say that correlation is strong whenever the points are tightly clustered around the line that kind of cuts through a lot of the data points. For example, in that left graph, you can see that all of these data points are actually really sort of all lined up such that there's a little bit of wiggle room here, but you can almost draw a line that cuts through most of these data points, and they're really tightly clustered around them. This is what we would say is strong correlation. So this is how we can visually tell whether two variables are strongly correlated. Now when this happens, the r value is going to be close to either negative one or positive one, depending on whether it slopes down or up.

All right. So, clearly, we can see here that this slopes downwards. Therefore, it's negative. But because the data points are tightly packed, it's going to be an r value that's very close to negative one. All right.

Now, as you get closer towards either one or negative one, that correlation gets stronger and stronger. On the opposite side, if you get closer towards zero, it means that the correlation is getting weaker. And when you have values that are close to zero, it means that there's no correlation and the data points are kind of scattered around everywhere.

Let's take a look at our remaining two values.

We have 0.13 and 0.64. So both of them are positive numbers, but which one do you think is going to represent each one of these two graphs? Well, if I take a look at the second graph over here and compare it to the third one, you can see that these data points are much more loosely scattered around. I can't draw any line that cuts through most of these data points, so there's no real correlation anywhere here. So which of the values do you think it's going to be?

Well, it's going to be 0.13, and this is what we would say is no correlation as we've seen before. Therefore, just by default, this final value over here, this r value, is going to be 0.64. We can still see that there's somewhat of a trend line that cuts through most of these numbers.

But unlike that left graph, those data points don't really cluster tightly around that line. So we say that this is weak correlation. Now there's no general consensus on what counts as strong versus weak, what the cutoff is. But generally, anything beyond 0.8 is considered strong here. So somewhere around 0.8 would be a good boundary for that.

Alright? That's really all there is to it, folks.

Thanks for watching. Let's get some practice.

In the last couple of videos, we were introduced to the linear correlation coefficient, this r value that tells you the strength and the direction of correlation between two variables. Now, most of the time, you're already given that r value, but in some cases, you may have to go and find it. Fortunately, you'll almost never have to calculate this by hand. The equations are really nasty and kind of complicated, and you'll be allowed to use a calculator to go and find it. That's what I want to show you how to do in this video, how to use a graphing calculator.

Got my TI-eighty four over here to find the correlation coefficients. We're going to plug some data into our calculator and then use some features in the stats menu. So let's go ahead and get started here. We're going to take a look at a familiar dataset that we've already seen before, and I've got a list of steps for you to follow along. But just go ahead, and the best way to practice this is to actually follow along with me.

Alright. Well, let's get started here. We're going to find the correlation coefficient for this data. So we have test scores versus time spent studying for an exam. This is data we've already seen before.

If you were to plot it on a scatterplot, it looks something like this. We've got time on the x-axis and we've got score on the y-axis. Right? How do we find the r value over here? Well, the first thing we usually do is we plug this data into our calculator, but there's actually one thing that we have to do first.

So let's take a look at our first step. We're going to go into the catalog of our calculator by hitting the second button, and then I'm just going to hit the zero key, which has this catalog right above it. It's going to show up with an alphabetized menu of different features in your graphing calculator. You're going to go all the way down to the 'd' because you're going to be looking for this one called 'diagnostic on.' It's going to be right over here.

And once you find it, you're just going to hit the enter key twice. You should see a little 'done' that pops up. And the good thing is you only have to do this one time. You don't have to do this every single time that you do this process. Alright?

So now that we're done with this, now we can actually go ahead and load this data onto our calculators. That's the second step. Alright? Let's go ahead and do this. We've done this a bunch of times before.

Open up the stat menu. Go to the edit button. If you already have data that's in here, you're going to have to clear it out. We're just going to load all this data in. Alright?

So again, what we're going to do here is we're going to use the time on the x-axis, and we're going to load that into L1. Alright? So let's go ahead and do this. This is kind of the boring part, but I'm just going to be plugging in a bunch of numbers. Alright?

So this is going to be 50, 60, zero, 40, 45, 30, 50, 10, and 20. Alright? Then just go ahead and hit the L2, plug in the exact sort of same in the same order, so you should have the same number on both columns. Alright?

So 86, 92, 67, 84, 83, and 75, 96, 65, and 73. Alright? Same number of entries in both columns, and you're done. Right? So now what we're going to have to do here is, how do we actually get an R value from this stuff?

We're going to go ahead and we're going to open up the stat menu, and we're going to go over to the calculator tab. So we're going to go ahead and hit the stat button, go to the calc tab, which has a bunch of different features and formulas depending on the data that you plug in. We're going to be taking a look at that fourth option, which is linreg ax+b. So we're going to be talking about this feature a little bit more in later videos. But for now, all you need to do is you just need to go ahead and plug in the correct xlist and ylist for your calculator to use.

Now we use the xlist in our L1, so that's exactly what we're going to use. And then for our ylist, we're just going to use L2. Right? This is, again, just by default because your calculator just is going to use x as L1 and y as L2. Once you're done with that, you're just going to go ahead and hit the Enter key a bunch of times and hit the calculator button.

And that's basically it. So if you look at this readout over here, there's going to be a bunch of different variables, like y, a, and b. Again, we'll talk about those later. But really, there's only one number that you need out of this readout, and it's the one at the very, very bottom. So notice how this readout here shows two values.

One is r squared and one is r. The r value, the one at the very bottom, is always the one that we use for this. This is the linear correlation coefficient. And clearly, we can see here that this is equal to 0.93. I'm just going to go ahead and round this up to 0.94.

All right. So what we can tell here is that this is a pretty strong linear correlation that's positive. And that kind of makes sense, again, if you look at the data set over here. Right? If you look at all of this data, this is actually pretty tightly packed together against the general trend.

If you were to draw a line that kind of cuts through most of the day, you'd see it looks like this. All of the dots form a really, really tight against that trend line. They're not spread out all over the place like this, which would give you a lower R value. Alright? So it's perfectly reasonable and perfectly makes sense that we get an R value that is really, really positive and also really close to one.

So that's really all there is to it, folks. So let me know if you have any questions. Let's take a look at some practice problems. Thanks for watching.

Welcome back, everybody. So let's take a look at this example problem, and we'll work it out together. A scientist is measuring the speed of sound in terms of feet per second at different altitudes in terms of thousands of feet. So those are our two variables here. We've got speed of sound versus altitude, and they record their measurements in the data table below.

We're going to use this data table with our graphing calculators to find a linear correlation coefficient. Remember, that is that little r variable that we've been talking about. Again, we never calculate these things by hand because we have technology to help us out with this. So let's go ahead and get our graphing calculators out because we're going to calculate r and then use this to answer this question, which is, is there a correlation between these variables? We should be able to tell just directly from that r value.

Let's get started here, get your calculators out. Remember, the first thing we have to do is get all this stuff into our calculator. So we're going to open up our stat menu. We're just going to go over here to the edit button. Remember, if you already have things that are here in these lists, go ahead and clear them out so we can populate this with the new data.

Basically, we're just going to go from left to right over here. And this is kind of the boring part where you're just going to enter a bunch of numbers into your calculator. So go ahead and feel free to skip if you need. So this is zero, five, ten. We're just going to increment by five.

So twenty-five, thirty, thirty-five, forty, forty-five, and fifty. Alright? So we got our values over here. That's eleven values. And let's go ahead and take a look at the y-values.

Alright? So we've got 1120.2. I'm just going to plug them in in the exact same order, 1094.7, 1076.9, 1058.1, 1034.5, 1015.4, 995, 968.2. Clearly, we can see here that there's a downward trend with the numbers, 966.5, and then 966.1, although that trend seems to slow down as the numbers get closer and closer towards the end.

The next thing we have to do as we get all this stuff into our calculators is open up that stat menu again. We're going to go over to the calculator tab, and we're going to go ahead and hit that fourth option, which is the linReg ax plus b. This just gives us a bunch of good data about our correlation and the two variables involved. The x list, remember, is always going to be L1 by default.

Y list is going to be L2. That's perfectly fine because it's exactly what our data table says. Our altitude was x and speed was y. And then we're just going to go ahead and hit the enter key a bunch of times so it calculates. It's going to give us a bunch of different information. We've got a couple of y variables a and b. You're just going to ignore those for now because the only one that we really need for right now is going to be this one right over here. It's going to be the r value. It's going to be the r squared.

We always use the r value. That's the linear correlation coefficient. And clearly, we can see here that this r value is negative 0.97, and I'll just round it to three decimal places, nine seven three. So clearly, we can see here that this is a number between negative one and positive one.

It's negative, which means that this correlation here is a negative correlation. As the altitude increased the x-values, our speed decreased. So, this went up. This went down.

So that shows us there's a negative correlation here. And moreover, what we also saw is that this correlation tends to be, or seems to be pretty strong. So we can see here that if we were to plot this out on a sort of scatterplot like this, you would see a bunch of data values, and they would be really, really close together in a trendline like this.

So the answer to this question over here is, do we have a correlation between these two variables? And that answer is absolutely we do. So, yes, you would just say that this is a strong negative correlation. This is a number that is negative and very, very close to negative one. So this would be strong negative correlation.

So then, again, that's just a little bit more practice about how you plug data into your calculator to get a correlation coefficient. Thanks for watching. Let me know if you have any questions.