### Elements of Research

The Pearson correlation coefficient is a number between -1 and +1 that measures both the strength and direction of the linear relationship between two variables. Apr 27, The statistical relationship between two variables is referred to as their Positive Correlation: both variables change in the same direction. Jul 3, Two or more variables considered to be related, in a statistical context, the size and direction of a relationship between two or more variables.

## Overview of Correlation

A negative relationship means that as values of one variable increase, or decrease, values of the other variable change in the opposite direction. The magnitude of a relationship between variables is also important when considering causality.

The magnitude of a relationship is the extent to which variables change together in one direction or the other. The highest magnitude of relationship is a perfect relationship, in which knowledge of the value of the independent variable determines the exact value of the dependent variable. On the other hand, the lowest magnitude of relationship, the zero relationship, occurs when systematic change between the values of an independent variable and a dependent variable is not discernable.

There are four formal criteria that are necessary to establish that a relationship is causative: Time Order A relationship is causal if an action is the result of another action. If A is the cause of B, then A must precede B in time. The Degree Strength of a Relationship Finally, a correlation coefficient measures the degree strength of the relationship between two variables.

The mesures we discuss only measure the strength of the linear relationship between two variables. Two specific strengths are: They are said to be perfectly linearly related, either positively or negatively. When two variables have no relationship at all, their correlation is 0. There are strengths in between Here are three examples: Weight and Horsepower The relationship between Weight and Horsepower is strong, linear, and positive, though not perfect.

Drive Ratio and Horsepower The relationship between drive ratio and Horsepower is weekly negative, though not zero. The Pearson correlation coefficient is. The Pearson correlation coefficient is. Correlations can be used to help make predictions. If two variables have been known in the past to correlate, then we can assume they will continue to correlate in the future.

We can use the value of one variable that is known now to predict the value that the other variable will take on in the future. For example, we require high school students to take the SAT exam because we know that in the past SAT scores correlated well with the GPA scores that the students get when they are in college.

Suppose we have developed a new test of intelligence. We can determine if it is really measuring intelligence by correlating the new test's scores with, for example, the scores that the same people get on standardized IQ tests, or their scores on problem solving ability tests, or their performance on learning tasks, etc.

## Correlation

And so, most of 'em are pretty close to the line. So I would call this a negative, reasonably strong linear relationship. Negative, strong, I'll call it reasonably, I'll just say strong, but reasonably strong, linear, linear relationship between these two variables.

Now, let's look at this one. And pause this video and think about what this one would be for you. I'll get my ruler tool out again.

And it looks like I can try to put a line, it looks like, generally speaking, as one variable increases, the other variable increases as well, so something like this goes through the data and approximates the direction.

And this looks positive. As one variable increases, the other variable increases, roughly. So this is a positive relationship.

But this is weak. A lot of the data is off, well off of the line.

But I'd say this is still linear. It seems that, as we increase one, the other one increases at roughly the same rate, although these data points are all over the place. So, I would still call this linear.

### Bivariate relationship linearity, strength and direction (video) | Khan Academy

Now, there's also this notion of outliers. If I said, hey, this line is trying to describe the data, well, we have some data that is fairly off the line.

So, for example, even though we're saying it's a positive, weak, linear relationship, this one over here is reasonably high on the vertical variable, but it's low on the horizontal variable. And so, this one right over here is an outlier. It's quite far away from the line. You could view that as an outlier. And this is a little bit subjective. Outliers, well, what looks pretty far from the rest of the data?

This could also be an outlier. Let me label these. Now, pause the video and see if you can think about this one.

Is this positive or negative, is it linear, non-linear, is it strong or weak? I'll get my ruler tool out here. So, this goes here. It seems like I can fit a line pretty well to this.

So, I could fit, maybe I'll do the line in purple. I could fit a line that looks like that. And so, this one looks like it's positive. As one variable increases, the other one does, for these data points. So it's a positive. I'd say this was pretty strong. The dots are pretty close to the line there.