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Welcome everyone to this section of the course discussing joint distributions, more specifically,

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covariance and correlation.

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When studying mathematics for data science, we're often interested in the relationships between two

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data dimensions or data features.

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Data driven organizations often need to understand interactions between two data features in order to

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make decisions about the features that are linked.

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Joint distributions are going to allow us to mathematically quantify the relationship between two distributions

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of data.

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So here we can see a visualization of the relationship between total bill that somebody leaves after

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a meal and the tip that they leave.

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And we can see that they seem to be linked somehow.

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So there's an intuition already that is total bill increases, so does the tip.

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So this section of the course is going to focus on two specific properties of joint distributions,

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covariance and correlation.

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In general, these terms measure how much two data features move together.

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For example, the total bill moving along with tip and likewise the tip moving along the total bill.

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So it's a common task to try to understand correlations between two data variables.

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For example, most businesses are going to keep track of how correlated price changes are with overall

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sales revenue.

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When thinking about covariance and correlation, we're typically discussing it in terms of two data

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features.

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Now, I want to point something out about the notation, especially if we're thinking about something

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like machine learning when we're talking about covariance and correlation and the formulas behind them.

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We commonly notate the two data features as X and Y.

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Since if one were to plot these features, you could plot one on the x axis and the other on the y axis.

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So, for example, if we look back at our TIPS data set here, we can see Total Bill is a feature and

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we're plotting it along the x axis.

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And Tip is also a feature that we can plot along the Y axis.

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So as we continue to actually show you the formulas for covariance and correlation, we can think of

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total bill as X and tip as Y.

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Now, before we do a deep dive into the details of covariance and correlation in this lecture, we want

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to do a high level overview of these concepts.

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We'll also understand the motivation behind understanding these concepts in the context of using data

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science for business.

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So let's begin with covariance.

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The formula for covariance is the following.

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The covariance between two features X and Y is equal to one over N and the sum between x ie minus the

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average value of x multiplied by y of T minus the average value of Y.

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And we're going to break this down in just a second, but I want you to start building out an intuition

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for covariance.

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And keep in mind covariance can actually go from negative infinity all the way to positive infinity.

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And you can see a visualization of the general relationship for something that has negative covariance.

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So you can see it's kind of a negative slope there for covariance hovering around zero, there really

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is no relationship and then covariance that is greater than zero.

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You can see positive slope relationship and you can kind of tell from the formula here that it feels

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similar to calculating the slope of a line.

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But let's break it down a little further.

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Before we continue a quick note.

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If you were to Google the definition or formula for covariance, you are going to sometimes see this

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as n minus one, and other times you may see it as just n like we're showing here to be a little more

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specific and minus one is for sample covariance, while n is for population covariance.

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But we'll discuss that in more details later on.

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But right now we'll just show you the formula for population covariance.

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So how do we break this down?

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Well, we're looking at the covariance of data feature X with data feature Y.

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For example, I could say, what's the covariance between the total bill and the tip?

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Then you'll notice we're dividing by MN, where MN is the actual number of data points.

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You should immediately know how this implies that X and Y need to have matching length essentially the

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same number of data points, and that makes sense.

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You can't really calculate the covariance between two data features that don't have matching data points.

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So for example, if you went to one restaurant and had a couple of tips and then you went to another

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restaurant that had a ton of instances of total bill, you can't calculate the covariance between those

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because they don't actually share the same length.

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And you should also notice that this feels a lot like the same calculation for an average.

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So I want you to keep in mind that you're calculating some sort of average value because you're taking

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a sum and then dividing it by the number of points.

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And remember, this is the sum notation.

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So what this is doing is it states to take the sum of the calculation on the right hand side for every

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point in X and Y.

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So you're going to start with x one and Y one, basically the first row of data, then x two and Y to

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the second row of data, and you're going to go on for every single row of data.

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So that would be every instance, for example, of total bill and tip all the way until you reach X,

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y and Y.

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Essentially that goes all the way to n.

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I is just a placeholder for one, two, three, four and so on all the way to end rows or end data points.

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Now let's take a look at what's happening inside this sum.

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It's an interesting calculation.

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Notice it's the difference between each individual point and the mean value for the data series.

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Then you're going to multiply that calculation between X and Y together.

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This entire formula should feel familiar if you've already seen the formula for just straight variance.

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Notice how it's actually variance, but with another data feature.

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So if you think back to the formula for variance, it was extremely similar except you squared it.

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In this case, you're essentially doing variance between two features.

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Thus the name covariance.

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So now that we've briefly explored the formula, what information does Covariance actually report back

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to us?

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So if we take the total bills and tips, take those values, plug them into the formula, how do we

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actually interpret this final value?

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Recall that variance reports back the squared difference of the average value from the mean.

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So thinking back to just variance, if we were to visualize this and calculate it, we're just talking

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about what's the squared difference between the average value from the mean.

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So visualizing this I think helps out a lot.

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Imagine that we have a bunch of values and some of them are all of them fall between values one and

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nine.

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Notice we have no values for one, one value for two, three values, four for two values for five,

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one value at seven and one value at nine.

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So again, just talking about variance for right now, we'll expand this concept into covariance in

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just a second, but I want you to visualize variance first.

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Then it's going to become super easy to visualize covariance.

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So thinking back on variance, what's the formula actually asking us to do?

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Well, first off, I got to calculate the average value.

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So I go ahead and calculate the average value here and notice it's hovering around five.

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Then what I'm going to do is for every single point, I'm going to take that point, subtract the average

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value, and then square that.

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And if you start thinking about this visually, taking the square of this is actually the same thing

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as taking the area of a square from that point to the average value.

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So here what we've done is we've shown you those actual squares visually.

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Notice how we're essentially constructing squares from each point all the way to the average value.

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And again, this is for variance.

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And then what you're going to end up having here are all those squares, and then you can end up summing

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them up and then taking the average value.

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And that's essentially showing you, Hey, for all these points, what's the average value of your distance

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from the mean after you perform this squaring operation?

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So that's variance.

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And hopefully this visual, if you take the time, gives you a little bit of intuition behind what the

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calculation is doing.

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From a visual standpoint, again, you're taking the points, calculating the mean.

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And then if you look inside the formula, what it's actually asking you to do is take each individual

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point, go to the mean, and then take that difference and square it.

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Remember, because you're squaring it, that's always going to be a positive value and then you're going

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to take the average there.

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So the same concept of area can actually be applied if covariance, it's just going to be in two dimensions

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X and Y instead of a singular dimension, just X.

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So let's take a look again at covariance and try to understand what's happening here.

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Remember, this time it's going to be pretty much the same calculation and you're building out now a

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rectangle instead of a perfect square and you're just doing it along two dimensions.

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So you're going to start off again with X and Y.

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So we can think of X feature as total bill and Y feature as tip.

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And then what we need to do is start off by calculating the mean.

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But instead of just a single mean like we did on that single line for X, now we're going to have two

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means one for X and one for Y.

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So I'll go ahead and plot that out.

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And conveniently here it falls in the middle.

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But obviously, depending on your actual data set, it may fall somewhere else.

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But here I'm just showing you this blue dot is going to represent the value of x mean lined up with

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a value at y mean.

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Then you're going to go through each of your data points.

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So let's go through a single data point X, Y, and y Y, and I go ahead and plot that out.

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Now take a look at the actual equation.

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This equation is just the area of a square in this particular case, and technically, it could be a

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rectangle, too.

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So before we had a perfect square, but this time we have a rectangle.

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It's just going to be x, y minus x mean multiplied by y, minus y mean.

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And again, that's just one side times the other.

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So that's again just calculating a rectangular area.

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This time it's in two dimensions.

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So it's not going to be a perfect square because last time it was x, y minus x average squared.

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And in this case we're just doing it for two dimensions.

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But the same sort of visualization and intuition applies.

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And then next, you simply look at the slope between the two lines to realize whether or not this is

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going to be negative versus positive.

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And therein lies one of the main differences between variants for a single feature versus covariance

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between two features.

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Notice this time the value could actually be negative.

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Unlike last time with the square operation, it was always going to be positive and this is going to

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be the same as the slope between the two points.

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So notice how this is a negative slope, Meaning if you were to actually perform this mathematical operation,

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you're going to have a negative value.

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So we can kind of treat this area as a negative area for covariance.

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So if it's a negative slope, like in this example, we can treat this total area as negative covariance.

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And keep in mind, eventually we're going to do this for a bunch of rectangular or square values and

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then you can add them up.

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So, for example, at this point we can see that the slope between Z Y and the actual values of the

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mean X mean and Y mean, that's going to be positive.

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So I'm going to fill in this rectangle as having positive covariance and notice the slope basically

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falls in line with that.

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And now what you're going to end up doing is you keep doing this for all the points XY and Y until you

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get to the last row.

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And so you just keep drawing these rectangles in the same way you drew the squares for variance earlier.

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And again, note how the formula indicates we're doing this for all X and Y points.

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And then what you're going to end up doing is you're going to have a set of series.

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So in this particular example, I'm showing you a series set where they're doing all those rectangles.

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And in this case, a lot of them actually have a positive slope or a positive covariance.

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Notice if you were just to look at the black points and kind of ignore those green rectangles.

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For right now, the general trend looks to be like a positive slope.

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So in this case you have a positive covariance and we can see for this particular example, the covariance

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is 8.04.

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There's also a correlation value, but we'll talk more about correlation later on.

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But I want you to get this idea that when you see a bunch of points here with kind of this general positive

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trend or positive slope, if you were to do this calculation of drawing those rectangles, which is

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literally what the formula is asking you to do, you end up just summing those up and then taking the

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average of the area of those rectangles, and that would supply you with the covariance value.

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So hopefully this intuition and visualization really helps you understand what's going on behind the

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scenes.

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Let's take a look at a series with a negative covariance.

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You can still see there are some points that have that positive slope, but the majority of them look

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to have a negative slope.

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So again, when you sum these all up and take the average, you end up getting a negative covariance.

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And in this case the negative covariance is -0.119.

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And again, we'll talk more about correlation later on.

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You'll notice that the correlation is also negative.

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And then four points where they're a little more flat.

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You can see here we have two data series with almost zero covariance.

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So essentially what's happening here is your positive rectangle area is going to start balancing out

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with the negative rectangle area and that ends up summing to close to zero.

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So again, we're going to take a deeper dive into covariance later on in this section.

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But I want you to keep in mind that visualization of those positive and negative rectangular areas to

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give yourself an intuition of how covariance is actually calculated.

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And I would highly encourage you to literally just grab a piece of pen and paper and diagram this out.

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And you'll notice the equation here is essentially the equation of the averages of the areas of rectangles.

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So even though this formula at first looks really intimidating, it's basically kind of elementary school

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arithmetic here.

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It's just calculating the average areas of all these rectangles.

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But it's being clever in the fact that it's using things like the average x value and the average Y

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value.

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So in review what covariance is, it's simply the sum of the area of these rectangles that you then

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average out where the slope between the lines is going to indicate whether it's a positive area value

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or a negative area value.

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And then just again, simply average out the sum of those areas.

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I would highly encourage you to try it out yourself on some simple examples of positive and negative

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covariance series.

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So just see if you can create your own little data set with two or three rows and then go by hand,

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maybe plot them out just generally by hand and then see how that actually ends up reflecting the formula

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of covariance.

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So the formula for covariance can seem very intimidating at first, but it's literally just drawing

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rectangles and then taking the average value of the area of those rectangles.

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So now that we understand the intuition behind covariance, let's explore the related topic of correlation.

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So what is correlation?

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Well, correlation is actually very related to covariance and even uses the formula for covariance or

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the resulting covariance in its calculation.

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And as we learn about correlation, know how its possible range differs from covariance.

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So remember that covariance could go from negative infinity all the way to positive infinity while correlation

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has to be between negative one and one.

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So essentially any data points that you feed into a correlation calculation are always going to spit

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out a value that's between negative one and one.

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So here is the correlation formula.

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Pretty simple, right?

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It's just using covariance.

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So it's simply the calculation for covariance divided by the product of the standard deviations of the

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two series.

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So on the bottom here, you have the standard deviation of the x theta series multiplied by the standard

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deviation of the Y series, and you're just taking that value and then dividing the covariance value

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by it.

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So why would you do this?

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Well, let's expand this formula by showing the calculations for standard deviations and covariance.

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So if I were to plug those back in, plugging in the formula for covariance and the formulas for the

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standard deviations, I get something that looks like this the full formula for calculating the correlation.

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Recall for covariance that X and Y are separate data series, meaning they could have very different

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ranges of min and max values.

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For example, the total bill could go from maybe $0 all the way to thousands of dollars, but the tip

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itself maybe only goes from $0 to maybe hundreds of dollars.

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So that's already an issue where tracking the covariance between two series that could have a relationship,

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but their values are super different.

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That's going to mean it's going to be hard to actually interpret the raw covariance number because again,

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it can go from negative infinity all the way to positive infinity.

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So instead it would be great to standardize the covariance so that it always has values between negative

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one and one.

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And I mentioned that the very start that businesses tend to just report correlation instead of covariance.

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And that's the reason is because you always know it's going to be between negative one and one and it's

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essentially standardized that way.

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So it makes it way, way easier to interpret without needing to understand the ranges of the original

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data feature values.

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This can cause issues in trying to determine the significance of covariance values, since it can take

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any value, as I just mentioned, which is why we use correlation.

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So the formula for correlation actually standardizes this for us.

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It uses the denominator to standardize the covariance between negative one and one.

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So this is all to say if you're following this formula, you'll realize by using the mathematical trick

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of dividing covariance, by the multiplication of the standard deviation between series X and series

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Y, you end up standardizing covariance to always fall between negative one and one.

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And when I say standardized covariance, I'm really just talking about correlation.

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That's all it is.

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It's standardized covariance so that it has to fall between negative one and one.

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You can easily then interpret that number and you don't even need to know anything about the ranges

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of X and Y.

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So this makes it much easier, again, to interpret a correlation value rather than a covariance value,

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which is why in the real world you pretty much always see correlation being reported rather than covariance

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being reported.

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And you can check out here what the difference correlations look like for different distributions or

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joint distributions of features.

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And notice that technically speaking, there's going to be some strange relationships between features

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that sometimes have zero correlation.

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So something to watch out for.

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For example, you can see that we kind of have this ring or x looking values or something that looks

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like a sine wave, and depending how you calculate correlation, when you standardize it, it's going

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to end up being zero.

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So it's just something to look out for.

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But in general, when you're talking about correlation and joint distribution between two features,

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you should think of one as perfectly correlated.

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So one value goes with the other.

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They're essentially equal to each other and negative one being perfectly correlated the other way.

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So as one increases, the other decreases and then the values in between.

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Notice again how certain data sets with interesting behaviors can still have a zero correlation value.

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So there is helpfulness and actually visualizing a joint distribution along with reporting its correlation

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value.

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So we've now explored covariance and correlation and understand their ability to inform us of the relationship

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between two data series.

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And as I keep mentioning, you're most commonly going to see correlation use and organizations as the

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metric being reported since it is standardized to be between negative one and one.

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Okay, so let's take a deeper dive into covariance and correlation.