# 2.5 linear vs non-linear okay so here’s the lesson for section 2.5 linear versus nonlinear relationships so in the last section we looked at how to make a scatter plot and we made a bunch of scatter plots for pairs of variables that all made linear relationships and we learned how to describe those linear relationships and we also learn how to draw a line of best fit that helped us describe the relationship and also helped us make estimations called interpolations and extrapolations in this section I just want to show you that all pairs of variables don’t form linear relationships okay they could form a nonlinear relationship so let’s take a look at what that looks like right here in this first example so I have a scatter plot of this table of values here okay so I’ve got my table of values and all ten of these points have been plotted on the grid below here just to remind you about the graph of a scatter plot the horizontal axis is the x axis which is the independent variable and the vertical axis is the y axis okay which is the dependent variable so vertical is the Y which is the dependent variable horizontal is X which is the independent variable okay so let’s just look at what it says down here it says the Gandolf predicts that when x is 11 y will be 11 so this guy again know if he predicts that when X is 11 he thinks that Y will be 11 he thinks that here it is y would be 11 if X is 11 that’s his prediction right there okay this other guy Merlin he predicts that when X is 11 Y will be 15 always a different color for him and so he thinks the next point on the graph would be right there so our job is to try and decide which one of these guys is correct in their production okay so if we remember from last section um on a scatter plot we can draw a line of best fit and if you remember how to draw a line of best best fit the properties of line best fit say that it should go through as many of the points as possible okay and it should have an even number of points on either side of the line that’s paw well let’s try and draw let’s try and draw a line that fits those properties we’ll try and draw a line that goes there as many of these points as possible it has an even number on either side so you know that’s that’s pretty close okay and you’ll notice that it goes through that prediction that Gandalf made saying that when X is 11 y is 11 so if I use this line of best fit J Gandalf’s prediction is correct but using logic when you look at the trend of this data the data isn’t isn’t increasing at a constant rate okay it doesn’t seem like it so veneer relationship you’ll notice as X is increasing the Y values start increasing by a little bit a little bit a little bit and then they continue to increase by by more each time so it’s actually increasing exponentially so this is a nonlinear relationship so even though Gandalf’s prediction is on line of best fit I don’t think the line of best fit represents this data at all so in this case what we can actually do is we can draw a curve of best fit so what I’m going to do is I’m gonna draw a smooth curve that will go through as many of these points as possible and it will hopefully evenly distribute the points on either side of the line as well so if I draw a curve instead of a straight line so it’s the curve of best bet it should be a smooth curve it shouldn’t zigzag back and forth like I did a bit there by oxidation smooth and should go through as many points as possible so if I have a a curve of best fit look it goes right through merlyn’s prediction and and this seems to represent the data better because it’s increasing exponentially so when X is 11 y is 11 if I use this curve of best fit to make an extrapolation yes so so who is correct and why let’s just write an answer below here okay so I think the Merlin is correct so Merlin is correct why because the data does not follow a linear trend linear trend okay so because this data does not follow any or trend a line of best fit is not an accurate representation of the deaths we can’t use it to make estimations what is more accurate in this case is a curve of best fit because the data is not the relationship between the variables is nonlinear okay so here we go many nonlinear relations can be modeled with what we just used a curve of best fit and a curve F it has similar rules to it a line of best fit has okay so accrue best fit should pass through or close to as many points as possible just like a line of best fit should and any points they’re not on the curve should be distribute evenly above and below it just like a line of that step okay so let’s practice describing scatter plots as either linear or nonlinear and having a strong or weak correlation and whether it’s a positive or negative relationship and also we’ll practice drawing our lines or curves the best fit so we’re going to do that for each of the following here so I’ll click on the first one the first one it seems to follow a linear relationship okay as X is increasing Y is decreasing so I’m gonna draw a line of best fit for this it’s gonna go through or as close to many of these points as possible and they should be evenly distributed above and below the curve or above and below the line sorry so there’s my line of best fit for that and how I would describe this relationship I would use a few words for this first of all it’s falling down to the right so it’s a negative relationship anything flying down to the right is negative all the points are very close to that line okay so it’s a it’s a pretty strong correlation so I’ll say it’s strong and we’ve already established that I used to wind a best-fit because it is a linear relationship okay so it’s a strong negative linear relationship next one II so this one it seems as X is increasing the Y the Y values are also increasing so it does seem like there is a trend here it is going up to the right so I draw a line of best fit here and how I would describe this relationship oh I used to line the best fit because I think it so many relationship the points are pretty spread out though they’re not very close to line so let me see it’s got a weak correlation and the line is going up to the right so that’s a positive relationship next one so in this one it seems like the points are kind of just scattered everywhere you know with the low value of X I have a low value of Y but also with a high value of X I have a low value y the car just scattered everywhere I’m gonna say this one has no relationship okay it’s not a linear relationship it’s not a nonlinear relationship it’s just I don’t you think there’s any relationship between our x and y variables here keep in mind Neil X is the horizontal axis Y is the variable axis okay so do the next one this one here this is the first one we have here that it doesn’t seem to follow a linear trend it does it seems to be nonlinear it seems like a curve best fit would be more appropriate for this data it seems like as X is increasing Y starts to increase a bit and then it starts to decrease so I’m gonna try and draw a smooth curve through or as close to as many of these points as possible and evenly distributing the points on either side of it okay so I’ve drawn a nice smooth curve here keep in mind this this isn’t a connecting the dots exercise so I can’t draw a line that goes like that you can’t just go you can’t do that okay that’s what people tend to want to do you can’t do it has to be a smooth curve like the one I drew the first time smooth curve okay so how I would describe this one describe this might have been on when your relationship okay and it’s kind of pretty strong correlation so I can add that if I wanted you as a strong nonlinear relation this one here once again these are kind of just scattered everywhere that doesn’t seem to be any trend trend so I’m gonna say no relationship the next one well this one looks like a very strong positive relationship and it seems to seem to be a linear relationship as X is increasing Y is increasing at a constant rate I’m gonna draw a line of best fit for this one you’ll be able to see it easier and draw my line of best fit fit there is many of the points as possible and this is all the points are really close to that line of best fit so it’s a strong   