Question Video: Using Straight Lines to Model Relationships between Two Quantitative Variables Mathematics • 8th Grade

Video Transcript

Does bivariate data suggest a linear relationship? (A) Yes. Otherwise, drawing a scatter plot would not be possible. (B) No, because some variables are not correlated linearly or at all. (C) No, because lines of best fit work for nonlinear relationships as well. And (D) yes, that is what bivariate means.

To answer this question, we’re going to recall the definition of two of our key terms. Our first is the term bivariate or bivariate data. We know the prefix bi- means two. For instance, if we’re riding a bicycle, we know that that has two wheels. Bivariate data then means data for two variables. For example, we might look to compare the amount of ice cream sold on a given day and the maximum temperature for that day.

So, what does the second term mean? That’s linear relationship. Well, if two things are correlated, they might have a linear relationship. This is a relationship of direct proportionality. And when we plot it on a graph, we trace a straight line. Consider the ice cream sales and temperature example. It might follow that the higher the temperature on a given day, the more ice creams are sold.

And so, a scatter graph or a scatter plot might look like this. We can draw a line of best fit on this scatter graph. This is a straight line. And so, ice cream sales and temperature might have a linear relationship. So, this is an example of bivariate data that might suggest the linear relationship. But does all bivariate data suggest this kind of relationship?

Well, let’s take another set of bivariate data. For instance, let’s look at cloud cover on a given day, measured in oktas, and the maximum wind speed in that day, measured in kilometers per hour. We cannot be convinced that there is any strong correlation between the amount of cloud cover and wind speed on a given day. And so, a scatter plot might look a little something like this. There appears to be no correlation at all, no relationship. And so, we wouldn’t be able to draw a line of best fit. And so, we see that this is an example of bivariate data which doesn’t have a linear relationship at all.

So, we compare these to our answers. Answer (A) says, “yes, it must be true. Otherwise, drawing a scatter plots would not be possible.” We’ve just shown that it is possible, and so it cannot be answer (A). (B) says, “no, because some variables are not correlated linearly or at all.” Now, we’ve shown that this is true. And so, the answer could be (B).

But then, we have option (C) and this says no again. But this time, the reason is because lines of best fit work for nonlinear relationships as well. Well, yes, that is true. We can have nonlinear correlation, but this doesn’t take into account the fact that some bivariate data will not have a relationship at all. And so, it cannot be (C). Option (D) says, “yes, that’s what bivariate means.” And we have actually shown that bivariate data simply means data for two variables. So, it’s not option (D).

And so, our answer must be (B) no, because some variables are not correlated linearly or at all.