# Measure of linear relationship between two variables is drawn

### The relationship between variables - Draw the correct conclusions

Chapter Relationships between variables. ▫ Definition. ▫ A Scatter Plot is a picture of Draw a Scatter Plot to represent the exists between two variables. ▫ But in numerical measure of how positive linear relationship between. Correlation is defined as the statistical association between two variables. Pearson's linear correlation coefficient only measures the strength and direction of a . of n bivariate observations drawn from a larger population of measurements. Scatterplots provide a visual display of the data and can be used to draw correlations the linear relationship between the two variables. There are three The measure of how much the change in one variable is caused by the other is .

You note down different values on a graph paper. Here, by taking data you are relating the pressure of the gas with its volume. Similarly, many relationships are linear in nature. Relationships in Physical and Social Sciences Relationships between variables need to be studied and analyzed before drawing conclusions based on it. In natural science and engineering, this is usually more straightforward as you can keep all parameters except one constant and study how this one parameter affects the result under study.

However, in social sciences, things get much more complicated because parameters may or may not be directly related. There could be a number of indirect consequences and deducing cause and effect can be challenging. Only when the change in one variable actually causes the change in another parameter is there a causal relationship.

Plot 2 with sample means The lines intersect atlocate it. This is our new center. The coordinates of X,Y relative to the new center are. Then it's easy to come up with many measures of linear relationships. A simple one is to count the number of points with the same sign those in quadrants I and III and subtract the number of points with different signs those in quadrants II and IV.

High values of this measure indicate a positive linear relationship while low values indicate a negative linear relationship. Instead of counting like and unlike signs, we consider a measure which takes the product of these new coordinates.

Thus we have n products, one for each point in the plot. Consider as a measure their average: Positive values of this measure indicate a positive linear relationship while negative values indicate a negative linear relationship.

Is this measure robust? No, you are catching on. For a given data set, we can always make this measure larger or smaller by changing the units. Suppose we have a positive linear relationship and X is measured in feet.

If we change the X's to inches then sXY increases by the factor If we change the X's to mm's then sXY increases by the factor Thus we need to standardize our measure. In this chapter we revisit this problem in Chapter 11we will insist on an absolute measure which in absolute value cannot exceed 1. As we said, for all data sets. The extreme values are interesting: Values of r close to zero indicate little or no linear relationship.

Scatter plots with values of r As we thought, the strongest relationships score 0 with our measure because they are both nonlinear.

The best linear pattern is Plot 2, although Plot 3 is close.

So let's just first think about whether there's a linear or non-linear relationship. And I'll get my little ruler tool out here. So, this data right over here, it looks like I could get a, I could put a line through it that gets pretty close through the data.

You're not gonna, it's very unlikely you're gonna be able to go through all of the data points, but you can try to get a line, and I'm just doing this. There's more numerical, more precise ways of doing this, but I'm just eyeballing it right over here. And it looks like I could plot a line that looks something like that, that goes roughly through the data.

So this looks pretty linear. And so I would call this a linear relationship.

- Bivariate relationship linearity, strength and direction

And since, as we increase one variable, it looks like the other variable decreases. This is a downward-sloping line.

## Relationship Between Variables

I would say this is a negative. This is a negative linear relationship. But this one looks pretty strong. So, because the dots aren't that far from my line.

This one gets a little bit further, but it's not, there's not some dots way out there. 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.

### Relationships Between Variables, Part 3: Measures of Relationships

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. 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.