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

positive it’s positive because it’s going up to the right linear relationship next one I can see you making an argument for this to have no relationship but it does seem like as X is increasing the Y values are decreasing okay so it does seem like there is a little bit of a relationship between these two variables looks like it’s a weak relationship but there it does seem to your relationship okay so I’m gonna say this has a weak negative negative because it’s falling down to the right weak negative linear relationship next one this one definitely seems to be a nonlinear relationship as X is increasing Y goes up then down then up then down again so let’s try and draw a curve through these points as well okay this curve is gonna also made backing up and down see if I can get one there’s many of these points as possible smooth occur if that curve but if this trend would continue my career would keep going in the same pattern if I had a big graph okay so this is definitely a nonlinear relationship okay next one this one here once again as X is increasing Y decreases for a while and then but the midway point it done starts to increase again so this is a nonlinear relationship don’t draw my curve a best bet you always draw a curve when it’s a nonlinear relation I could probably do better but that’s pretty close so that’s a nonlinear relation as well all right let’s move on now let’s actually make some graphs ourselves so let’s test the hypothesis the older you are the more money you earn so I’ve got my table of values what I want to do is graph this table of values so what I’m gonna do I’ve got my graph ready here I’ve got remember in the table of values column on the left is your independent variable you use X for that and gave in the column on the right is your dependent variable we use Y for that earnings depends on age earnings is the dependent variable so we’ll plot earnings a very dumbness Forest earnings is on the y axis vertical axis age is on the horizontal axis the x axis and I’ve already made my scale I should put my brief in here I’ve already made my scale two to fit all of the data so what we can do is actually plot these points and what’s test hypothesis that as age increases you earn more money okay so what’s the plot that so we can visually see if this trend is true so we plot the point at age 25 you’re in 22,000 let’s plot that so at age 25 you’re in 22,000 so here’s that point there next one would be at age 30 you are in twenty six point five 26500 so age thirty twenty six thousand five hundred twenty-six thousand five hundred there and if I plot all of the points you know keep going back whipping dot applying the point this is what my plane should look like okay so these are all the data points plotted remember the hypothesis said older you are the one when you learn well let’s look at this graph that does seem to be true for a while it’s as we get older you start you earn more and more money until a certain point and then it seems like the data starts to go down again okay so with this data it doesn’t it doesn’t seem to be a linear relationship because as x increases Y increases but then it decreases so I don’t want to draw a line of best fit for this because it’s it’s not a linear relation it seems to be a nonlinear relation because it curves like this so I’m going to draw a curve above it and if I draw a curve in best fit through as many of these points as possible John nice smooth curve evenly distributing points above and below the curve it should look something like this okay no answer some questions about this okay so we already drew the curve of best bet okay we already drew that because it’s a nonlinear relation you draw a curve when it’s nonlinear if it was a linear relation I would have drawn a line of that set okay so we’ve drawn record with best fit already let’s describe the relationship so we know it’s gonna be we’re gonna describe doesn’t non linear because we’d you’re a curve a best bet what else we can say it seems as though

the earnings increase up until age let’s take a look at our graph seems like the increase up until about age you know up until about age 65 around that area and then they start to tail back down so up until about age between somewhere between 60 and 65 I’ll just say 65 okay then burnings then them earrings start to decrease so then they decrease so just the data support the hypothesis no in fact it doesn’t that bothers you all the hypotheses said was that let’s go back and look at it older you are the one one you earn and if we look at the graph at a certain point the older you are the less money you start to earn so no it doesn’t support the hypothesis no um I’ll be more specific no after the age of 65 you are in less money okay now let’s try and explain why the data for ages or 69 do not correspond with hypothesis well probably the simplest explanation would be around 65 that some people tend to retire right and so if they’re not working anymore you’re not going to be making as much money just by collecting your pension so let’s write a little one-word answer down here just retire Minh that would be my explanation for why the dad doesn’t support the hypothesis for ages over 65 okay let’s do another example so a skydiver jumps from an airplane the distance fallen and time taken are reported in the table so I’ve got my table of values I’ve got my column on the Left which I know it’s my independent variable and it’s X column on the right is my dependent variable I use Y so it shows the data so this is the time after he’s jumped from the airplane in seconds and this is the distance the person that’s fall so first thing we would want to do is we want to graph that data on our table here and I already have the scale down here I know on the x-axis he’s my independent variable so time time is always independent so I would label that time in seconds on the right that’s distance fallen so distant and it’s in meters okay so I’ve lived my graph the next thing I wanna do I want to start plotting these points so obviously when he like before he’s even jumped he hasn’t fallen any so first point is just at the origin there zero zero after he’s been falling for one second he’s falling five meters so you know roughly about here after two seconds he’s falling 19 meters and then if I continue applying these points my graph should look something like this okay next thing I’m gonna want to do I’m gonna well look at this data I wonder side does this look like a linear relation is it increasing at a constant rate or is it increasing exponentially okay and in this case it looks like a nonlinear relation to me I think it’s increasing exponentially okay as x increases Y is increasing but an accelerated rate okay so what I would do for a nonlinear relationship I would draw a curve a best fit if I draw my curve of best fit for this it should look something like this okay that’ll be my curve in best fit and that seems to represent the data pretty well it represents the data better than if I drew a line investment if I thought this was a linear relation I have to draw a line of best fit and a line of best fit would look something like this and that does not seem to represent the data nearly as well as the curve the best fit does so this is clearly a nonlinear relation so let’s classify the relation is layer nonlinear so we’ve already established that we’re going to classify this as a nonlinear relationship so let’s write that nonlinear and then let’s explain it so as time increases the rate of distance fallen is increasing so he’s

not falling at a constant rate as time is increasing his great of falling is increasing and that and that’s due to the acceleration of gravity okay so this causes the data to form a curve how far will the will be died with the skydiver have fallen after in 3.5 seconds well let’s quickly look at our graph here at 3.5 seconds he’s fallen about 60 meters so that’s an interpolate interp interpolation sorry um because it’s data within its enough nation within our data set between two values does interpolation if I ask you how far he’ll fall and after six seconds I would go all the way up here and okay I’d be you know somewhere up there so now be an extrapolation because of the Ellen would be beyond our set of data so if you have any questions um make sure to ask and you can get the work heat for this section from Jenson math dots yeah