Transcription
All right, we're going to look at how to interpret correlation coefficients in terms of our variables now. So, we've looked at a general understanding of what do the numbers mean, and what does that mean in terms of is it strong, is it moderate, is it weak? But now we're going to look at what does that mean about the variables themselves. So, what we're going to do is just go through some examples where I've already calculated the correlation coefficient for some relationships, and we're just going to explain what that correlation coefficient means.
So, this first example, we are looking at the correlation coefficient for the relationship between calories and sodium in a sample of hot dogs, and that correlation coefficient was found to be 0.94. So, explain what this means about the relationship between the number of calories and the amount of sodium in hot dogs. So, think about these variables. I have calories and sodium. So, I have to think about first, which one of these would be my explanatory variable, and which one of these would be my response variable? Well, because the sentence says, "explain what this means about the relationship between the number of calories and the amount of sodium," I'm going to go ahead and say that the calories was the X, and the sodium was the Y. I don't have a lot of information to go off here, that's what I'm going to say.
So, now with a correlation coefficient of 0.94, 0.94 falls in the range of a strong positive linear relationship. So, that's the first thing that I can write down for sure. But I haven't said anything about what this means about the relationship of these specific things. This is a general statement about what this relationship is, but I want to be a little more specific here. So, what this is telling me is that as the number of calories in a hot dog increases, the amount of sodium increases because it's positive in a strong linear relationship. So, all I've done is taken the information and the fact that 0.94 means we have a strong positive relationship, and I've taken that and put it in a single sentence to summarize it.
So, the next one, the correlation coefficient for the relationship between the cost of a Chevy vehicle and the gas mileage is found to be 0.647. Explain what that means in terms of the vehicles. So, we're looking at the cost of the Chevy vehicle and the gas mileage, and that correlation coefficient is 0.647. So, 0.647 is a moderate positive linear relationship. And then again, I want to figure out which is my explanatory and which is my response. So, would the cost affect the gas mileage, or would the gas mileage affect the cost? I think the gas mileage would affect the cost. So, the gas mileage would be my X, and the cost would be my Y. So, as the gas mileage increases, so that's like how many miles to the gallon are you getting, are you getting a lot or are you getting a little, the cost of a Chevy vehicle increases in a moderate linear relationship. So, it's not a strong relationship, but it's moderate. So, it, it's definitely linear, but we've got quite a few stray dots on that scatter plot.
All right, and then the last one that we have here says, the correlation coefficient for the relationship between shoe size and number of books read was found to be 0.002. Explain what this means in terms of the variables. So, we're looking at shoe size and number of books read, and that correlation coefficient was 0.002. So, is shoe size affecting the number of books you read, or is the books you read affecting your shoe size? Well, you can't really control your shoe size, so that has to be X, and then the number of books you read would be Y. And 0.002 would be very weak, possibly even none. And it's still positive. I did not put any negative ones on here. However, what I will tell you is this R value is so small. This is, this is none. There is no relationship here. I was inconsistent in my colors. Let me, let me fix that real quick. This should have been in purple. There we go. I'm consistent again.
So, then the interpretation, because this R value is so small, there is no linear relationship between shoe size and the number of books a person has read, which makes sense. People with bigger feet don't read more or less than people with smaller feet.
And then my last thing on here is just a word of warning. Correlation does not imply causation. So, just because there appears to be a correlation between two variables, that does not mean that the explanatory variable causes the response variable. It just means that there is a linear relationship between the two. And we've already talked about what does it mean or what has to happen in order to have cause to occur, and it's a very specific type of experiment. So, that is ways to interpret the results from the correlation coefficient.