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The first kind of plot we want to talk about is the scatterplot also called a scatter graph or scatter

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chart, which shows the relationship between two dimensions or characteristics or measurements of a

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group.

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So as we walk through this idea, let's just take a set of data.

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Let's say we're running a small business here and we ship out products to customers as they order them

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from our website.

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And maybe we keep track of lots of different pieces of data about our shipments.

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And let's say this is just a snapshot here of some of that data.

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So we record this state that we're shipping to.

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We categorize those states by regions.

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So we group states into the West, Midwest, Southeast, northeast, etc. regions.

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And then we also record the number of items we send in each shipment.

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So for this first order here in this first row, the person ordered three items.

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So the item count is three.

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Their order total is $2.08.

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So they bought three very inexpensive items.

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And with just that data alone, we can create a scatterplot that compares the number of items sent and

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the order total, the total amount of money for that item count.

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So here's what that scatterplot might look like if we put item count along the horizontal axis here.

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So all the way from zero items up to ten items, we can see in our data here that the most items we've

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recorded for any shipment is ten items, and then we place order total along the vertical axis here.

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So starting at zero along the bottom up to $100.

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So item count here is a number order total is in dollars and we can see here if we skim through order

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total that in fact this data is sorted by order total from smallest to largest order with the most expensive

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order being $99.69, which is why this vertical axis spans from $0 to $100.

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And then if we plot each order as a data point, what we create is a scatterplot.

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So taking this first order here, the customer ordered three items and the order total was $2.08.

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So we could think about along the horizontal axis here, locating three items.

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If this is two items and this is four items, then three items should be about halfway between.

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And then if this is $0 and this is $20, then $2.08 should be very close to the bottom of our vertical

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axis here, very close to $0.

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And in fact, we see that data point right there.

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That is the data point for this first order.

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And we could take any other order in our data set here and locate its point on the scatter graph.

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For instance here, this order that got sent to Georgia, the person ordered nine items and spent $45.24.

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So nine items should be right about halfway between eight and ten.

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And if we go up to 45, 24, if this is $40 and $60, $50 should be right about here.

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$45 should be right about here.

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It looks like this is the point associated with that order of nine items for a total of $45.24.

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So all of these orders have been plotted as individual points in this scatterplot.

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Now, a couple of things we really want to remember as we're creating scatter plots.

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The first is the most general point that what we're doing here is essentially comparing two characteristics

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of the same variable, if you will.

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So here in this particular example, we're comparing item count and order total for individual orders.

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And we could just as easily compare different characteristics of the same set of orders.

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For instance, maybe we also keep track of the time of day when the order was placed.

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We could make a different scatterplot that shows time of day along the horizontal axis and item count

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along the vertical axis or time of day along the horizontal axis and order total along the vertical

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axis, regardless of which two characteristics we pick about our variable.

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When we create a scatterplot, we want to if we can think about putting the independent variable along

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the horizontal axis and the dependent variable along the vertical axis.

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Now that's not going to be a perfect science because there may not be direct causation between the two

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characteristics that we're comparing.

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But if we can think about in general one characteristic causing the other, we want to put the characteristic

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that initiates the cause along the horizontal axis and the outcome or the cause along the vertical axis.

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So an easy example might be height versus weight of people.

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So if we think about.

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Height.

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Versus.

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Wait.

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This is a good example because it reveals the shortcomings of this idea.

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But in general, we have some intuition that the taller someone is, the more they're going to weigh.

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Of course, that's a very broad generalization.

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We often see taller people who weigh less than shorter people or shorter people who weigh more than

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taller people.

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So obviously, there's not perfect causation here.

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Being taller doesn't necessarily mean that you will be heavier, but it probably does make a little

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more sense to put height along the horizontal axis.

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Because in general, as a very broad idea as height increases, weight is probably likely to increase.

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So height is cause weight is the effect or the outcome.

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Now, oftentimes if we switch these so if we were to put weight along the horizontal axis and height

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along the vertical axis, it's usually not going to cause a big problem.

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Same thing with this example here.

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We could flip item count and order total putting item count along the vertical axis in order total along

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the horizontal axis, and we'd still be able to do the math that explores this relationship.

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Same with height and weight.

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But if possible, if we can somewhat identify that a height increase tends to cause a weight increase,

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we would put height along the horizontal axis, weight along the vertical axis in the same way that

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maybe in general as item count increases or total increases.

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We like to generally put that independent variable along the horizontal axis and the dependent variable

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along the vertical axis.

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But again, this is not a perfect science and switching these two isn't usually going to cause us trouble.

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We also want to say along these lines that saying we've heard before that.

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Correlation.

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Does not necessarily equal.

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Causation.

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So just because weight tends to increase as height increases doesn't mean that height perfectly dictates

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weight or just because order total tends to increase as item count increases doesn't mean that increasing

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the item count is guaranteed to increase the order total.

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Two things can be correlated, meaning that they can increase together or they can decrease together,

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or they can move in opposite directions.

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But the relationship between the two characteristics, the correlation between the two characteristics

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doesn't necessarily mean that a change in one characteristic is actually the cause for the change.

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In the other characteristic, there could be some third confounding variable or some other outside force

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that's causing the correlation between the two characteristics or the correlation we're seeing might

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just be coincidental.

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There's no guarantee necessarily of causation.

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And so we need to be really careful that just because we plot our data on a scatterplot and maybe we

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see some general trend does not prove that we somehow have causation between the two characteristics.

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All we can really say is that we see some kind of a trend in the data.

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In this case, the example we're using, it appears as though as item count increases, order, total

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increases or as order total increases, item count increases because that's the direction we're seeing

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in this trend line that we've plotted.

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Now, that being said, let's talk about this trend line a little more.

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We call this when we're dealing with a scatterplot, we call this the trend line, sometimes the regression

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line, the best fit line line of best fit, even the least squares line, those are all basically names

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for the same thing, which is this line that indicates the trend through the data.

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And we actually have mathematical formulas for calculating this line.

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Now, we can do this by hand with this set of formulas here, but of course it's going to be much easier

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for us to plug our data into a calculator or software and have it calculate the equation of this trend

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line for us.

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But even if we do that, these are the formulas that our software or our calculator will use to find

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the equation of this line.

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Now, if you've ever taken algebra, you know that the equation of a line is given by Y equals m, x

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plus B.

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Now M, this value here in front of the X is the slope of the line.

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It tells us the rate at which the line is increasing or decreasing.

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So a higher value for M means the line is steeper, it's increasing faster, whereas a lower value for

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M means the line is more shallow, it's increasing more slowly.

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A negative value for M indicates that the trend is decreasing.

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The line would be going in the opposite direction this way, this value.

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B Here is the point at which the line intersects the vertical axis.

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So if we picture here this vertical axis, then it b is this value right here.

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So in our case it looks like B is a little less than ten because we have here zero and 20 along the

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vertical axis, this looks like it's a little less than half way from 0 to 20.

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So I'm guessing that the value of B is a little less than ten, and we'll actually calculate this value

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to find out.

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So if we have the slope and we have the y intercept, then we can plug in any value of x and this equation

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will return to us a value for y.

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So while this formula is normally in algebra, just y equals mx x plus B, this little hat on the Y,

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we call this y hat.

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This indicates that we're talking specifically about a trend line and we want to distinguish between

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the equation of a trend line and the equation of just a regular line, because we want to be super aware

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of the fact that the equation of the trend line is sort of this predictive trend.

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And what we notice here, if we look at this trend line, is that it actually doesn't pass through a

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single point in our data set, right?

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We plotted all these points.

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This little point down here was the 0.3 $2.08.

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This point right here was the 0.9 and $45.24.

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This line is supposed to predict the value for order total when we pick a particular value for item

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count.

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So what this trend line is really saying is that let's say we consider an item count of eight.

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If we come here to item count of eight, we come straight up from the horizontal axis until we meet

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the trend line and then we head straight over to the vertical axis.

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From that point it looks like we get to a value here of maybe roughly 65, let's say.

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If that's the case, then what this trend line is telling us is that the expected order total for an

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order with eight items is about $65.

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But of course, if we look at our raw data, we don't see any.

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Orders where the item count is eight and the total is $65.

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In fact, if we scan through here, the only time where we have an item count of eight is these three

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orders right here.

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And we can see that the order total for each of those is over $95 each time.

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In fact, we see those three data points right here, and those values aren't anywhere close to $65.

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We can also scan through this list and see orders that have a total near $65.

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What we find is that the closest two orders are these right here where the totals were $58.37 and $70.01

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and the item counts.

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There were ten and six items respectively, not eight items.

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So what we need to realize is that the trend line is not predictive at all in terms of actual specific

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data values in our data set.

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What it does instead is compute what it believes are the expected values for order total based on item

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count or vice versa, and gives us a trend line that is the most efficient path through the data, which

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means that if we're trying to make real world decisions using a trend line, we need to be really careful

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because this gives us a guideline that we can use to estimate what we might see.

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Of course, using this example, we know that the trend line estimates that if someone orders eight

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items, we can probably expect that order to be around $65 on average.

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But as we can see from our data, we are certainly not guaranteed that exact order total if we pick

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an eight item order.

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So we can somewhat use this trend line through a scatterplot to make predictions.

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But we just have to be aware of the fact that it's only an expected value.

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It's only an average for order total based on item count.

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It's not perfectly predictive of all of the orders will get.

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If it was perfectly predictive, then every single dot in our scatterplot would be exactly along this

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line.

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So that being said, let's just walk through this math using the table we've created here so that we

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can see how to calculate M and B, and then use this equation to plot the trend line.

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So if we have our raw data here, we need a few things.

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We need the value of N, which is the count of data points in our set.

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Here we have 25 orders if we count here, the number of rows we have, we have 25 orders.

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We've indicated that here.

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So when our case end is equal to 25.

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So let's look at what this looks like to plug everything in here.

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We'll get 25.

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Then we multiply that by the sum of all of the products of X and Y.

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So we're going to say here that item count represents X and that order total represents Y, which makes

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sense because we put item count along the horizontal axis, which is the x axis, and we put order total

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along the vertical axis, which is the Y axis.

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So what we're doing here with this sum of all the products of X and Y is we're multiplying X by Y for

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every order.

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So we're multiplying item count and order total for each order.

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And we see that here in this column, the product of count and order, total item count and order total.

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So if we multiply three by $2.08, we get $6.24 or just 6.24 without the units of dollars.

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If we multiply six by five, we get 30.

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If we multiply one by 9.65, we get 9.65.

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So we're just multiplying X and Y or these two values to get the product.

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Then this here, the sum of all the products means we take all these products and we sum them, we add

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them together.

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And if we sum all of these values, we get 88 6.38.

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So we multiply this by 80 806.38.

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Then we'll subtract the sum of all of the x values.

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So we've said x is item count.

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If we sum all of these item counts, we get a sum of 138 right here.

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So 138 multiplied by the sum of all the y's well y is order total.

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So if we sum all of these order totals, we get a sum of 1,229.38.

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So we multiply this by 12 29.38, and then we divide by n, which we already said was the count of orders

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25 multiplied by the sum of all of the x squared values.

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In other words, X is item count.

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So we want to take every item count and square it.

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So we take three, we square it, we get nine.

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That's this column here, count squared, we take six, we square it and we get 36 one.

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Word is one zero squared is zero, four squared is 16.

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So we square all of the X's and once we've squared them all, we add them together and the total is

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1038.

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That is the sum of all the x squared.

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So we multiply by 1038 and then we subtract from this the sum of all the X's.

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And once we get the sum of all the X's, then we square that value.

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Well, the sum of all the X's we already found, we take all the X's, we add them together, we get

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138.

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So that is the sum of all the X's.

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But then we square that.

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So we're taking 138 and squaring it.

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And here's the sum squared value 138 squared is 19,044.

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So we're subtracting 19,000.

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44.

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And if we use a calculator to do the math here, what we get is approximately 7.3132.

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When we round a four decimal places, that's the value of M, That is the slope of the trend line,

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which means that for every one unit we move to the right along the horizontal axis will move up about

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7.3 units along the vertical axis, meaning that for every one item we add to item count, we increase

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order total by about $7.31.

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That's what that slope is telling us.

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Now all we have to do is calculate B using this formula, and when we do, we get the sum of all the

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y's, which we know is 12 2938 So we get 1229.

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38.

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And then we subtract.

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We use the value of em that we just calculated.

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So M is about 7.3132.

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So we say 7.3132, and then we multiply that by the sum of the x's, which we know is 138, so 138.

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And then we divide this whole thing by the number of orders and equals 25, so we divide by 25.

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And the result there is approximately 8.8063 when we round to four decimal places, which means that

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the equation of our trend line using this formula here is y hat approximately equal to 7.3132 times

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x plus 8.8063.

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Now, we've already plotted the trend line here, but if we wanted to use this equation to plot it because

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let's say we didn't have it yet, we just had the scatterplot and now we want to sketch in this trend

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line what we would do.

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Remember that B here is the y intercept.

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This is the value where the trend line intersects the Y axis.

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So we would come here along the y axis, the vertical axis, and we would locate about 8.8, and that's

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this value we saw earlier.

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It's about 8.8.

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Remember before we said we thought it was a little less than ten?

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Well, it's about 8.8.

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So we would start there with this point.

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We'd put a point there along the axis and then this slope here of 7.313, two tells us, like we said

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before, that for every one unit we move over along the horizontal axis, we move up seven units along

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the vertical axis, which means that when we move out to X equals one from 8.8, we should go up about

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7.3.

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So at x equals one, we should get about 8.8 plus 7.3 is 16.1.

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So we would come up and we would locate about 16.

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Point one right about here.

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We could see that that's maybe at 16 here along the vertical axis.

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Then we would move over one more unit to X equals two and we would move up another 7.3.

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So from 16.1, adding 7.3, again we get about 23.4 and we can see here at two a value of about 23.4.

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So you can see that line starting to sketch out.

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That's how to do it by hand.

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Of course, if we just plug this line into any graphing calculator, computer algebra system or software

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program, it will plot this line for us without us having to find points to plot it out manually.

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But once we have even just two points, then we can sketch in a line that passes through those two points

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and we see that we get the trend line.

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Now we'll talk much more about scatter plots and trend lines when we talk later in the course about

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regression.

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But this is the general idea behind using a set of raw data to create a scatterplot.

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How we can read and interpret a scatterplot, and then how to use these formulas to actually build the

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equation of the trend line that runs through the data in our plot.